Discrete-Time Approximations and Limit Theorems: In Applications to Financial Markets 9783110654240, 9783110652796

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Yuliya Mishura, Kostiantyn Ralchenko Discrete-Time Approximations and Limit Theorems

De Gruyter Series in Probability and Stochastics

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Edited by Itai Benjamini, Israel Jean Bertoin, Switzerland Michel Ledoux, France René L. Schilling, Germany

Volume 2

Yuliya Mishura, Kostiantyn Ralchenko

Discrete-Time Approximations and Limit Theorems |

In Applications to Financial Markets

Mathematics Subject Classification 2010 Primary: 60-02, 91-02, 60F17; Secondary: 91G15, 91G20, 91G30 Authors Prof. Dr. Yuliya Mishura Taras Shevchenko National University of Kyiv Department of Probability Statistics and Actuarial Mathematics Volodymyrska str., 60 Kiev 01601 Ukraine [email protected]

Dr. Kostiantyn Ralchenko Taras Shevchenko National University of Kyiv Department of Probability Statistics and Actuarial Mathematics Volodymyrska str., 60 Kiev 01601 Ukraine [email protected]

ISBN 978-3-11-065279-6 e-ISBN (PDF) 978-3-11-065424-0 e-ISBN (EPUB) 978-3-11-065299-4 ISSN 2512-9007 Library of Congress Control Number: 2021943975 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2022 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Introduction This book is devoted to various approximations of financial markets, both in time and in relation to basic parameters. Regarding approximation in time, everyone understands that markets operate in discrete time; however, from a computational point of view, it is much easier to work with them if viewed in continuous time, for which the Black–Scholes formula is one of the first and famous pieces of evidence. Therefore, the following questions immediately arise. How wrong are we if we replace discrete time with continuous? And what is the rate of convergence of asset and option prices in this case? Moreover, if we let the time be continuous in the entire sequence of models, including the limiting model, but the parameters change, tending to a certain limit, then what will happen to prices and capitals? The question of convergence itself has been studied for many years, and for initial acquaintance we recommend to the interested reader, for example, the book [139]. Also, for a preliminary acquaintance with the functional limit theorems and theory of financial markets, we recommend the books [17, 58, 62, 65, 110, 120, 136, 152, 153, 171]. However, the question of choosing a suitable model and a suitable approximation is so wide that there is an urgent need to describe new models from the point of view of limit transitions, for example, models with stochastic volatility and new approximations, and to estimate, if possible, the rate of convergence of prices of basic securities. In general, this is the subject of this book. The book consists of four chapters, each of which solves its own problem related to approximations of financial markets. Chapter 1 is devoted to the approximation of financial markets with continuous time by markets with discrete time. To begin with, we introduce the discrete-time multi-period market and define the basic concepts including the structure of asset prices, self-financing strategies, arbitrage, and completeness. We then consider discrete-time sequences of financial markets as an intermediate step in the transition to continuous time. We consider the traditional Cox– Ross–Rubinstein market model, but then we move away from this model and study arbitrage-free markets with the jump distribution concentrated on some interval and incompleteness of the non-Bernoulli market. After that, we introduce the basic concepts related to financial markets with continuous time. Then we describe the simplest passage to the limit, in which the geometric Brownian motion is the limiting process. In this case we have the Black–Scholes formula as the result of limit transition, weak convergence holds in the Skorokhod topology, and we can immediately conclude that the convergence of option prices holds for some options. But option price is not a unique object whose convergence follows from the weak convergence of the pre-limit markets. Other possible objects are the so-called Greeks, of which we have chosen to consider the Delta of European call options written both in discrete and continuous time and state the conditions of its convergence. After that, we go beyond the limit geometric Brownian motion and consider a much more general situation where the limiting process of the stock price is a geometric diffusion process. https://doi.org/10.1515/9783110654240-201

VI | Introduction Here, as usual, certain technical difficulties are caused by the need to consider multiplicative pre-limit schemes. We overcome this difficulty in the general diffusion case and then apply general results to recurrent schemes for the diffusion approximation when the limit process is a geometric Ornstein–Uhlenbeck process, a Cox–Ingersoll– Ross process, and, finally, a process that can be non-Markov and have a memory like fractional Brownian motion. All these questions are considered in the most detailed way and all general schemes are written out with examples. Chapter 2 is devoted to the rate of convergence of asset and option prices in the discrete-time approximation. We estimate the rate of convergence of options step by step for most of the schemes discussed in the first chapter, using various methods. In many cases, various types of approximations of the distribution of sums of independent random variables by the normal standard distribution function help. Although this approximation procedure requires a fairly thorough analysis, the results obtained are easy to simulate. In addition, we estimate the rate of convergence in models with stochastic volatility, including stochastic volatility represented by the Ornstein–Uhlenbeck standard and, separately, the Ornstein–Uhlenbeck fractional process. In the latter case, we apply elements of Malliavin calculus in order to realize three approaches to estimate the rate of convergence. Most of the results for stochastic volatility are provided with simulations and numerical calculations, although, due to the limited volume of the publication, we did not set ourselves the goal of providing all theoretical results with calculations. The third chapter is devoted to the limit theorems for markets with non-random time-varying coefficients. We do not discretize time here but instead consider the scheme of series of financial markets with coefficients that are non-random but depend both on time and the number of series. Since the coefficients are non-random, it is possible for each number of the series to write down the generalized Black–Scholes equation with variable coefficients and find its solution, and this plan is realized. We then investigate the limit behavior of a sequence of markets and prices in two ways: by purely probabilistic methods and using the generalized Black–Scholes equation. Then we investigate the conditions of convergence of barrier option prices with time-varying parameters. Continuing to consider barrier options, we return to time discretization and investigate the rate of convergence of barrier options at such discretization. This requires some additional assumptions and invoking a Komlós–Major–Tusnády approximation. As a result, the rate of convergence is less than for standard vanilla options. Then we ask ourselves the following question. It is easy to establish that the price of vanilla and barrier options is differentiable in standard parameters, such as asset price, time, etc. What about differentiability of the barrier price in barrier? The answer is positive, barrier price is differentiable in barrier, and it is received with the help of Malliavin calculus. Chapter 4 is devoted to convergence of stochastic integrals in application to financial markets. We still consider the sequences of financial markets with varying coefficients depending on time; however, now these coefficients are random, and it does

Introduction

| VII

not allow us to write Black–Scholes formula or something like this. To analyze the situation where both the strategy and the price process tend to some limit values and to study how the capitals corresponding to these strategies behave, we use the functional limit theorems, especially the conditions of weak convergence of stochastic integrals. In contrast to the previous chapters, which mainly dealt with two-dimensional markets, consisting of one non-risky and one risky asset, in this chapter assets are multidimensional. We also solve the inverse problem: under the condition of convergence of capitals, we establish the convergence of the corresponding strategies. Finally, we return to the barrier options but written on the assets with stochastic drift and volatility coefficients and establish the convergence of the European barrier option prices in the generalized Black–Scholes model. Appendix A concludes, containing an extended description of essentials of calculus, probability, and stochastic processes. We tried to make the book as self-contained as possible, defining almost all the concepts that are used in it and citing most of the results on which it relies. However, the reader should be familiar with the basics of analysis, probability theory, and, preferably, the basics of the theory of stochastic processes and stochastic analysis. We hope that this book will be useful to everyone who is interested in the real behavior of financial markets, researchers, teachers, and graduate and undergraduate students, as well as practitioners. During the research itself and the preparation of the book, we consulted with many experts to whom we are very grateful for their help and support. We are especially grateful to Professor Alexander Novikov (University of Sydney), who for the first time drew our attention to the importance of assessing the rate of convergence in financial markets, to Professors Alexander Melnikov (University of Alberta), Oleg Seleznjev (Umeå University), and Kęstutis Kubilius (Vilnius University), who give Yuliya Mishura an opportunity to deliver relevant lecture courses at these universities in 2014–2019, and to Professor Łukasz Stettner, who is a specialist in discrete-time approximations and with whom Yu. Mishura had a chance to discuss the related questions at AMaMeF conferences and during Simons Semester in Banach Center in 2019 in Warsaw and Będlewo. Also, we are grateful to all our colleagues and graduate students who were involved in this activity and addressed some problems presented in the book (see, in particular, the related papers [16, 98, 99, 101, 121, 122, 127, 155]): Alexei Kulik, Georgiy Shevchenko, Petro Slyusarchuk, Sergiy Shklyar, Viktor Bezborodov, Luca Di Persio, Sergii Kuchuk-Iatsenko, Yevheniya Munchak, Olena Soloveiko, and Yuri Yukhnovskii. We are thankful to Hanna Zhelezniak for the assistance in the preparation of manuscript. Kyiv, July 2021

Yuliya Mishura Kostiantyn Ralchenko

Contents Introduction | V Abbreviations and notations | XV 1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.2 1.2.1 1.2.2

1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.5 1.5.1 1.5.2 1.5.3 1.6

Financial markets. From discrete to continuous time | 1 Financial markets with discrete time | 2 Description of asset prices as stochastic processes with discrete time | 2 Description of investors’ strategies. Self-financing strategies | 4 Arbitrage-free multi-period markets | 5 Contingent claims. Complete and incomplete markets | 6 The sequence of discrete-time markets as an intermediate step in the transition to continuous time | 7 Description of the sequence of financial markets | 7 No-arbitrage and completeness of the sequence of markets with discrete time, created by independent random variables. Multiplicative scheme | 10 Preliminaries for the financial markets with continuous time | 19 The notion of self-financing strategy for the models in continuous time | 19 Arbitrage and martingale measures | 22 Hedging strategies | 25 Complete markets | 26 From discrete to continuous time. The limit process is a geometric Brownian motion | 27 Pre-limit sequence of the models with discrete time in the multiplicative scheme | 27 Geometric Brownian motion | 28 Functional central limit theorem for the financial markets with discrete time represented by the multiplicative scheme | 29 Black–Scholes formula as the result of limit transition. Option pricing | 35 “Delta” as an example of Greek functionals | 39 Weak convergence of Greek symbol “Delta” for prices of European options: from discrete time to continuous | 40 Pre-limit “Delta” and the method of the common probability space | 41 Some preliminary results | 41 Convergence of Δkn to Δ(x, T − t) | 44 General schemes of diffusion approximation | 50

X | Contents 1.6.1 1.6.2 1.7 1.7.1 1.7.2 1.7.3 1.8 1.8.1 1.8.2 1.8.3 1.9 1.9.1 1.9.2 1.9.3 1.9.4 2 2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.2 2.2.1 2.2.2

General functional limit theorem for diffusion approximation | 50 Functional limit theorem for diffusion approximation of the sums and the products of random variables | 53 A recurrent scheme for the diffusion approximation when the limit process is a geometric Ornstein–Uhlenbeck process | 58 Geometric Ornstein–Uhlenbeck process and construction of discrete scheme | 60 Pre-limit and limit Ornstein–Uhlenbeck markets are arbitrage-free and complete | 62 Convergence of the asset prices in the geometric Ornstein–Uhlenbeck model | 65 Functional limit theorems for additive and multiplicative schemes in the Cox–Ingersoll–Ross model | 68 “Non-truncated” and “truncated” Cox–Ingersoll–Ross processes and some of their properties | 69 Discrete approximation schemes for “non-truncated” and “truncated” Cox–Ingersoll–Ross processes | 75 Multiplicative scheme for the Cox–Ingersoll–Ross process | 81 General conditions of weak convergence of discrete-time multiplicative schemes to asset prices with memory | 84 General conditions of weak convergence | 86 Fractional Brownian motion as a limit process and pre-limit coefficients taken from Cholesky decomposition of its covariance function | 92 Possible perturbations of the coefficients in Cholesky decomposition | 101 Riemann–Liouville fractional Brownian motion as a limit process | 104 Rate of convergence of asset and option prices | 109 The rate of convergence of option prices when the limit is a Black–Scholes model | 109 Introduction | 109 The rate of convergence in the binomial model | 110 The rate of convergence of option prices in the model with uniformly distributed asset jump | 120 The rate of convergence of option prices when a general martingale-type discrete-time scheme approximates the Black–Scholes model | 133 The rate of convergence of option prices on the asset following the geometric Ornstein–Uhlenbeck process | 145 Brief discussion of the limit Ornstein–Uhlenbeck asset price process | 146 Description and properties of the pre-limit discrete-time price processes | 147

Contents | XI

2.2.3 2.2.4 2.2.5 2.2.6 2.3 2.3.1 2.3.2 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5 2.4.6 2.4.7 2.4.8 2.4.9 2.5 2.5.1

2.5.2 2.5.3

2.5.4 2.5.5 2.5.6 3 3.1

Incompleteness of the pre-limit market | 149 Weak convergence of asset price, with the rate of convergence | 151 The rate of convergence of objective option prices | 153 From objective measure to martingale measure. The rate of convergence of fair option prices | 160 Estimation of the rate of convergence of option prices by using the method of pseudomoments | 166 Rate of convergence in the CLT for i. i. d. random variables by the method of pseudomoments; rate of convergence of asset prices | 166 The rate of convergence of put and call option prices | 172 Market model with stochastic Ornstein–Uhlenbeck volatility: option pricing and discretization | 174 Diffusion model with stochastic volatility governed by the Ornstein–Uhlenbeck process | 177 Definitions and auxiliary results | 178 Absence of arbitrage in the market with stochastic volatility | 181 The case of independent Wiener processes | 183 Derivation of an analytic expression for the option price | 186 Discrete approximation of volatility processes | 192 The price of European call options | 193 Numerical examples | 196 Approximation precision check for the case of deterministic volatility | 200 Option pricing with fractional stochastic volatility and discontinuous payoff function of polynomial growth | 202 Payoff function: additional assumptions, auxiliary properties. Discussion of asset price model, absence of arbitrage, martingale measures, incompleteness | 204 Malliavin calculus with application to option pricing | 210 The rate of convergence of approximate option prices in the case when both the Wiener process and fractional Brownian motions are discretized | 215 The rate of convergence of approximate option prices in the case when only fractional Brownian motion is discretized | 223 Option price in terms of the density of the integrated stochastic volatility | 231 Simulations | 235 Limit theorems for markets with non-random time-varying coefficients | 243 Convergence of European option prices in the Black–Scholes model with time-varying parameters | 243

XII | Contents 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.3 3.3.1

3.3.2 3.4

4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.4 4.4.1 4.4.2 4.4.3

Model | 244 Explicit form of call and put option prices with time-varying parameters | 245 Robustness of asset and European option prices | 250 Convergence of barrier option prices with time-varying parameters | 254 Robustness of the barrier option price | 255 The price of the barrier option as a solution to the boundary value problem. Limit pricing theorem | 258 The rate of convergence of barrier option prices under a discretization of time | 260 Description of the model. The rate of convergence of the barrier option fair price in the discrete binomial market to the corresponding price in the continuous-time market | 260 Modeling | 268 The differentiability of a barrier option price as a function of the barrier | 270 Convergence of stochastic integrals in application to financial markets | 279 Multi-dimensional financial market, self-financing strategies | 279 Functional limit theorems for the integrals with respect to semimartingales | 284 Weak convergence of integrals with respect to processes of bounded variation | 284 Weak convergence of stochastic integrals with respect to martingales | 289 Weak convergence of stochastic integrals with respect to semimartingales | 292 Weak convergence of integrands under the condition of convergence of stochastic integrals | 295 Application of functional limit theorems for stochastic integrals to financial investment | 299 Weak convergence of capitals of self-financing strategies | 299 Convergence of risk minimizing strategies | 301 Limit behavior of capitals and barrier option prices in the Black–Scholes model with stochastic drift and volatility | 307 Description of the model | 308 Weak convergence of capitals in the generalized Black–Scholes model | 308 Weak convergence of European barrier option prices in the generalized Black–Scholes model | 312

Contents | XIII

A Essentials of calculus, probability, and stochastic processes | 319 A.1 Essentials of calculus | 319 A.1.1 Some inequalities for exponential functions | 319 A.2 Essentials of probability | 320 A.2.1 Conditional expectation and its properties | 320 A.2.2 Equivalent probability measures | 321 A.3 Essentials of stochastic processes | 321 A.3.1 Martingales, local martingales, and semimartingales | 323 A.3.2 Wiener process | 331 A.3.3 Fractional Brownian motion | 332 A.3.4 Essentials of stochastic calculus | 333 A.3.5 Elements of Malliavin calculus | 337 A.3.6 Convergence of stochastic elements | 339 A.3.7 Weak convergence to Wiener process with a drift | 347 A.3.8 Central limit theorems in the scheme of series | 347 A.3.9 The rate of convergence in the central limit theorem | 348 A.3.10 The rate of convergence to the normal law in terms of pseudomoments | 350 A.3.11 Stochastic differential equations and the approximations of solutions | 358 A.4 Some algebra related to matrices in the Cholesky decomposition | 362 Bibliography | 363 Index | 371

Abbreviations and notations a. s. CLT d = D([0, T]) fBm r. v. EMM ELMM

Almost surely Central limit theorem Equality in distribution the space of càdlàg functions defined on [0, T], i. e., functions continuous from the right and with the left limits Fractional Brownian motion Random variable Equivalent martingale measure Equivalent local martingale measure

󳨀 →

Weak convergence in distribution of the random variables

d

ℙ

󳨀 →

ℚ(n) , ℚ, d

󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒

ℚ(n) , ℚ, fdd

󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ ℚ∼ℙ ℚ(n) , ℚ

Convergence in probability of the random variables

Weak convergence in distribution of the random variables, where the measures ℚ(n) and ℚ correspond to the pre-limit r. v. and the limit r. v., respectively Weak convergence of finite-dimensional distributions of the stochastic processes Measures ℚ and ℙ are equivalent

Weak convergence of the measures corresponding to the stochastic processes ⌊a⌋ The biggest integer number not exceeding a ⌈a⌉ The fractional part of a, i. e., the difference a − ⌊a⌋ Birt (L, l, p) The right-tailed binomial distribution, i. e., the probability that at least l successes occur in L Bernoulli trials with parameter p 2 x Φ(x) = √12π ∫−∞ e−y /2 dy, the standard normal distribution function (standard normal cumulative distribution function, standard Gaussian distribution function) n! n (k ) = k!(n−k)! , binomial coefficient 󳨐󳨐󳨐󳨐󳨐⇒

dℚ | dℙ ℱt

i. i. d. i. i. d. r. v. a⋅b ℂ ℙ πtcall (x)

on the σ-field ℱt The restriction of the Radon–Nikodym derivative dℚ dℙ Independent identically distributed Independent identically distributed random variables Inner product in any finite-dimensional space Call option and/or any European contingent claim that is not specified Put option Fair (non-arbitrage, calculated with respect to the unique martingale measure) option price of a discounted call option as the function of the initial price x of the underlying asset, with intermediate expiry date t ∈ [0, T] and calculated at zero moment of time in the complete market

https://doi.org/10.1515/9783110654240-202

XVI | Abbreviations and notations πtput (x)

π call (x) π put (x) 𝕋 ℤ+

Fair (non-arbitrage, calculated with respect to the unique martingale measure) option price of a discounted put option as the function of the initial price x of the underlying asset, with intermediate expiry date t ∈ [0, T] and calculated at zero moment of time in the complete market = πTcall (x) if the market is considered at the time interval [0, T] and T is an ultimate expiry date = πTput (x) if the market is considered at the time interval [0, T] and T is an ultimate maturity date The time set on which the financial market is considered. It can be a finite, countable set, some interval, or ℝ+ . 0, 1, 2, . . .

1 Financial markets. From discrete to continuous time The problem of convergence of discrete-time to continuous-time financial models is developed very well, starting from the central limit theorem (CLT) for approximation of the Black–Scholes model by the Cox–Ross–Rubinstein model and continuing with more involved models. Of course, they will be the subject of this book, but the interested reader can also be referred to the papers [26, 30, 39, 44, 50, 79]. The evident questions here are the following. Does the weak convergence of the stock price process imply convergence of the option price processes? What about convergence of the hedging portfolios and the optimal portfolio strategies? Attempts to go ahead from simple binomial schemes were made using the results of the weak convergence of stochastic processes. These results were generalized with the help of the functional limit theorems. The most general versions of functional limit theorems are collected in [83]; see also [110]. Some applications of functional limit theorems to financial markets were summarized in [139]. The rate of convergence of the option prices in the framework of weak convergence was also widely discussed, e. g., in [30, 76]. However, for the case when the limiting stock price process is a semimartingale of more general structure, for example, if it is a solution of the diffusion stochastic differential equation, the results are formulated under comparatively restricted conditions. For instance, the approximation by Markov chain is studied and the conditions using analytical terms are formulated in [139]. The binomial and trinomial models are also widely used in [139, 168]. In particular, the approximation described in Chapter 2 of [139] is based on the binomial model and the sequence of real numbers that one needs to define. The concept of the present chapter is “from discrete to continuous time at financial markets in the framework of diffusion approximation.” We consider in detail the main properties of the markets with discrete (Sections 1.1 and 1.2, the latter for the sequence of markets with discrete time) and continuous (Section 1.3) time. Then we take the first step in the sense of approximating the market with continuous time by markets with discrete time, namely, we consider the Black–Scholes model together with the convergence of option prices. A specific feature that distinguishes such financial considerations from standard functional limit theorems is that pre-limit schemes are multiplicative, due to which it is necessary, as a rule, to pass to logarithms, and this entails the appearance of additional terms in the expansions, the asymptotic behavior of which must be carefully analyzed. As an illustration, we consider in Section 1.5 the convergence of one of the Greeks, namely, “Delta,” of the call option. Starting with Section 1.6, we consider general multiplicative schemes of diffusion approximation, and then we apply it consequently to the limit geometric Ornstein–Uhlenbeck (Section 1.7) and Cox–Ingersoll–Ross (CIR) (Section 1.8) schemes. All these considerations concern the diffusion Markov memoryless models. To include into consideration non-Markov models with memory as the limit schemes, we formulate in Section 1.9 the general conhttps://doi.org/10.1515/9783110654240-001

2 | 1 Financial markets. From discrete to continuous time ditions of convergence of additive and multiplicative schemes with memory to some limit process and illustrate these considerations with the fractional limit processes. In what follows we denote by c and C various constants whose values are not so important and can be different from line to line. Symbol ℂ is reserved for the value of a non-discounted contingent claim, in particular, for the value of a call option.

1.1 Financial markets with discrete time In this section we briefly recall basic notions and facts from the theory of financial markets with discrete time. We introduce asset prices as stochastic processes with discrete time (stochastic sequences) and we describe investors’ strategies, in particular, self-financing strategies, the notion of arbitrage and arbitrage-free multi-period markets, complete and incomplete markets, and contingent claims, where everything is related to the discrete time. For more details on this topic we refer, e. g., to [58, 120].

1.1.1 Description of asset prices as stochastic processes with discrete time To give a mathematical description of the random (i. e., stochastic) nature of asset prices and other components of financial markets, we introduce and fix a probability space (Ω, ℱ , ℙ). The initial probability measure ℙ is called objective or physical because the investors suppose that it corresponds to the objective visible situation on the financial market. Other measures on the measurable space (Ω, ℱ ) will be discussed later on. Elements ω ∈ Ω are called scenarios since they correspond to different scenarios of the possible changes of the asset prices. Furthermore, we consider the changes of the asset prices in time. To start with the simple case, we consider discrete- and finite-time observation. We introduce the set 𝕋 = {t0 , t1 , . . . , tn } and suppose that all the asset prices are observed in the moments of time {ti , 0 ≤ i ≤ n}. Without loss of generality we can assume that 𝕋 = {0, 1, . . . , T}, where T ∈ ℕ is some integer. Let 𝔽 = {ℱt , t ∈ 𝕋} be a filtration on (Ω, ℱ , ℙ), i. e., a sequence of non-decreasing σ-fields ℱ0 ⊂ ℱ1 ⊂ ⋅ ⋅ ⋅ ⊂ ℱT ⊂ ℱ . The family (Ω, ℱ , 𝔽, ℙ) is called a stochastic basis with filtration or the filtered probability space. All processes Z considered on (Ω, ℱ , 𝔽, ℙ) are assumed to be adapted to this filtration, i. e., Zt is ℱt -adapted (ℱt -measurable). Let us have a finite number (Sti , 0 ≤ i ≤ m, t ∈ 𝕋) of asset prices. We suppose that all prices are non-negative on any ω ∈ Ω. One of the assets, say St0 , plays a special role. It is the so-called numéraire, that is, a non-risky (risk-free), or at least less risky, asset that can even be non-random. It is supposed to be positive. It corresponds to the interest rate of bank accounts, or to the rate of inflation (often it is assumed that

1.1 Financial markets with discrete time

| 3

the interest rate of the bank account equals the rate of inflation). Otherwise, it is the asset price that demonstrates the strongest stability. For example, if Sti correspond to currency exchange rates, then St0 corresponds to the most stable currency on the time interval {0, 1, . . . , T} (if it loses stability, it can be replaced with another one). The price St0 is considered as a discounting factor at time t. For technical simplicity, we suppose from now on that ℱ0 = {0, Ω}. If the prices St0 are random, we suppose that they are predictable. Definition 1.1. A stochastic process X = {Xt , t ∈ 𝕋} on the stochastic basis (Ω, ℱ , 𝔽, ℙ) is called predictable if X0 is non-random and for any 1 ≤ t ≤ T random variable Xt is ℱt−1 -measurable. Predictability means that we can reconstruct Xt by the past information up to moment t−1, and this is suitable for the numéraire if it is a stochastic process, because predictability means that the process is less random and more sustainable. Summarizing, the numéraire St0 can be written in the form St0 = S00 ∏ti=1 (1 + ri ), where ri =

Si0 0 Si−1

− 1 > −1

are ℱi−1 -measurable random variables which can be interpreted as the subsequent interest rates on the intervals of the form [i − 1, i). Wanting to make things easier, we can assume that the subsequent interest rates are non-random, and in the simplest case they are assumed to be equal, i. e., St0 = S00 (1 + r)t , r > −1. It is often supposed for simplicity that S00 = 1 and that r ≥ 0, because the real rate of inflation is in most cases at least non-negative. As to the other asset prices (Sti , 1 ≤ i ≤ m), we suppose that they change their value at times 1, 2, . . . , T, and we can observe their initial values S0i and all other values (Sti , 1 ≤ i ≤ m, 1 ≤ t ≤ T). Any interval [i − 1, i), 1 ≤ i ≤ T, is called a period, so we consider a model with initial values S0i and T periods. The assets (Sti , 1 ≤ i ≤ m) are called risky assets, so we have the market consisting of one non-risky and m risky assets. We exclude from the consideration the case of Sti ≡ 0 for some 1 ≤ i ≤ m and all t ∈ 𝕋. In order to compare the values of the asset prices at different times, we need to take into account the rate of inflation. Our approach is that in any particular model the rate of inflation is described by the risk-free asset St0 . Therefore, the discounted price process has the form Xti =

Sti

St0

,

1 ≤ i ≤ m.

Evidently, Xt0 = 1, t ∈ 𝕋. We introduce the following notations. Denote St = (St1 , . . . , Stm ),

t ∈ 𝕋,

the vector of risky assets, St = (St0 , St1 , . . . , Stm ),

t ∈ 𝕋,

4 | 1 Financial markets. From discrete to continuous time the vector of all assets, Xt = (Xt1 , . . . , Xtm ),

t ∈ 𝕋,

the vector of discounted risky assets, and X t = (1, Xt1 , . . . , Xtm ),

t ∈ 𝕋,

the vector of all discounted assets. 1.1.2 Description of investors’ strategies. Self-financing strategies Now we have the model of the financial market described above and consider an investor who buys and sells the shares of the assets on this market. We assume that it is possible to buy and sell any quantity (integer, fractional) of any asset. Denote by ξti the quantity of the ith asset between t − 1 and t, and let ξ t = {ξti , 0 ≤ i ≤ m},

t ∈ 𝕋,

be the vector consisting of all shares. This vector is called a strategy, or an investor’s portfolio, between t − 1 and t. We suppose that the decision at moment t concerning i the value of any ξt+1 that will act between t and t + 1 is made due to the information available up to the moment t. It can be explained so that during the period between t − 1 and t the investor analyzes the information coming until the moment t and at i the moment t the investor announces her/his decision. So, any component ξt+1 of the i portfolio is ℱt -adapted, i. e., predictable. Moreover, since ℱ0 = {0, Ω}, ξ1 , 0 ≤ i ≤ m is non-random. Now, consider the total capital of the investor at any moment t, immediately after her/his decision. The capital consists of the components corresponding to the investment to any asset: m

m

i=0

i=1

0 0 1 m m i 0 0 i Ut (ξ ) = ξt+1 St + ξt+1 St1 + ⋅ ⋅ ⋅ + ξt+1 St = ∑ ξt+1 Sti = ξt+1 St + ∑ ξt+1 Sti .

Denote by a ⋅ b the inner product of vectors a and b situated in any finite-dimensional ̄ Then space (it can be, e. g., ℝm or ℝm+1 ; a vector can be denoted by both a and a). 0 0 Ut (ξ ) = ξ t+1 ⋅ St = ξt+1 St + ξt+1 ⋅ St ,

where ξ t = (ξt0 , ξt1 , . . . , ξtm ), ξt = (ξt1 , . . . , ξtm ). In particular, we have the equalities m

U0 (ξ ) = ξ 1 ⋅ S0 = ∑ ξ1i S0i , i=0

m i i i i and, in addition, we put ξ0i = ξ1i and ∑m i=0 ξ0 S0 := ∑i=0 ξ1 S0 .

1.1 Financial markets with discrete time

| 5

Now we make supposition that the investor changes the portfolio in such a way that the total capital does not change. It means that there are no additional financial sources except the initial capital U0 (ξ ) and the change of capital is only due to the change of market prices. Also, there are no additional outflow and inflow of capital. Such strategies are called self-financing strategies. Definition 1.2. Mathematically, a strategy (ξ t , t ∈ 𝕋) is self-financing if for any t = 0, 1, . . . , T − 1, ξ t+1 ⋅ St = ξ t ⋅ St .

(1.1)

1.1.3 Arbitrage-free multi-period markets Now we consider the multi-period financial market consisting of m + 1 assets {Sti , 0 ≤ i ≤ m, t ∈ 𝕋},

𝕋 = {0, 1, . . . , T}.

Definition 1.3. A financial market does not admit arbitrage (is arbitrage-free, arbitrage-free in the “global” sense) if there is no such self-financing strategy ξ = {ξti , 0 ≤ i ≤ m, t ∈ 𝕋} that the initial discounted capital V0 (ξ ) ≤ 0 and the final discounted capital VT (ξ ) ≥ 0 with probability 1 and VT (ξ ) > 0 with positive probability. Definition 1.4. The market is arbitrage-free (in the “local” sense) between the moments of time t and t + 1 if there is no such self-financing strategy ξ = {ξ s , 0 ≤ s ≤ t + 1} that Vt+1 (ξ ) ≥ Vt (ξ ) a. s. and Vt+1 (ξ ) > Vt (ξ ) with positive probability. Theorem 1.5. The financial market is arbitrage-free in the “global” sense if and only if it is arbitrage-free on each step in the “local” sense. Corollary 1.6. Let a financial market be arbitrage-free in the sense of Definition 1.3. Then for any 0 < t < T there is no such self-financing strategy ξ = {ξsi , 0 ≤ i ≤ m, t ≤ s ≤ T} that the discounted capital Vt (ξ ) ≤ 0 and the final discounted capital VT (ξ ) ≥ 0 with probability 1 and VT (ξ ) > 0 with positive probability. Definition 1.7. A probability measure ℙ∗ on the probability space (Ω, ℱ ) is called a martingale measure if the adapted discounted price process X = {Xt = (Xt1 , . . . , Xtm ), ℱt , t ∈ 𝕋} is a martingale with respect to this measure. The martingale measure ℙ∗ is called an equivalent martingale measure if ℙ∗ ∼ ℙ. We denote by 𝒫 the set of all equivalent martingale measures. Theorem 1.8. A multi-dimensional multi-period financial market is arbitrage-free if and only if the set 𝒫 of equivalent martingale measures is non-empty. In this case there exists a measure ℙ∗ ∈ 𝒫 such that the Radon–Nikodym derivative dℙ∗ /dℙ is bounded.

6 | 1 Financial markets. From discrete to continuous time 1.1.4 Contingent claims. Complete and incomplete markets Consider a multi-period financial market S = (Sti , 0 ≤ i ≤ m, t ∈ 𝕋) on the stochastic basis (Ω, ℱ , 𝔽 = {ℱt , t ∈ 𝕋}, ℙ) with filtration. We consider T as the final moment of trading and suppose that we stop at this moment. Evidently, ℱT ⊂ ℱ , and the inclusion can be strict. Definition 1.9. Any random variable ℂ which is defined on the probability space (Ω, ℱ , 𝔽 = {ℱt , t ∈ 𝕋}, ℙ) is called a (European) contingent claim. Contingent claim ℂ is called a derivative contingent claim (derivative contract, derivative security), or simply a derivative, of the assets {St = (Sti , 0 ≤ i ≤ m), t ∈ 𝕋} if ℂ is measurable with respect to the σ-field GT := σ{St , 0 ≤ t ≤ T} ⊂ ℱT . In what follows we consider an arbitrary ℱT -measurable non-negative contingent claim ℂ and suppose that the financial market is arbitrage-free, i. e., 𝒫 ≠ 0. Definition 1.10. A discounted contingent claim that corresponds to contingent claim ℂ is the random variable of the form 𝔻=

ℂ . ST0

Specific cases of non-discounted contingent claims are a European call option of the form ℂ = (ST − K)+ or a European put option of the form ℙ = (K − ST )+ , where ST is the price of the asset on which this option is signed at moment T and K ≥ 0 is a strike price. Definition 1.11. Contingent claim ℂ is called hedgeable (attainable, attainable payoff, replicable payoff) if there exists a self-financing strategy ξ whose capital at the maturity date T coincides with ℂ a. s., i. e., ℂ = ξ T ⋅ ST a. s. This strategy is called a strategy that hedges (replicates, generates, attains) contingent claim ℂ. The corresponding strategy is called a hedging strategy or briefly hedge of the contingent claim. Definition 1.12. A financial market is ℱT -complete if any ℱT -measurable contingent claim is hedgeable. Theorem 1.13. An arbitrage-free market is ℱT -complete if and only if all restrictions of all martingale measures to ℱT coincide. Remark 1.14. Suppose that ℱ = ℱT . Then instead of the notion of ℱT -complete financial market we consider the notion of complete financial market and the latter theorem can be simplified to the following statement. Theorem 1.15. An arbitrage-free market is complete if and only if the martingale measure is unique.

1.2 Intermediate step | 7

1.2 The sequence of discrete-time markets as an intermediate step in the transition to continuous time In order to construct approximations, we consider the sequence of the financial markets with discrete time under the assumption that the diameter of discretization tends to zero. In what follows any market in this sequence will consist of two assets: a nonrisky asset and a risky one. We shall number the markets by index n. Our aim is to study in detail their properties including the no-arbitrage property and completeness depending on their construction via random drivers. The following notations will be used: O(n) will denote the objects in the nth series (it can be a probability set, the flow of σ-fields, a probability measure, etc.). For the series of stochastic processes we shall use both notations of the form Xn or X (n) , while for the price of some asset Y (n) at time k we shall use the notation Ynk . 1.2.1 Description of the sequence of financial markets Let (Ω(n) , ℱ (n) ) be a sequence of measurable spaces. Consider the sequence of financial markets with discrete time, each with one risk-free asset B(n) and one risky asset S(n) (bond and stock), defined on the corresponding probability space. Our aim is to include a mathematical description of this model in the so-called scheme of series. So, let T > 0 be a fixed number and let parameter n (number of series) take integer values from ℕ. For n ≥ 1 consider the partition of the interval 𝕋 = [0, T] having the form π (n) = {0 = tn0 < tn1 < ⋅ ⋅ ⋅ < tnn = T}. We assume that the points of the partition are the trading moments on the corresponding financial market. Now, let {rnk , 1 ≤ k ≤ n} be a set of non-negative numbers that we will treat as the subsequent values of the interest rate, so the price of the risk-free asset B(n) at time tnk has the form k

Bkn = B0n ∏(1 + rni ), i=1

B0n > 0,

n ≥ 1.

(1.2)

In what follows we assume for technical simplicity that B0n = 1. Now, let for any n ≥ 1 the sequence of random variables {Rkn , 1 ≤ k ≤ n} be defined on the space (Ω(n) , ℱ (n) ) and have a financial interpretation as the sequence of the gains in the respective trading periods. We assume that these random variables satisfy the following condition of boundedness: there exists such number 0 < c < 1 that |Rkn | ≤ c, n ≥ 1, 1 ≤ k ≤ n. Denote by {ℙ(n) , n ≥ 1} the sequence of objective (physical) probability measures defined on (Ω(n) , ℱ (n) ) so that each (Ω(n) , ℱ (n) , ℙ(n) ), n ≥ 1, is a probability space. Introduce the sequence of the flows of σ-fields 𝔽(n) = {ℱnk = σ{Rin , i = 1, . . . , k}}, generated by random

8 | 1 Financial markets. From discrete to continuous time variables Rkn . Then for any n ≥ 1, σ-fields {ℱnk , 1 ≤ k ≤ n} create a filtration on the corresponding probability space. Assume that the price of the risky asset S(n) at time tnk has the form k

Snk = Sn0 ∏(1 + Rin ), i=1

(1.3)

where Sn0 , n ≥ 1, is a sequence of positive numbers. Then the price of the corresponding discounted risky asset at time tnk equals k

Xnk = Sn0 ∏ i=1

1 + Rin . 1 + rni

(1.4)

The sequence of collections of random variables is often referred to as the scheme of series. So, we call our model the sequence of markets in the scheme of series, or simply the scheme of series. Examine the conditions that ensure the no-arbitrage property of financial markets with discrete time in the scheme of series. As known from Section 1.1, the financial market in the nth series will be arbitrage-free if and only if there exists an equivalent martingale probability measure ℙ(n,∗) ∼ ℙ(n) , with respect to which {Xnk , 1 ≤ k ≤ n} is an {ℱnk , 1 ≤ k ≤ n}-martingale, or simply 𝔽(n) -martingale. According to Theorem A.26, all equivalent martingale measures ℙ(n,∗) have Radon–Nikodym derivatives of the form n dℙ(n,∗) = (1 + ΔMnk ), ∏ dℙ(n) k=1

(1.5)

where {Mnk , 1 ≤ k ≤ n} is some 𝔽(n) -martingale, ΔMnk := Mnk − Mnk−1 > −1. Let us consider one simple sufficient no-arbitrage condition in the nth series. Denote by 𝔼n and Varn the expectation and variance with respect to the objective measure ℙ(n) and by 𝔼∗n and Var∗n the expectation and variance with respect to the martingale measure ℙ(n,∗) . Lemma 1.16. The financial market (1.2)–(1.3) is arbitrage-free in the nth series if there exists a set of measurable functions {φkn (x), x ∈ ℝk , 1 ≤ k ≤ n} such that |φkn (x)| < 21 and the following equalities hold for 1 ≤ k ≤ n: 𝔼n (Rkn | ℱnk−1 ) + 𝔼n (φkn (R1n , . . . , Rkn )Rkn | ℱnk−1 )

− 𝔼n (φkn (R1n , . . . , Rkn ) | ℱnk−1 )𝔼n (Rkn | ℱnk−1 ) = rnk .

(1.6)

Proof. The no-arbitrage property is guaranteed by the existence of the measure ℙ(n,∗) satisfying the following properties: (i) measure ℙ(n,∗) is indeed a probability measure, (ii) ℙ(n,∗) ∼ ℙ(n) , (iii) ℙ(n,∗) is a martingale measure.

1.2 Intermediate step |

9

Condition (ii) of equivalence has the form n dℙ(n,∗) = (1 + ΔMnk ) > 0 ∏ dℙ(n) k=1

(1.7)

a. s., the martingale condition (iii) can be written as 𝔼∗n (Xnk | ℱnk−1 ) = Xnk−1 ,

1 ≤ k ≤ n,

(1.8)

a. s., and the condition for ℙ(n,∗) to be a probability measure has the form 𝔼n (

n dℙ(n,∗) ) = 𝔼n ∏(1 + ΔMnk ) = 1. (n) dℙ k=1

(1.9)

The martingale condition can be verified based on Lemma A.8, and it follows that the martingale condition can be rewritten as 𝔼n ( dℙ X k | ℱnk−1 ) dℙ(n) n (n,∗)

(n,∗) 𝔼n ( dℙ dℙ(n)

|

ℱnk−1 )

= Xnk−1 .

With (1.5) in mind, this relation can be reduced to the equality 𝔼n ((1 + ΔMnk )(1 + Rkn ) | ℱnk−1 ) = 1 + rnk , i. e., 𝔼n (ΔMnk + (1 + ΔMnk )Rkn | ℱnk−1 ) = rnk . Finally, taking into account the martingale property of M (n) , we transform the latter equality to 𝔼n ((1 + ΔMnk )Rkn | ℱnk−1 ) = rnk ,

1 ≤ k ≤ n.

(1.10)

Recall that the random variable ΔMnk is measurable with respect to the σ-field ℱnk . Therefore ΔMnk can be presented as ΔMnk = ψ(Rin , i = 1, . . . , k), where ψ is some Borel function. But the martingale property allows to present ΔMnk in the more precise form ΔMnk = φkn (R1n , . . . , Rkn ) − 𝔼n (φkn (R1n , . . . , Rkn ) | ℱnk−1 ).

(1.11)

If we additionally assume that |φkn (x)| < 21 , then inequalities (1 + ΔMnk ) > 0 are valid, i. e., condition (1.7) and consequently condition (1.9) will be satisfied. Note that (1.10) and (1.6) are equivalent, whence the proof follows.

10 | 1 Financial markets. From discrete to continuous time Remark 1.17. Obviously, it is a highly non-trivial problem to check condition (1.6) in the general case. But in some cases it can be simplified. Recall that there exists such number 0 < c < 1 that |Rkn | ≤ c, n ≥ 1, 1 ≤ k ≤ n. Now, for example, let us try to present k k k−1 k k k−1 ΔMnk as ΔMnk = νnk (R1n , . . . , Rk−1 → ℝ is a n )(Rn − 𝔼n (Rn | ℱn )), where νn = νn (x): ℝ Borel function, bounded in the absolute value by 1/2c. Then conditions (1.7) and (1.9) k k 2 k−1 obviously are met. Denote the conditional variance Vark−1 n (Rn ) := 𝔼n ((Rn ) | ℱn ) − k k−1 2 (𝔼n (Rn | ℱn )) . Then equality (1.6) is reduced to the following one: k−1 k k 𝔼n (Rkn | ℱnk−1 ) + νnk (R1n , . . . , Rk−1 n ) Varn (Rn ) = rn .

Therefore, the condition 󵄨󵄨 r k − 𝔼 (Rk | ℱ k−1 ) 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 n n n n 󵄨󵄨 ≤ 󵄨󵄨 k) 󵄨󵄨 2c 󵄨󵄨 Vark−1 (R n n supplies the no-arbitrage property of the market. 1.2.2 No-arbitrage and completeness of the sequence of markets with discrete time, created by independent random variables. Multiplicative scheme 1.2.2.1 Arbitrage in the markets with discrete time. Completeness and incompleteness. General no-arbitrage conditions Let in any series the random variables {Rkn , 1 ≤ k ≤ n} be mutually independent with respect to the corresponding objective measure ℙ(n) and bounded as before: |Rkn | ≤ c < 1. We will try to present ΔMnk in the form (1.11), taking into account that now we have an equality 𝔼n (φkn (R1n , . . . , Rkn ) | ℱnk−1 ) = 𝔼kn φkn (x1 , . . . , xk−1 , Rkn )|x1 =R1 ,...,xk−1 =Rk−1 , n

n

where, according to Lemma A.5, the expectation 𝔼kn and covariance Covkn are taken with respect to the random variable Rkn , and then all previous random variables are substituted. We introduce the following random variables: Covkn (φkn , Rkn ) := (𝔼kn (φkn (x1 , . . . , xk−1 , Rkn )Rkn )

− 𝔼kn (φkn (x1 , . . . , xk−1 , Rkn ))𝔼n (Rkn ))|x1 =R1 ,...,xk−1 =Rk−1 . n

n

Then condition (1.6) can be reduced to the following one: 𝔼n Rkn + Covkn (φkn , Rkn ) = rnk .

(1.12)

It is a challenge to determine the general form of the functions φkn for which equalities (1.12) hold. Consider two particular situations, in a sense opposite to each other. Namely, we consider a binomial distribution and an absolutely continuous distribution of Rkn . In the first case we prove the completeness of the market, while in the second, on the contrary, it is incomplete.

1.2 Intermediate step |

11

1.2.2.2 Bernoulli distribution. Cox–Ross–Rubinstein model We now consider a particular case of the pre-limit discrete-time model, namely, the Cox–Ross–Rubinstein model. It is also called binomial model or binomial scheme. To describe this model of a market in the nth series, we introduce the particular case of the probability space Ω(n) := {−1, +1}n = {ωyn1 ,...,ynn = (yn1 , . . . , ynn ) | yni ∈ {−1, +1}} and two sequences an and bn , n ≥ 1. For ωyn1 ,...,ynn = (yn1 , . . . , ynn ), denote by Ynk (ωyn1 ,...,ynn ) := ynk the projection of the kth coordinate. Let Rkn (ωyn1 ,...,ynn ) := an

1 + Ynk 1 − Ynk + bn . 2 2

So, a Cox–Ross–Rubinstein pre-limit model in the nth series can be characterized by the following properties that are collected in the Cox–Ross–Rubinstein property: (i) the number of periods in the nth series equals n, (ii) the interest rate rn = rT determines the evolution of the non-risky asset B(n) as n follows: Bkn = (1 + rn )k , k = 0, . . . , n, (iii) a single risky asset S(n) is characterized by the subsequent values Snk , k = 0, . . . , n, and gains defined by Rkn = (Snk − Snk−1 )/Snk−1 ,

k = 1, . . . , n.

The gains are assumed to be independent and take only two values, either akn or bkn , k = 1, . . . , n. In the simplest case it is assumed that akn = an and bkn = bn , k = 1, . . . , n. In the general case the random variables Rkn have a Bernoulli distribution, taking only two values akn and bkn , akn < bkn , with probabilities pkn > 0 and qnk = 1 − pkn > 0, respectively. Since this market is very well known, we only state without proof that it is arbitrage-free in the nth series if and only if akn < rnk < bkn . The martingale measure is unique and determined by the relations ℙ(n,∗) (Rkn = akn ) =

bkn − rnk

bkn − akn

together with (1.5), where we put ΔMnk =

rnk − μkn (σnk )2

(Rkn − μkn ),

(1.13)

12 | 1 Financial markets. From discrete to continuous time and μkn = 𝔼n Rkn , (σnk )2 = Varn Rkn . This means that the market is complete. We can establish directly that the condition akn < rnk < bkn supplies the relations ΔMnk > −1 a. s. In the case when akn = an and bkn = bn , rnk = rn , k = 1, . . . , n, the market is arbitrage-free in the nth series if and only if an < rn < bn , the martingale measure is unique and determined by the relations p∗n := ℙ(n,∗) (Rn = an ) =

bn − rn , bn − an

qn∗ = 1 − p∗n .

(1.14)

Remark 1.18. Let, additionally to the Cox–Ross–Rubinstein property, the random variables Rkn can attain two values, an and bn , where −1 < an < bn , and these values

have a symmetric form in the following sense: an = e−σ√δn − 1 and bn = eσ√δn − 1, where δn = Tn for some T > 0. Also, let rn = rδn . Assumptions on the asset price mean that it goes up or down, Snk

Snk−1

= 1 + an or 1 + bn ,

and (1 + an )(1 + bn ) = 1. We refer to such a model as a symmetric binomial model. It will be considered in more detail in Remark 1.44 and Section 1.4.5. It is possible to consider the modifications of the symmetric binomial model with an = e− σ2 δ − 2n

2 (r− σ2

and bn = e − 1, or an = e − 1 and bn = e δn = Tn . The latter one will be considered in Section 2.1.2. +σ√δn

)δn −σ√δn

2 (r− σ2

σ 2 δn 2

)δn +σ√δn

−σ√δn

−1

− 1, where

1.2.2.3 Self-financing and replicating strategies on the Bernoulli market Consider the Cox–Ross–Rubinstein arbitrage-free and complete market described by the Cox–Ross–Rubinstein property from Section 1.2.2.2. Assume that its starting point is the same in any series, so, in the nth series it starts from a non-random value Sn0 = Xn0 = X0 . Assume that the maturity date corresponding to the last period equals T and the values of risky assets in the nth series equal an and bn , while the interest rate equals ∈ (an , bn ). As usual, denote the probabilities that correspond to the martingale rn = rT n b −r

r −a

measure by p∗n = b n−an and qn∗ = 1 − p∗n = bn −an . Let ℂ be some (non-discounted) conn n n n tingent claim (an at this moment not obligatory call option), measurable with respect to the σ-algebra Fnn = σ{Rkn , 1 ≤ k ≤ n} = σ{Snk , 0 ≤ k ≤ n} = σ{Xnk , 0 ≤ k ≤ n}. Then there exists a Borel function h such that the discounted contingent claim ℍ = ℂ(1 + rn )−n = ℂ(1 +

admits the following representation: ℍ = h(Xn0 , . . . , Xnn ).

rT ) n

−n

1.2 Intermediate step |

13

Since the market is complete, it admits the unique martingale measure, P (n,∗) , say. Let x̄ and ȳ be two m-dimensional vectors. As usual, denote by x⋅̄ ȳ their inner product, that k k ̄k is, x̄ ⋅ ȳ = ∑m i=1 xi yi . Furthermore, consider the pair Sn = (Bn , Sn ) and the corresponding discounted pair X̄ nk = (1, Xnk ). Let us adapt Definition 1.2 of the self-financing strategy to our situation. Of course, this notion is valid in the general situation, not only on our binomial market. Definition 1.19. A two-dimensional trading strategy ξn̄ = {(ξnB,k , ξnS,k ), 1 ≤ k ≤ n} is called self-financing if ξn̄ k ⋅ X̄ nk = ξn̄ k+1 ⋅ X̄ nk ,

1 ≤ k ≤ n − 1.

(1.15)

The self-financing property of the strategy means that there is neither an inflow of capital from the outside, nor an outflow of capital outward on the market, and all the redistribution of the portfolio is carried out exactly within the framework of available resources. This property can be considered for more general market models, and for models with discrete time it has the form of (1.15). For the models with continuous time this property will be considered in Section 1.3.1. Definition 1.20. A self-financing trading strategy ξn̄ = {(ξnB,k , ξnS,k ), 1 ≤ k ≤ n} is called a replicating strategy for an attainable discounted contingent claim ℍ if ℍ = ξn̄ n ⋅ X̄ nn a. s. a. s.

For the non-discounted contingent claim ℂ, attainability means that ℂ = ξn̄ n ⋅ S̄nn

Definition 1.21. The discounted value process V = {Vnk , 0 ≤ k ≤ n} associated to a trading strategy ξn̄ is defined by Vn0 := ξn̄ 1 ⋅ X̄ n0 ,

Vnk := ξn̄ k ⋅ X̄ nk ,

1 ≤ k ≤ n,

where X̄ nk = (1, Xnk ), 0 ≤ k ≤ n, are the discounted price processes for the non-risky and risky assets. The following result can be found in [58, p. 250]. As usual, denote by ℙ(n,∗) the unique martingale measure in the nth series and by 𝔼∗n expectation with respect to this measure. Proposition 1.22. The value process of a replicating strategy for a contingent claim ℍ is equal to Vnk = 𝔼∗n [ℍ | ℱnk ],

0 ≤ k ≤ n,

14 | 1 Financial markets. From discrete to continuous time and admits the following representation: Vnk (ω) = vnk (X0 , Xn1 (ω), . . . , Xnk (ω)), where the function vnk is given by vnk (x0 , . . . , xk ) = 𝔼∗n [h(x0 , . . . , xk , xk

X n−k Xn1 , . . . , xk n )]. X0 X0

(1.16)

Functions vnk are called the capital functions. Note that the capital functions can be defined by using a backward recursion vnn (x0 , . . . , xn ) = h(x0 , . . . , xn ),

vnk (x0 , . . . , xk ) = p∗n vnk+1 (x0 , . . . , xk , xk b̂ n ) + (1 − p∗n )vnk+1 (x0 , . . . , xk , xk â n ),

where b̂ n =

1+bn , 1+rn

â n =

1+an . 1+rn

Definition 1.23. A replicating strategy for a contingent claim ℍ is called a delta-hedge and is denoted by Δkn := Δkn (Xn0 , Xn1 (ω), . . . , Xnk−1 (ω)). According to Proposition 5.46 from [58], function Δkn , participating in delta-hedge in the case of the binomial model, equals Δkn (xn0 , xn1 , . . . , xnk−1 ) =

vk (xn0 , xn1 , . . . , xnk−2 , xnk−1 , xnk−1 b̂ n ) − vk (xn0 , xn1 , . . . , xnk−2 , xnk−1 , xnk−1 â n ) . xk−1 (b̂ − â ) n

n

(1.17)

n

Below we obtain the expression for a delta-hedge for a discounted European call option + ̃n )+ , (1 + rn )−n (Snn − K) = (Xnn − K

̃n = K(1 + rn )−n K

in a market with discrete time. Since in this case the option price depends on the asset price only at the last moment, all expressions are greatly simplified for any moment ̃n−k = K(1 + rn )−n+k . of time. Indeed, denote K Lemma 1.24. A replicating strategy for a discounted European call option in the binomial model is given by Δkn (Xnk−1 ) =

vnk (Xnk−1 b̂ n ) − vnk (Xnk−1 â n ) , X k−1 (b̂ − â ) n

n

n

(1.18)

where n−k

n−k i n−i−k )(p∗n ) (1 − p∗n ) i

̃n−k ) ( vnk (y) = ∑ (â in b̂ n−i−k y−K n +

i=0

n−k

=

b̂ in y ∑ (â n−i−k n i=0

̃n−k ) (n − k )(q∗ )i (1 − q∗ )n−i−k . −K n n i +

(1.19)

1.2 Intermediate step |

15

Proof. Equality (1.18) immediately follows from (1.17), where a general form of a replicating strategy Δkn is obtained. Indeed, in the case under consideration, ℍ depends only on the price of an asset at the exercise time. In this case, Vnk depends only on the current price of an asset, Vnk (ω) = vnk (Xnk (ω)). Moreover, the right-hand side of relation (1.16) for the capital function equals the expectation of a function determined by a binomial random variable with parameter p∗n and therefore has the form of (1.19), whence the proof follows. 1.2.2.4 Arbitrage-free market with distribution of the jump concentrated on some interval Consider the case where the distribution of any random variable Rkn is concentrated on the interval [akn , bkn ]. Let, as before, μkn = 𝔼n Rkn , (σnk )2 = Varn Rkn , and let rnk be an interest rate at the kth period of the nth series. Lemma 1.25. Let the random variables {Rkn , 1 ≤ k ≤ n} be mutually independent under the objective measure ℙ(n) and have a non-degenerate distribution concentrated on some interval [akn , bkn ]. The financial market is arbitrage-free in the nth series if at least one of the following conditions holds: (i) rnk = μkn , 1 ≤ k ≤ n, (ii) μkn −

(σnk )2 bkn −μkn

< rnk < μkn , 1 ≤ k ≤ n,

(iii) μkn < rnk < μkn +

(σnk )2 , μkn −akn

1 ≤ k ≤ n.

Proof. We shall try to present ΔMnk as k k ΔMnk = φk−1 n (Rn − μn ),

(1.20)

k−1 k−1 where the random variable φk−1 n is supposed to be ℱn -measurable. Since φn is ink dependent of Rn , the relation (1.12) will be reduced to the following one: 2

k k μkn + φk−1 n (σn ) = rn ,

whence we immediately get that φk−1 n is non-random and equals φk−1 n =

rnk − μkn (σnk )2

.

In turn, it means that ΔMnk admits the representation (1.13). The no-arbitrage property of the market is equivalent to the relation ΔMnk > −1 a. s., or 2

(rnk − μkn )(Rkn − μkn ) + (σnk ) > 0 We consider three cases.

a. s.

(1.21)

16 | 1 Financial markets. From discrete to continuous time (i) If rnk = μkn , then the relation (1.21) obviously holds with probability 1. (ii) Let rnk > μkn . Then (1.21) holds with probability 1 if 2

(rnk − μkn )(akn − μkn ) + (σnk ) > 0,

(1.22)

and moreover we have akn < μkn , since the distribution of Rkn is non-degenerate and concentrated on the interval [akn , bkn ]. Therefore inequality (1.22) is equivalent to the inequality rnk < μkn +

(σnk )2

μkn − akn

.

(iii) The case when the inequality rnk < μkn holds can be treated similarly. The lemma is proved. Remark 1.26. It is obvious that under the assumptions of Lemma 1.25, one of the martingale measures can be defined according to relations (1.7) and (1.13). It is useful to reformulate Lemma 1.25 in terms of i. i. d. r. v. {Rkn , 1 ≤ k ≤ n}. Lemma 1.27. Assuming that the random variables {Rkn , 1 ≤ k ≤ n} are identically distributed with 𝔼n Rkn = μn and Varn Rkn = σn2 , mutually independent and have a distribution concentrated on [an , bn ], the financial market is arbitrage-free in the nth series under any of the additional assumptions: (i) rn = μn , (ii) μn −

(σn )2 bn −μn

< rn < μn ,

(iii) μn < rn < μn +

(σn )2 . μn −an

1.2.2.5 Incompleteness of the non-Bernoulli market Incompleteness of the financial market generated by mutually independent random variables {Rkn , 1 ≤ k ≤ n} with non-Bernoulli distribution will be established under additional conditions that simplify the situation from a technical point of view. Note that the following assumptions are natural for the model of the sequence of financial markets in the scheme of series taking into account potential unbounded increasing of the parameter n. Lemma 1.28. Let the random variables {Rkn , 1 ≤ k ≤ n} be mutually independent under the objective measure ℙ(n) and have non-degenerate distribution. Assume additionally that μkn = 0, 𝔼n (Rkn )3 = 0, and assume there exist 0 < α < β such that for any 1 ≤ k ≤ n we have β α 󵄨 󵄨 < 󵄨󵄨Rk 󵄨󵄨 < √n 󵄨 n 󵄨 √n

17

1.2 Intermediate step |

β

(for example, the distribution is symmetric), and also |rnk | < n . If the distribution is symmetric, assume additionally that it is concentrated not at two symmetric points. Then the sequence of financial markets is arbitrage-free and incomplete starting with n >

β8 . α8

Proof. We try to seek a representation for ΔMnk in two ways: (i) as it was done in Lemma 1.25 and (ii) in the form 3

k k ΔMn,1 = ψk−1 n (Rn ) ,

where the random variable ψk−1 should be ℱnk−1 -measurable. Under the current asn sumptions on the distribution of Rkn we have 4

𝔼n (Rkn ) > 0 Furthermore, 𝔼n (Rkn )4 >

α4 , n2

and ψk−1 n =

and for n >

rnk

𝔼n (Rkn )4

.

β8 α8

|rnk | 󵄨󵄨 k 3 󵄨󵄨 β β3 n2 β4 󵄨󵄨 k 󵄨󵄨 (R ) < ≤ < 1. 󵄨 󵄨󵄨ΔMn,1 󵄨󵄨 = 󵄨 n 󵄨 󵄨 n n√n α4 α4 √n 𝔼n (Rkn )4 Consider the equality k ΔMn,1 = ΔMnk ,

where ΔMnk is taken from (1.20). Now this equality can be transformed to rnk

𝔼n (Rkn ) that is, Rkn = 0 or (Rkn )2 =

3

(Rkn ) = 4

rnk

𝔼n (Rkn )2

Rkn ,

𝔼n (Rkn )4 . If the distribution is asymmetric or it is symmetric and 𝔼n (Rkn )2

not concentrated at two symmetric points, the latter equality is impossible. Therefore, for n >

β8 α8

the market will be both arbitrage-free and incomplete because there exist

at least two martingale measures ℙ(n,∗) and ℙ(n,∗,1) which are given by the relations n dℙ(n,∗) = ∏(1 + ΔMnk ) and (n) dℙ k=1

n dℙ(n,∗,1) k = ∏(1 + ΔMn,1 ), (n) dℙ k=1

respectively. 1.2.2.6 How to maintain independence with respect to the martingale measure It is interesting to formulate assumptions supplying independence of the random variables {Rkn , 1 ≤ k ≤ n} under martingale measure, too.

18 | 1 Financial markets. From discrete to continuous time Lemma 1.29. Let the random variables {Rkn , 1 ≤ k ≤ n} be mutually independent under the objective measure ℙ(n) and have non-degenerate distribution concentrated on some interval [akn , bkn ]. Assume additionally that the assumptions of Lemma 1.25 are fulfilled and so the market is arbitrage-free. Let us choose ℙ(n,∗) according to relations (1.7) and (1.13). Then the random variables {Rkn , 1 ≤ k ≤ n} are mutually independent with respect to ℙ(n,∗) also. Moreover, 𝔼∗n Rkn = rnk . Proof. Indeed, for any Borel sets A1 , . . . , An ℙ(n,∗) (Rkn ∈ Ak , 1 ≤ k ≤ n) = 𝔼n (

dℙ(n,∗) n ∏1 k ) dℙ(n) k=1 Rn ∈Ak n

= 𝔼n (∏((1 + n

k=1

rnk − μkn (σnk )2

rnk − μkn

= ∏ 𝔼n ((1 +

(σnk )2

k=1

(Rkn − μkn ))1Rk ∈Ak )) n

(Rkn − μkn ))1Rk ∈Ak ). n

(1.23)

Furthermore, dℙ(n,∗) 1 k ) dℙ(n) Rn ∈Ak n r i − μi = ∏ 𝔼n (1 + n i 2n (Rin − μin )) (σn ) i=1,i=k̸

ℙ(n,∗) (Rkn ∈ Ak ) = 𝔼n (

× 𝔼n ((1 + = 𝔼n ((1 +

rnk − μkn

rnk

(σnk )2

(Rkn − μkn ))1Rk ∈Ak )

− μkn k (Rn (σnk )2

n

− μkn ))1Rk ∈Ak ). n

(1.24)

It follows from (1.23) and (1.24) that n

ℙ(n,∗) (Rkn ∈ Ak , 1 ≤ k ≤ n) = ∏ ℙ(n,∗) (Rkn ∈ Ak ), k=1

which means that indeed {Rkn , 1 ≤ k ≤ n} are mutually independent with respect to the measure ℙ(n,∗) . Furthermore, since the process Xnk = Sn0 ∏nk=1 we can conclude that

k

𝔼∗n (Xnk ) = Sn0 𝔼∗n (∏ i=1

whence 𝔼∗n Rkn = rnk .

1 + Rin ) = Sn0 , 1 + rni

1+Rkn 1+rnk

is ℙ(n,∗) -martingale,

1.3 Preliminaries for the financial markets with continuous time

| 19

Remark 1.30. Compare the distributions of Rkn with respect to ℙ(n,∗) and ℙ(n) . Let the random variables {Rkn , 1 ≤ k ≤ n} satisfying the assumptions of Lemma 1.25 be identically distributed and have a density {fn (x), x ∈ ℝ}. Also, let rnk = rn , 1 ≤ k ≤ n. Then for any x ∈ ℝ ℙ(n,∗) (Rkn ≤ x) = 𝔼n ((1 +

rn − μn k (Rn − μn ))1Rk ≤x ) n (σn )2

= ℙ(n) (Rkn ≤ x) + =

rn μn − μ2n rn − μn k (n) k 𝔼 (R 1 (R ≤ x) k ≤x ) − ℙ n n n R n σn2 σn2

𝔼n (Rkn )2 − rn μn (n) k r −μ ℙ (Rn ≤ x) + n 2 n 𝔼n (Rkn 1Rk ≤x ) n σn2 σn x

x

= cn(1) ∫ fn (y)dy + cn(2) ∫ yfn (y)dy, −∞

where cn(1) =

−∞ k 2 𝔼n (Rn ) − rn μn , σn2

cn(2) =

rn − μn . σn2

This means that with respect to the measure ℙ(n,∗) random variables Rkn have a distribution density of the form fn∗ (x) = cn(1) fn (x) + cn(2) xfn (x).

1.3 Preliminaries for the financial markets with continuous time In this section we introduce the basic notions of financial markets with continuous time: self-financing strategies, the arbitrage property and martingale measures, completeness, and hedgeability. Mathematical preliminaries of the theory of martingales and semimartingales as well as the other aspects of stochastic calculus are presented in Appendix A.

1.3.1 The notion of self-financing strategy for the models in continuous time Let 𝕋 = [0, T], where T ≤ ∞, and let (Ω, ℱ , ℙ) be a complete probability space. This means that for all events B ∈ ℱ with ℙ(B) = 0 and all A ⊂ B one has A ∈ ℱ (and consequently ℙ(A) = 0). Let a financial market consist of two assets, a risk-free asset B = {B(t), t ∈ [0, T]} and a risky one S = {S(t), t ∈ [0, T]}. Recall that such market is called (B, S)-market, and this designation can be interpreted both as a “bond and stock” market and as a “Black–Scholes” market. Assume for technical simplicity that T = +∞ (however, for

20 | 1 Financial markets. From discrete to continuous time T < ∞ the situation is absolutely the same) and define the stochastic basis, that is, probability space, with filtration Ωℱ = (Ω, ℱ , 𝔽 = {ℱt , t ≥ 0}, ℙ). The notion of stochastic basis with filtration is introduced by Definition A.16. Also, assume that the stochastic process {S(t), t ≥ 0} like all processes considered below is adapted to the filtration 𝔽 = {ℱt , t ≥ 0}. For adaptedness, see Definition A.17. Definition 1.31. Strategy is the couple of stochastic processes (φ, ψ) = {φ(t), ψ(t), t ≥ 0} which correspond to the number of units of risk-free and risky assets available to the investor at time t. The values of these processes may be either positive or negative with some positive probabilities (a short sale of any asset is allowed). Obviously, the capital U(t) of the investor at time t equals U(t) = φ(t)B(t) + ψ(t)S(t). For technical simplicity, we can assume that the risk-free asset has a simple structure, for example, it is a process of T bounded variation, and that ∫0 |φ(t)|d|B|(t) < ∞ a. s., where symbol d|B|(t) stands for the integration with respect to the variation of the process B. The situation with the risky asset is more complicated. Assume additionally that the stochastic process S(t) admits a stochastic differential of the form dS(t) = α(t)dW(t) + β(t)dt, where W is a Wiener process (see Definition A.66), the processes α and β are adapted to the filtration 𝔽 = {ℱt , t ≥ 0}, and t

t

2

󵄨 󵄨 𝔼(∫ α (s)ds + ∫󵄨󵄨󵄨β(s)󵄨󵄨󵄨ds) < ∞ 0

0

t

for any t > 0. Note that M(t) := ∫0 α(s)dW(s) is a square-integrable martingale and t

A(t) = ∫0 β(s)ds is an integrable process of bounded variation. Assume that t

𝔼 ∫ ψ2 (s)α2 (s)ds < ∞, 0

t

󵄨 󵄨 𝔼 ∫󵄨󵄨󵄨ψ(s)β(s)󵄨󵄨󵄨ds < ∞. 0

t

Then, according to Section A.3.4, there exists a stochastic integral ∫0 ψ(s) dM(s) that is a square-integrable martingale, and also there exists the Lebesgue–Stieltjes intet gral ∫0 ψ(s) dA(s) that is a process of integrable variation. Therefore, there exists an

1.3 Preliminaries for the financial markets with continuous time

| 21

integral t

t

t

∫ ψ(s) dS(s) = ∫ ψ(s) dM(s) + ∫ ψ(s) dA(s) 0

0

0

t

t

= ∫ ψ(s)α(s) dW(s) + ∫ ψ(s)β(s) ds. 0

0

Remark 1.32. It is possible to assume only that t

2

t

2

ℙ{∫ ψ (s)α (s) ds < ∞} = 1 0

󵄨 󵄨 and ℙ{∫󵄨󵄨󵄨ψ(s)β(s)󵄨󵄨󵄨 ds < ∞} = 1. 0

t

In this case the stochastic integral ∫0 ψ(s) dM(s) exists but is only a locally squareintegrable martingale, which means that there exists a sequence of non-decreasing stopping times {τn , n ≥ 1} such that τn → ∞ with probability 1 and the stopped t∧τ processes ∫0 n ψ(s) dM(s) are the square-integrable martingales. Furthermore, in this t

case the stochastic process ∫0 ψ(s)β(s) ds is a process of a. s. locally bounded variation, i. e., its total variation is bounded a. s. on any interval.

Definition 1.33. The strategy (φ(t), ψ(t)) is called self-financing if its capital can be pret t sented as U(t) = U(0) + ∫0 φ(s) dB(s) + ∫0 ψ(s) dS(s), or, in differential form, dU(t) = φ(t) dB(t) + ψ(t) dS(t). The self-financing strategy is a strategy in which a change of the portfolio occurs only due to changes in asset prices, without capital inflows and outflows. Example 1.34. Let S(t) = W(t), B(t) = 1. (i) Strategies {φ(t) = 1, ψ(t) = 1} are self-financing if U(0) = 1. Indeed, we have dB(s) = 0, dS(s) = dW(s), and t

U(t) = φ(t)B(t) + ψ(t)S(t) = 1 + W(t) = 1 + ∫ dW(s) t

t

0

= U(0) + ∫ φ(s) dB(s) + ∫ ψ(s)dS(s). 0

0

Strategies {φ(t) = −t − W 2 (t), ψ(t) = 2W(t)} are self-financing if U(0) = 0. Indeed, we can use the relation t

∫ 2W(s) dW(s) = W 2 (t) − t, 0

22 | 1 Financial markets. From discrete to continuous time which is a simple application of the Itô formula, and obtain U(t) = φ(t)B(t) + ψ(t)S(t) = −t − W 2 (t) + 2W 2 (t) = W 2 (t) − t t

t

t

= ∫ 2W(s) dW(s) = U(0) + ∫ φ(s) dB(s) + ∫ ψ(s) dS(s). 0

0

0

(ii) To find the class of all self-financing strategies, write the equation of selft financing. In our case, U(t) = φ(t) + ψ(t)W(t) = U(0) + ∫0 ψ(s) dW(s), whence for the strategy to be self-financed on this market, it should satisfy the relation t φ(t) = U(0) − ψ(t)W(t) + ∫0 ψ(s) dW(s). 1.3.2 Arbitrage and martingale measures The concept of arbitrage in financial markets with continuous time can be in principle formulated and treated as in the case of discrete time. However, there are many varieties of this concept, depending on the form of the underlying assets, on the different classes of admissible strategies, and on the different approaches to the notion of arbitrage. A detailed discussion of this issue is contained, e. g., in the books [48, 152], and [153]. We assume now that T < ∞. Definition 1.35. A (B, S)-market is called arbitrage-free if there is no such selffinancing strategy (φ, ψ) for which the initial capital U(0) = φ(0)B(0) + ψ(0)S(0) ≤ 0, while U(T) = φ(T)B(T) + ψ(T)S(T) ≥ 0 with probability 1 and U(T) > 0 with positive probability. In order to establish the arbitrage-free property, we specify the market itself and the class of the self-financing strategies. Consider a (B, S)-market consisting of the bond (non-risky asset) with the exponential representation t

B(t) = exp{∫ r(s) ds},

(1.25)

0

where r = {r(t), t ∈ [0, T]} is a non-negative bounded adapted process and a stock (risky asset) with an exponential representation of the form t

t

S(t) = S(0) exp{∫ α(s) ds + ∫ σ(s) dW(s)}, 0

(1.26)

0

where α = {α(t), t ∈ [0, T]} is a bounded adapted process, σ = {σ(t), t ∈ [0, T]} is a bounded adapted strictly positive process, 0 < σ0 ≤ σ(t) ≤ σ1 , t ∈ [0, T], and σ0 and

1.3 Preliminaries for the financial markets with continuous time

| 23

σ1 are some constants. So, the risky asset is presented by the geometric Brownian motion with non-constant adapted bounded coefficients. The asset satisfies the following linear stochastic differential equation: σ 2 (t) )S(t) dt + σ(t)S(t) dW(t). 2

dS(t) = (α(t) +

The corresponding discounted price process has the form t

t

0

0

S(t) = S(0) exp{∫(α(s) − r(s)) ds + ∫ σ(s) dW(s)} X(t) = B(t) and satisfies the following linear stochastic differential equation: dX(t) = (α(t) − r(t) +

σ 2 (t) )S(t) dt + σ(t)S(t) dW(t). 2

Note that dB(t) = B(t)r(t) dt and t

dX(t) = exp{− ∫ r(s) ds}(dS(t) − S(t)r(t)dt). 0

We call such a market (B, S)-market with bounded coefficients. Now, we restrict ourselves to the class SFS of strategies (φ, ψ) satisfying the following assumptions. T T (i) The Lebesgue–Stieltjes integrals ∫0 |φ(s)| ds and ∫0 |ψ(s)| ds exist a. s. Then the Lebesgue–Stieltjes integrals T

T

󵄨 󵄨 ∫󵄨󵄨󵄨φ(s)󵄨󵄨󵄨B(s)r(s) ds and

󵄨 󵄨 ∫󵄨󵄨󵄨ψ(s)α(s)󵄨󵄨󵄨S(s) ds

0

0

exist a. s. because the processes r, α, and B are bounded and S is continuous and therefore bounded on almost all trajectories. T (ii) We assume that 𝔼 ∫0 ψ2 (s)S2 (s) ds < ∞. Then the stochastic integral T

∫ ψ(s)S(s)σ(s) dW(s) 0

exists and is a square-integrable martingale. (iii) The capital admits the following representation: t

t

U(s) = U(0) + ∫ φ(s) dB(s) + ∫ ψ(s) dS(s) 0

0

24 | 1 Financial markets. From discrete to continuous time t

t

= U(0) + ∫ φ(s)B(s)r(s) ds + ∫ ψ(s)(α(s) − r(s) + t

0

0

σ 2 (s) )S(s) ds 2

+ ∫ ψ(s)σ(s)S(s) dW(s). 0

Applying the above relations, the Itô formula (Theorem A.67), and Definition 1.33, = we can calculate the stochastic differential of the discounted capital V(t) = U(t) B(t) t

U(t) exp{− ∫0 r(s) ds} in the case of the self-financing strategy (φ, ψ) ∈ SFS: t

dV(t) = exp{− ∫ r(s) ds}(dU(t) − U(t)r(t) dt) 0

t

= exp{− ∫ r(s) ds}(φ(t)dB(t) + ψ(t) dS(t) 0

− φ(t)B(t)r(t) dt − ψ(t)S(t)r(t) dt)

= ψ(t) dX(t).

Theorem 1.36. A (B, S)-market with bounded coefficients is arbitrage-free if we restrict ourselves to the class SFS of self-financing strategies. Proof. We introduce the notation β(s) = r(s)−α(s) − 21 σ(s). Note that this stochastic proσ(s) cess is adapted and bounded. Consider the new probability measure ℙ∗ with the restriction on [0, T] of its Radon–Nikodym derivative having the form T

T

0

0

1 dℙ∗ = exp{∫ β(s) dW(s) − ∫ β2 (s) ds}. dℙ 2

(1.27)

Then obviously Novikov’s condition (see Theorem A.70) T

1 𝔼 exp{ ∫ β2 (s) ds} < ∞ 2 0

holds (it was one of the main reasons to assume boundedness of the coefficients), and the relation (1.27) really sets a Radon–Nikodym derivative of the probability measure ℙ∗ ∼ ℙ. Applying the Girsanov theorem (see Theorem A.69), under measure ℙ∗ the discounted risky asset has the form t

t

0

0

̃ − 1 ∫ σ 2 (s) ds}, X(t) = S(0) exp{∫ σ(s)dW(s) 2

1.3 Preliminaries for the financial markets with continuous time

| 25

t

̃ = W(t)−∫ β(s) ds is a Wiener process with respect to the measure ℙ∗ . Prowhere W(t) 0 ̃ cess X satisfies the stochastic differential equation dX(t) = σ(t) dW(t), i. e., is a square∗ integrable martingale with respect to the measure ℙ . Moreover, discounted capital ̃ under measure ℙ∗ admits a stochastic differential dV(t) = ψ(t) dX(t) = ψ(t)σ(t) dW(t). T Note additionally that 𝔼∗ ∫0 ψ2 (t)σ 2 (t) dt < ∞, where 𝔼∗ stands for the expectation with respect to the measure ℙ∗ . It means that V(t) is also a square-integrable martingale with respect to the measure ℙ∗ . Furthermore, suppose that V(0) ≤ 0. Then for any t ∈ [0, T] t

t

̃ = V(0) ≤ 0, 𝔼∗ V(t) = V(0) + 𝔼∗ ∫ ψ(s) dX(s) = ∫ ψ(s)σ(s) dW(s) 0

0

and this obviously means that the market is arbitrage-free. Remark 1.37. Similarly to the discrete time case, denote by 𝒫 the set of equivalent martingale measures, i. e., the measures that transform the risky asset into a martingale. As easily follows from the proof of Theorem 1.36, for a (B, S)-market described by the relations (1.25)–(1.26), the set 𝒫 consists of the unique probability measure ℙ∗ , described by the relation (1.27).

1.3.3 Hedging strategies Consider a (B, S)-market (1.25)–(1.26) and let ℂ ≥ 0 be a European contingent claim with maturity date T > 0 on this market. Definition 1.38. Strategy (φ, ψ) is called a hedging (replicating) strategy for the European contingent claim ℂ ≥ 0 if it is self-financing and its capital at maturity equals U(T) = φ(T)B(T) + ψ(T)S(T) = ℂ a. s. Lemma 1.39. If (φ(t), ψ(t)) is a hedging strategy and the market is arbitrage-free, then the arbitrage-free price of the (non-discounted) claim ℂ at time t equals U(t) := φ(t)B(t) + ψ(t)S(t) and the price of the corresponding discounted claim 𝔻 at time t equals V(t) := φ(t) + ψ(t)X(t). Proof. Not going deeply into the details, we present two arguments in favor of the statement of the theorem. (i) We can give an outline of the proof, arguing “ω by ω.” Consider the nondiscounted claim. If its price at time t is less than U(t) and equals U ′ (t), then it is possible at this moment to buy the security (claim, contingent claim) ℂ for the price U ′ (t) and sell φ(t) units of the share B and ψ(t) units of the share S. If the buyer of the portfolio will follow the strategy (φ, ψ), then at time T, φ(T)B(T) + ψ(T)S(T) = ℂ. This means that the claim which the investor has in hands will be

26 | 1 Financial markets. From discrete to continuous time of the same price as the portfolio that he sold, they “annihilate,” and no additional costs between moments t and T are necessary. Therefore, the investor has a positive income (U(t) − U ′ (t))

B(T) B(T) = (φ(t)B(t) + ψ(t)S(t) − U ′ (t)) B(t) B(t)

and no risk. Similarly, if the price of ℂ at time t exceeds U(t) and equals U ′′ (t), then it is necessary to sell ℂ and buy portfolio U(t); the income at time T will be equal to (U ′′ (t) − U(t)) B(T) without risk. B(t) (ii) Consider the discounted claim. Similarly to the discrete time case, we can prove that for ℙ∗ ∈ 𝒫 , the discounted hedgeable contingent claim 𝔻 ≥ 0, and the discounted capital V = {V(t), t ∈ [0, T]} of some self-financing strategy (φ, ψ) ∈ SFS that hedges 𝔻, we have 𝔼∗ (𝔻) < ∞ and V(t) = 𝔼∗ (𝔻 | ℱt ) ℙ-a. s., t ∈ [0, T], where, as usual, 𝔼∗ means expectation taken with respect to martingale measure ℙ∗ . Furthermore, T

󵄨󵄨 𝔼 (𝔻 | ℱt ) = 𝔼 (V(T) | ℱt ) = V(0) + 𝔼 (∫ ψ(s) dXs 󵄨󵄨󵄨 ℱt ) 󵄨 ∗

∗

∗

t

0

= V(0) + ∫ ψ(s) dXs = Vt = φ(t) + ψ(t)X(t). 0

1.3.4 Complete markets As in the discrete case, a financial market with continuous time is called complete if any contingent claim on it is hedgeable. A criterion of the completeness of the arbitrage-free market is the uniqueness of the equivalent martingale measure, which means that 𝒫 consists of a unique point. As we claimed in Remark 1.37, this is the case when the market is described by the relations (1.25)–(1.26). In particular, a Black–Scholes market with constant coefficients is arbitrage-free and complete. Completeness can be described as the unique source of randomness. Of course, in the case of multi-dimensional markets the situation is different; see [153]. In that case completeness is equivalent to the coincidence of the number of assets and the “rank” of random sources; for the exact statements see also [65]. A lot of attention is paid now in the literature to markets with jumps, jumpdiffusion, and Lévy market models; see, e. g., [7] for the purpose of in-depth study of the theory and various applications and [103] for applications to option pricing. The markets with stochastic volatility are also widely treated; see, e. g., [61, 62]. In this book models of the markets with stochastic volatility are considered in Sections 2.4 and 2.5. Such markets are incomplete, as a rule.

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion

| 27

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion Now our goal is to formulate conditions that ensure the convergence of prices of primary assets in financial markets with discrete time to certain limits when the length of the interval partition tends to zero (high-frequency approach). Of course, the Black– Scholes price in the limit is the simplest case, but later on we will consider much more general schemes. 1.4.1 Pre-limit sequence of the models with discrete time in the multiplicative scheme We consider the relation between the multiplicative scheme of the financial market that operates in discrete time and the famous Black–Scholes–Merton model of the financial market operating in continuous time. Let, as in Section 1.2, (Ω(n) , ℱ (n) ) be a sequence of measurable spaces. Consider the sequence of financial markets with discrete time, with one risk-free and one risky asset each, each defined on the corresponding probability space. More precisely, let T > 0 be a fixed number and let parameter n take integer values from ℕ. For n ≥ 1 consider the partition of the interval 𝕋 = [0, T] having the form π(n) = {0 = tn0 < tn1 , . . . < tnn = T}. As before, we assume that the points of the partition are the trading moments on the corresponding financial market. Let our goal be to calculate the option price of some European option at time T. On the one hand, in the case when the number n of trading periods is large enough and even in the simplest case of the binomial model, the calculations can be very technical and can take a lot of time even with powerful computers. The situation is further complicated if the distribution of jumps in stock prices is not a binomial one; see Section 1.2. On the other hand, it is well known that the binomial distribution can be with sufficient accuracy approximated using the Gaussian distribution. Moreover, there is quite a powerful theory of functional limit theorems which allows a Gaussian distribution to approximate a sufficiently large class of pre-limit distributions. Therefore, we can hope to get a much simpler formula for calculation of the option price by moving to the limit with an unlimited increase in the number of periods. Summarizing in a few words, real financial markets operate in discrete time; however, it is desirable to produce calculations in continuous time. We describe a mathematical model that allows passage to the limit. Consider the multiplicative scheme of the financial market described in Section 1.2. Recall that it consists of one non-risky and one risky asset, described by equations (1.2) and (1.3), and the corresponding discounted risky asset is given by (1.4).

28 | 1 Financial markets. From discrete to continuous time These formulas are given in discrete time. However, it is easy to consider the stepwise extensions of the asset prices to continuous time. For the purpose of technical . Then, for example, the discounted risky asset can simplicity, assume that tnk = kT n be rewritten as a sequence Xn = {Xn (t), t ∈ [0, T]} of the stochastic processes with continuous time of the form Xn (t) =

Sn0

⌋ ⌊ nt T

∏ k=1

1 + Rkn 1 + rnk

,

(1.28)

where ⌊a⌋ is the biggest integer part not exceeding a, ∏0k=1 = 1. 1.4.2 Geometric Brownian motion Consider the following model of the market with continuous time t ∈ 𝕋 = [0, ∞). Let a financial market consist of two assets, bond and stock. A bond is a risk-free asset with constant interest rate r > 0 described by the equation B(t) = exp{rt},

(1.29)

and a stock is a risky asset described by the equation S(t) = S(0) exp{μt + σW(t)},

(1.30)

where W = {W(t), t ≥ 0} is a Wiener process (recall that Section A.3.2 contains the definition and basic properties of the Wiener process). Process S = {S(t), t ≥ 0} is called the geometric Brownian motion. Coefficient μ ∈ ℝ is called a drift coefficient while σ ∈ ℝ is a volatility coefficient. Since a Wiener process is symmetric in the sense that −W is also a Wiener process, it is always supposed that σ > 0. The model described by equations (1.29) and (1.30) is called the Black–Scholes–Merton model. It was originally introduced in the papers [19] and [115]. Why is the geometric Brownian motion selected as the basic risky asset? One argument is that the geometric Brownian motion is the limiting process for binomial and similar models with discrete time. Another argument is that the relative increments S(t)−S(s) S(t) (or simply the ratios S(s) ) in this model are stationary and additionally the S(s) stochastic process {log S(t), t ≥ 0} has stationary increments. These properties lead to the simplicity of the model, but they are criticized if in reality stationarity is violated. Another argument is the arbitrage-free property and completeness of the market, which was established in Section 1.3. The corresponding discounted risky asset is described by the equality X(t) = S(0) exp{(μ − r)t + σW(t)},

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion

| 29

and it satisfies the following linear stochastic differential equation: dX(t) = (μ − r +

σ2 )X(t) dt + σX(t) dW(t). 2

Without going into details just now, we note only that under the unique martingale measure the discounted risky asset gets the form X(t) = S(0) exp{σW(t) −

σ2 t} 2

(1.31)

and satisfies the following stochastic differential equation: dX(t) = σX(t) dW(t),

X(0) = S(0),

or, in the equivalent integral form, t

X(t) = S(0) + σ ∫ X(s) dW(s). 0

Both of the latter equations can be interpreted in the sense that under the unique martingale measure X is a martingale. Another very useful fact is that denoting expectation with respect to the unique equivalent martingale measure by 𝔼∗ , we get from the latter equations that for any t ≥ 0, 𝔼∗ X(t) = S(0). 1.4.3 Functional central limit theorem for the financial markets with discrete time represented by the multiplicative scheme Now, for the convenience of further considerations, we collect conditions ensuring the passage to the limit. (A1) There exists c0 > 0 such that the random variables {Rkn , 1 ≤ k ≤ n} admit the c following bound: |Rkn | ≤ √n0 . Assume also that rnk ≥ 0.

(A2) There exists n0 ∈ ℕ such that for n ≥ n0 {Rkn , 1 ≤ k ≤ n} satisfy conditions supplying the arbitrage-free property of the market in the nth series. (A3) With respect to the measure ℙ(n,∗) the random variables {Rkn , 1 ≤ k ≤ n} are mutually independent and moreover 𝔼∗n Rkn = rnk . (A4) There exist constants S0 > 0, r > 0, and σ > 0 such that lim max

n→∞ 1≤k≤n

and

⌊ nt ⌋ T

rnk

= 0,

lim S0 n→∞ n

lim ∑ Var∗n Rkn = σ 2 t > 0,

n→∞

k=1

= S0 ,

0 < t ≤ T.

⌊ nt ⌋ T

lim ∑ rnk = rt > 0,

n→∞

k=1

30 | 1 Financial markets. From discrete to continuous time (A5) There exists a constant c1 > 0 such that ⌊

nt2 T

⌋

∑

k=⌊

nt1 T

rnk

⌋+1

≤ c1 (t2 − t1 ),

⌊

nt2 T

⌋

∑

k=⌊

nt1 T

Var∗n Rkn ≤ c1 (t2 − t1 )

⌋+1

for any 0 ≤ t1 ≤ t2 ≤ T. Remark 1.40. (i) Some of the sufficient conditions supplying the arbitrage-free property of the market defined by the multiplicative scheme (1.2)–(1.4) are described in Lemmas 1.16, 1.25, and 1.28. The arbitrage-free property implies the existence of an equivalent martingale measure ℙ(n,∗) , described by relations (1.7) and (1.9). Denote, as before, by 𝔼∗n and Var∗n the expectation and variance with respect to the measure ℙ(n,∗) , respectively. We do not discuss here the question of the uniqueness of ℙ(n,∗) . The results are valid for any ℙ(n,∗) , satisfying conditions formulated below. (ii) Some of the sufficient conditions supplying the mutual independence of the random variables {Rkn , 1 ≤ k ≤ n} with respect to the measure ℙ(n,∗) are formulated in Lemma 1.29. (iii) Condition (A5) means in particular that max1≤k≤n rnk → 0, n → ∞. ℚ(n) , ℚ, d

We introduce the following notations: 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ stands for weak convergence in disℚ(n) , ℚ, fdd

tribution of the random variables, 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ stands for weak convergence of finiteℚ(n) , ℚ

dimensional distributions of the stochastic processes, and 󳨐󳨐󳨐󳨐󳨐⇒ stands for weak convergence of the measures corresponding to the stochastic processes. Each notation corresponds to the measures ℚ(n) (for the pre-limit processes) and ℚ (for the limit process). Concerning the weak convergence of probability measures see Section A. Let the pre-limit sequence of the processes with discrete time be described by (1.28), the arbitrage-free property is supplied by conditions (A1)–(A3), and the measures ℙ(n,∗) are any martingale measures that correspond to the pre-limit processes. Let the limit process have a form X(t) = S(0) exp{σW(t)− 21 σ 2 t} and denote by ℙ∗ the measure corresponding to this process. Once again, ℙ∗ is the unique equivalent martingale measure on the limit market, and with respect to this measure X is a martingale. For a brief description of this process see Section 1.4.2 above. Theorem 1.41. (i) Let the random variables {Rkn , 1 ≤ k ≤ n} satisfy conditions (A1)–(A4). Then with respect to the sequence of martingale measures ℙ(n,∗) , the weak convergence of finite-dimensional distributions on the interval [0, T] holds: ℙ(n,∗) , ℙ∗ , fdd

Xn 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ X.

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion

| 31

(ii) Let conditions (A1)–(A5) hold. Then the weak convergence of the measures corresponding to the stochastic processes Xn and X on the interval [0, T] holds: ℙ(n,∗) , ℙ∗

Xn 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ X. Proof. Without loss of generality and for technical simplicity assume that T = 1 and Sn0 = 1. Recall that tnk = nk . (i) At first, prove the weak convergence of one-dimensional distributions. To begin with, note that Xn (t) > 0; therefore, for any 0 ≤ t ≤ 1 we can consider the random variable log Xn (t). Now, recall the Taylor expansion for the function f (x) = log(1 + x), x > −1: log(1 + x) = x −

x2 1 x3 , + 2 3(1 + θ)3

1 3 where |θ| ≤ |x|. If |x| ≤ c < 1, then | 3(1+θ) 3x | ≤ c0 √n

1 , where c0 2

< has the form

1 |x|3 . 3(1−c)3

Choose n1 so that for n ≥ n1

is taken from condition (A1). The additive representation of log Xn (t) ⌊nt⌋

log Xn (t) = ∑ (log(1 + Rkn ) − log(1 + rnk )). k=1

(1.32)

Applying the Taylor expansion to any term in (1.32), we get for n ≥ n1 the following equality: 1 2 3 log(1 + Rkn ) = Rkn − (Rkn ) + α(n, k) ⋅ (Rkn ) , 2 where |α(n, k)| ≤

Then for n ≥ n2 ,

1 c0 3 3(1− √n )

(1.33)

≤ 83 . Similarly, choose n2 so that for n ≥ n2 , max1≤k≤n rnk < 21 .

1 2 3 log(1 + rnk ) = rnk − (rnk ) + β(n, k)(rnk ) , 2 where |β(n, k)| ≤ 83 . Taking into account these expansions, we get ⌊nt⌋

log Xn (t) + ∑

k=1

Var∗n Rkn ⌊nt⌋ k 1 1 1 2 2 = ∑ (Rn − rnk + (rnk ) − (Rkn ) + Var∗n Rkn ) 2 2 2 2 k=1 ⌊nt⌋

+

3 ∑ (α(n, k)(Rkn ) k=1

+

3 β(n, k)(rnk ) ).

Now, denote 1 1 1 2 2 ηkn = Rkn − rnk − (Rkn ) + (rnk ) + Var∗n Rkn 2 2 2

(1.34)

32 | 1 Financial markets. From discrete to continuous time and 3

3

ηkn = α(n, k)(Rkn ) + β(n, k)(rnk ) . Concerning the random variables ηkn , according to condition (A3), they are centered and orthogonal, or more precisely, 𝔼∗n ηkn = 𝔼∗n ηkn ηjn = 0, j ≠ k. Concerning the “residuals”, they can be bounded as follows: 󵄨󵄨⌊nt⌋ 󵄨󵄨 3 n 󵄨 󵄨󵄨 󵄨󵄨 ∑ ηk 󵄨󵄨󵄨 ≤ n ⋅ 8 ( c0 ) + 8 max (r k )2 ∑ r k → 0 n n n 󵄨󵄨 󵄨󵄨 3 √n 3 1≤k≤n 󵄨󵄨 k=1 󵄨󵄨 k=1

(1.35)

as n → ∞ a. s., and therefore these values do not affect the rest of the proof. Now, the variance can be transformed as 1 2 Var∗n ηkn = Var∗n (Rkn − (Rkn ) ) = Var∗n Rkn + δnk , 2 where 3

δnk = −𝔼∗n (Rkn ) +

1 ∗ k 4 1 ∗ k 2 2 k ∗ k 2 𝔼 (R ) − (𝔼n (Rn ) ) + rn 𝔼n (Rn ) . 4 n n 4

Taking into account the upper bound for some c > 0,

c0 √n

for Rkn , we obtain the following bound for δnk :

cr k c 󵄨󵄨 k 󵄨󵄨 󵄨󵄨δn 󵄨󵄨 ≤ 3/2 + n . n n

(1.36)

Together with condition (A4), it means that ⌊nt⌋

∑ Var∗n ηkn → σ 2 t.

k=1

Now, it follows from the CLT (see Theorem A.110) that with respect to the measures ℙ(n,∗) the weak convergence ⌊nt⌋

ℙ(n,∗) , ℙ∗ , d

( ∑ ηkn , t ∈ [0, 1]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (𝒩 (0, σ 2 t), t ∈ [0, 1]) k=1

holds and in turn implies the weak convergence ℙ(n,∗) , ℙ∗ , d 1 (log Xn (t), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (𝒩 (− σ 2 t, σ 2 t), t ∈ [0, 1]). 2

Secondly, via similar calculations and estimations applied to log

Xn (t2 ) = log Xn (t2 ) − log Xn (t1 ), Xn (t1 )

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion

| 33

we can prove that ℙ(n,∗) , ℙ∗ , d 1 log Xn (t2 ) − log Xn (t1 ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ 𝒩 (− σ 2 (t2 − t1 ), σ 2 (t2 − t1 )). 2

(1.37)

Finally, the weak convergence of finite-dimensional distributions follows now from (1.37) and Theorem A.109. (ii) Weak convergence of the measures corresponding to Xn is equivalent to weak convergence of the measures corresponding to log Xn . To establish the latter weak convergence, it is sufficient to prove weak relative compactness of the corresponding measures; see Definition A.90 and corresponding theorems. According to Theorem A.99, taking into account mutual independence of the summands and Remark A.100, to establish weak relative compactness, it is sufficient to establish that there exists a constant c > 0 such that for any n ≥ n0 and 0 ≤ t1 ≤ t2 ≤ 1 the following relation holds: 2

𝔼∗n (log Xn (t2 ) − log Xn (t1 )) ≤ c(t2 − t1 ). Taking into account the evident inequality ⌊nt2 ⌋ − ⌊nt1 ⌋ ≤ n(t2 − t1 ), expansions (1.32)–(1.34), bounds (1.35)–(1.36), and condition (A5), we get that there exists a constant c > 0 such that 2

𝔼∗n (log Xn (t2 ) − log Xn (t1 )) = 𝔼∗n (

⌊nt2 ⌋

∑

k=⌊nt1 ⌋+1

≤

ηkn +

⌊nt2 ⌋

∑

k=⌊nt1 ⌋+1

⌊nt2 ⌋ 2𝔼∗n ( ∑ ηkn ) k=⌊nt1 ⌋+1

ηkn )

2

2

+ 2(t2 − t1 )(

c 2 + c max (rnk ) ). √n 1≤k≤n

2 Therefore it is sufficient to bound an (t1 , t2 ) := 𝔼∗n (∑k=⌊nt ηkn )2 . However, according to 1 ⌋+1 condition (A5), taking (1.36) into account, we arrive at

⌊nt ⌋

an (t1 , t2 ) =

⌊nt2 ⌋

∑

k=⌊nt1 ⌋+1

2

𝔼∗n (ηkn ) =

⌊nt2 ⌋

∑

(Var∗n Rkn + δnk )

k=⌊nt1 ⌋+1

c(t − t ) ≤ c(t2 − t1 ) + 2 1 + crnk (t2 − t1 ), √n whence the proof follows. Remark 1.42. Elementary transformations using Taylor expansion for log lead us to the conclusion that under condition (A4) there is a convergence ⌊ nt ⌋ T

∏ (1 + rnk ) → exp{rt} as n → ∞. k=1

34 | 1 Financial markets. From discrete to continuous time Corollary 1.43. Weak convergence of measures means that for any bounded and continuous (in the Skorokhod topology defined on the space D([0, T]) of càdlàg functions) functional f : x ∈ D([0, T]) → ℝ, we have 𝔼∗n f (Xn ) ⇒ 𝔼∗ f (X). Furthermore, the following functionals are continuous in the Skorokhod topology (function x = x(t) ∈ D([0, T])): T

f (x) = x(T),

max x(t),

min x(t),

0≤t≤T

0≤t≤T

∫ x(t) dt. 0

Therefore, any of the functionals +

f (x) = (K − x(T)) ,

(K − max x(t)) ,

+

0≤t≤T T

+

+

(K − ∫ x(t) dt)

(K − min x(t)) , 0≤t≤T

0

and some others that correspond to vanilla, look-back, and Asian options are bounded and continuous in the Skorokhod topology. For such functionals the weak convergence of probability measures immediately implies the convergence of option prices. Remark 1.44. Consider the sequence of symmetric binomial markets. Let T = 1, r > 0, and σ > 0 be fixed values. The random variables Rkn can attain two values, an and bn , where −1 < an < bn , and these values have a symmetric form in the following sense: an = e−σ√δn − 1 and bn = eσ√δn − 1, where δn = n1 . Also, let rn = rδn . Assumptions on the

asset price mean that it goes up or down,

Skn n Sk−1

= 1 + an or 1 + bn , and (1 + an )(1 + bn ) = 1.

Compare asymptotic behavior of an , rn , and bn : an ∼ −σ √δn ,

bn ∼ σ √δn

as n → ∞, and this means that −1 < an < rn < bn for sufficiently large n, and for such n condition (A1) evidently holds. Consider only such n. Then the model is arbitragefree and complete, and condition (A2) holds. Calculate the unique martingale measure ℙ(n,∗) : p∗n = ℙ(n,∗) (Rkn = an ) =

σ√δn bn − rn eσ√δn − 1 − rδn 1 = ∼ → bn − an 2 σ√δn −σ√δn 2σ√δ n e −e

as n → ∞. With respect to the measure ℙ(n,∗) we have 𝔼∗n Rkn = rn , and condition (A3)

holds since the model is binomial. Furthermore, ∑⌊nt⌋ r = k=1 n

⌊nt⌋r n

→ rt, a2n ∼

σ2 , b2n n

∼

σ2 , n

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion

| 35

and therefore ⌊nt⌋

∑ Var∗n Rkn = ⌊nt⌋(a2n

k=1

r − an bn − rn nt + b2n n − rn2 ) ∼ (a2n + b2n ) ∼ σ 2 t. bn − an bn − an 2

2 Therefore, condition (A4) holds. Furthermore, ∑k=⌊nt r ≤ r(t2 − t1 ). The random ⌋+1 n

⌊nt ⌋

1

variables Rkn are equally distributed in k. Consider Var∗n R1n . From elementary inequalities 0 ≤ 1 − e−x ≤ x and 0 ≤ ex − 1 ≤ ex x for x > 0 (see Lemma A.2(iv) in the appendix) we immediately deduce that a2n ≤ σ 2 δn and b2n ≤ e2σδn σ 2 δn . Therefore, Var∗n R1n ≤ a2n + b2n ≤ σ 2 δn (1 + e2σδn ) ≤

3σ 2 n

2 for such n that e2σδn ≤ 2. Therefore, for such n the sum ∑k=⌊nt Var∗n Rkn admits the 1 ⌋+1 following bound:

⌊nt ⌋

⌊nt2 ⌋

∑

k=⌊nt1 ⌋+1

Var∗n Rkn ≤ n(t2 − t1 ) Var∗n R1n ≤ 3σ 2 (t2 − t1 ).

We get that conditions (A1)–(A5) are satisfied, and consequently Theorem 1.41 holds for this symmetric binomial market.

1.4.4 Black–Scholes formula as the result of limit transition. Option pricing Consider the model of the financial market with continuous time, described by equations (1.29) and (1.30), and whose discounted risky asset price with respect to the martingale measure is described by equation (1.31). Recall also that according to the notations introduced in Section 1.4.1, the formula n

Xn (T) = Sn0 ∏ k=1

1 + Rkn 1 + rnk

describes the risky asset price at time T. Similarly, the formula Xn (t) =

Sn0

⌋ ⌊ nt T

∏ k=1

1 + Rkn 1 + rnk

describes the risky asset price at any time 0 ≤ t ≤ T, and under conditions (A1)–(A5) Theorem 1.41 holds, supplying weak convergence in the distribution of the corresponding random variables: ℙ(n,∗) , ℙ∗ , d

Xn (t) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ X(t) = S(0) exp{σW(t) −

σ2 t}. 2

36 | 1 Financial markets. From discrete to continuous time Furthermore, let f : ℝ → ℝ+ be a bounded Borel function and let ℂ = f (Xn (t)) be the derivative contingent claim whose payoff function f depends only on the asset price at the maturity date, which can vary, and therefore from now on it is denoted by t. Then the Lebesgue dominated convergence theorem guarantees the convergence of arbitrage-free prices: 𝔼∗ f (X(t)) = lim 𝔼∗n f (Xn (t)),

(1.38)

n→∞

where the expectation in the left-hand side is taken with respect to the martingale measure for the limit market, i. e., such measure with respect to which X(t) has a form as described by equation (1.31) and consequently has log-normal distribution with parameters log S(0) − σ 2 /2T and σ 2 T. In particular, equality (1.38) holds for the European put option with strike price K because in this case f (x) = (K − x)+ , x ≥ 0, i. e., 0 ≤ f (x) ≤ K. For technical simplicity, redenote by x the initial price S(0) of a risky asset. The initial price S(0) = x is also called spot price. Let πtcall (x) and πtput (x) be the arbitragefree prices of the discounted European call and put options, respectively, for the asset with initial price x, with expiry date t, and calculated at zero moment of time. Then it follows from equation (1.38) and the boundedness of the payoff function of the put option that πtput (x) = lim 𝔼∗n (Ke−rt − Xn (t)) = 𝔼∗ (Ke−rt − X(t)) +

n→∞

= 𝔼∗ (Ke−rt − x exp{σW(t) −

+

σ2 t}) . 2 +

(1.39)

Sometimes this formula is used especially for the expiry date T and is written without the subscript T: π put (x) = πTput (x) = e−rT 𝔼∗ (K − S(T)) ,

(1.40)

+

where S(T) is a price of a non-discounted risky asset with initial price x at time T. Now we can use the put–call parity relation for the continuous time, which has the form πtcall (x) − πtput (x) = x − Ke−rt . Therefore, πtcall (x) = x − Ke−rt + lim 𝔼∗n (Ke−rt − Xn (t))

+

n→∞

= lim 𝔼∗n (Xn (t) − Ke−rt ) = 𝔼∗ (X(t) − Ke−rt ) n→∞

= 𝔼∗ (x exp{σW(t) −

+

+

σ2 t} − Ke−rt ) . 2 +

(1.41)

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion

| 37

Remark 1.45. Formulas (1.38)–(1.41) demonstrate that the option price is invariant under taking limit in the framework of Theorem 1.41. But the real situation is the opposite: we approximate the pre-limit option price by the limit one because the limit one can be easily calculated. Immediately the question of the rate of convergence of option prices in the pre-limit model to the limit option price arises. We discuss it below, in Section 2.1.4, and also refer the reader to papers and books [26, 30, 76, 117–119, 139, 167, 168]. Now we calculate πtcall (x) via the right-hand side of (1.41) and get the famous Black–Scholes formula for the arbitrage-free (fair) price of the European call option as a result. Recall again that both prices, πtcall (x) and πtput (x), substantially depend on the maturity time (maturity date, exercise time, exercise date) t. Theorem 1.46. Let the discounted asset price at time t have the form X(t) = x exp{σW(t) −

σ2 t}, 2

where W(t) is a random variable that is a value of the Wiener process at time t. Then the arbitrage-free (fair) price of the discounted European call option for this asset has the form πtcall (x) = xΦ(d+ (t, x)) − e−rt KΦ(d− (t, x)), x

(1.42)

2

where Φ(x) = √12π ∫−∞ e−y /2 dy is a standard normal distribution function, or standard cumulative Gaussian distribution function, log(x/K) + (r + σ 2 /2)t , σ √t log(x/K) + (r − σ 2 /2)t d− (t, x) = d+ (t, x) − σ √t = . σ √t d+ (t, x) =

Proof. Obviously, πtcall (x) =

2 2 1 + √ ∫(xeσ ty−σ t/2 − Ke−rt ) e−y /2 dy √2π

ℝ

1 = √2π x = √2π

∞

∫ (xeσ

√ty−σ 2 t/2

− Ke−rt )e−y

2

/2

dy

−d− (t,x) ∞

∫

e−(y−σ

√t)2 /2

dy − e−rt K(1 − Φ(−d− (t, x)))

−d− (t,x)

= xΦ(d+ (t, x)) − e−rt KΦ(d− (t, x)).

(1.43) (1.44)

38 | 1 Financial markets. From discrete to continuous time Remark 1.47. Let 𝔽 = {ℱt , t ∈ [0, t2 ]} be a filtration generated by the Wiener process W participating in the asset price on the Black–Scholes market with exercise date t2 . It is well known that the fair price at time t1 ≤ t2 of the discounted call option equals πtcall (x) = ert1 𝔼∗ ((X(t2 ) − Ke−rt2 ) | ℱt1 ). 1 ,t2 +

Now we can prove a more general result concerning the option price when the asset price has a general log-normal distribution. Lemma 1.48. Let the non-discounted asset price has at time t the form xeY , where Y is a Gaussian random variable with m = 𝔼(Y), σ 2 = Var Y. Then, under the same measure, the price πtcall (x, m, σ) of the discounted European call option with strike price K and maturity date t, calculated at moment zero, equals 1

πtcall (x, m, σ) = xem+ 2 σ

2

−rt

Φ(

m − log K m + σ 2 − log K ) − Ke−rt Φ( ). σ σ

(1.45)

Proof. Let us express the discounted option price in terms of density of the distribution: πtcall (x, m, σ) = e−rt 𝔼(xeY − K) =e

∞

−rt

+

∫ (xey − K) log K

1 (y − m)2 exp{− } dy := e−rt (xI1 + I2 ). 2σ 2 σ √2π

Now we calculate either of the integrals: ∞

I1 = ∫ ey log K

(y − m)2 1 exp{− } dy 2σ 2 σ √2π

= exp{m + = exp{m +

σ2 log K − m − σ 2 }(1 − Φ( )) 2 σ

σ2 m + σ 2 − log K }Φ( ), 2 σ

where Φ, as before, is a standard cumulative Gaussian distribution function. Similarly, ∞

I2 = −K ∫

log K

1 (y − m)2 1 − y22 exp{− } dy = −K ∫ e dy 2 √2π 2σ σ √2π log K−m ∞

σ

log K − m m − log K = −K(1 − Φ( )) = −KΦ( ). σ σ Thus equality (1.45) holds.

| 39

1.4 From discrete to continuous time. The limit process is a geometric Brownian motion

We investigate the behavior of the European call option price (1.45) as a function of the mean m and the variance σ 2 (in some sense, it is simpler than in the Black–Scholes model, because now m and σ are not connected and in the Black–Scholes model we 2 have corresponding values of parameters log x − σ2 t and σ 2 t. Lemma 1.49. The option price (1.45) increases in m and in σ 2 . Proof. We omit the multiplier e−rt , put for technical simplicity x = 1, denote the result simply π call (m, σ), and calculate the derivatives with respect to m and to s := σ 2 . The derivative in m is equal to 𝜕 call s m + s − log K π (m, σ) = exp{m + }Φ( ) √s 𝜕m 2 +

2

s 1 1 m + s − log K exp{m + } exp{− ( )} √2π√s √s 2 2

(1.46)

2

K 1 m − log K − exp{− ( )} √2π√s √s 2 s m + s − log K ). = exp{m + }Φ( √s 2 The derivative in s is equal to 𝜕 call 1 s m + s − log K π (m, σ) = exp{m + }Φ( ) √s 𝜕s 2 2 +

2

s − m + log K s 1 m + s − log K exp{m + } exp{− ( )} √ √s 2 2 2 2πs√s 2

K(m − log K) 1 m − log K exp{− ( )} √s 2 2√2πs√s 1 s m + s − log K = exp{m + }Φ( ) √s 2 2 +

2

+

1 m − log K K exp{− ( ) }. √ √s 2 2 2πs

(1.47)

From equalities (1.47) and (1.46) it follows that both derivatives are positive, and therefore the option price increases in m and in σ 2 .

1.4.5 “Delta” as an example of Greek functionals Now we consider the fair price of the call option as the function of the current asset price. So, we introduce a different notation. Let V(X, t) = ert 𝔼∗ {(X − Ke−rT ) | ℱt } +

40 | 1 Financial markets. From discrete to continuous time be the fair price at time t of a European call option with strike price K and exercise time T given the current price X of the discounted risky asset represented by the geometric Brownian motion X(t) = exp{σW(t) −

σ2 t}. 2

The Black–Scholes partial differential equation in terms of discounted asset prices is written as follows ([58]): 𝜕V 1 2 2 𝜕2 V 𝜕V + σ X + rX − rV = 0, 2 𝜕t 2 𝜕X 𝜕X

t ∈ [0, T].

The so-called Greek functionals, or just Greeks, correspond to different partial derivatives of the function V. In particular, let Δ = 𝜕V . We consider this representative 𝜕X of the Greek functionals and call it “Delta.” The “Delta” characterizes the change of the price of a separate option or a portfolio of options with the price of the asset S. The “Delta” describes the correlation between the changes of option prices and changes of the prices of the corresponding assets. The Black–Scholes formula for the option price for any intermediate moment of time can be rewritten as V(X(t), t) = X(t)Φ(d+ ) − Ke−rT Φ(d− ), which implies the known expression for Δ in terms of a fixed X(t) = x, namely Δ(x, T − t) =

𝜕V = Φ(d+ (x, T − t)), 𝜕x

where Φ(x) is the standard normal distribution function, d± (x, T − t) =

log Kx + (r ± 21 σ 2 )(T − t) σ √T − t

.

Thus, we note that starting with a variable value of the asset price, we again come to a fixed value x.

1.5 Weak convergence of Greek symbol “Delta” for prices of European options: from discrete time to continuous In this section we consider the conditions of convergence of the pre-limit “Delta” having the form Δkn and introduced in Lemma 1.24 for the symmetric Cox–Ross–Rubinstein pre-limit model, to Δ(x, T −t) for the Black–Scholes model, presenting all intermediate calculations.

1.5 Convergence of Greek symbols |

41

1.5.1 Pre-limit “Delta” and the method of the common probability space Conditions for the weak convergence of the measures generated by the binomial model as well as by more general pre-limit models to the measure generated by the geometric Brownian motion immediately entail the question of the convergence of some known functionals, in particular of Greek symbols. We shall consider this question only for

the symmetric Cox–Ross–Rubinstein pre-limit model with an = e−σ√δn − 1, bn = eσ√δn − 1, and rn = rδn , where δ = δn = Tn . Also, we shall consider only the Greek symbol “Delta.” The problem is to introduce an analogue of the “Delta” for the Cox–Ross– Rubinstein model of a financial market and to study its convergence to Δ(x, T − t). In view of Definition 1.23, one can expect that the delta-hedge can be viewed as the desired analogue of the “Delta.” The fraction on the right-hand side of (1.18) is a certain discrete analogue of the capital function and thus there is a reason to expect that one obtains the “Delta” by passing to the limit provided the capital function converges to a limit value. Since the delta-hedge and “Delta” of an option are defined for markets kn T with different structures of times, denote by ktn the integer number such that tn ≤ t < (ktn +1)T . n

Here 0 ≤ ktn ≤ n − 1. Hence, ktn = ⌊ nt ⌋, the biggest integer not exceeding T Consider the following sequence of stochastic processes:

nt . T

kn

Xn (t) = Xnt . In other words, Xn are defined by formula (1.28). Also, let the limit process X(t) = S(0) exp{σW(t) − 21 σ 2 t} be a geometric Brownian motion. As before, all the processes can be defined on different probability spaces (Ω(n) , ℱ (n) , ℙ(n) ). According to Theorem 1.41 and Remark 1.44, weak convergence of the measures corresponding to the stochastic processes Xn and X on the interval [0, T] holds: ℙ(n,∗) , ℙ∗

(Xn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, T]),

n → ∞.

Now we shall apply Theorem A.108 and consider the common probability space for ̃n , X, ̃ n ≥ 1, with the same finite-dimensional distributions as the all the processes X ̃n , original ones Xn , X, n ≥ 1. For technical simplicity, let the subsequence again be X such that ℙ ̃n (t) 󳨀→ ̃ X X(t),

n → ∞, t ∈ [0, T]. In what follows we shall omit symbol ̃ as this will not cause confusion. So, in what follows we assume that Xn (t) 󳨀→ X(t), t ∈ [0, T]. ℙ

1.5.2 Some preliminary results We start with a result that allows one to find an alternative representation for Δkn being more convenient than (1.18). As we just mentioned, we can assume that all price

42 | 1 Financial markets. From discrete to continuous time processes are defined on a common probability space and for any fixed t converge in probability; consequently, again for any fixed t, they contain a subsequence converging with probability 1. This allows one to drop the random variable Xnk−1 in the definition of Δkn (Xnk−1 ) and to study the convergence of Δkn (xn ) under the assumption that we simply have the real-valued strictly positive sequence xn that converges to some fixed x > 0. So, we start to transform Δkn (x) at every fixed point x > 0. Theorem 1.50. For every x > 0 and any t ∈ (0, T) there exists a number n = n(x) such that one can find integer numbers ma = ma (x) and mb = mb (x) with 0 ≤ ma (x) ≤ mb (x) ≤ n − ktn and 0 < qn1 < 1 for which the delta-hedge for a European call option admits the following representation: kn

kn

Δnt (x) = Birt (n − ktn , ma (x), qn1 ) + Mnt (x, qn∗ )1ma (x)=m ̸ b (x) ,

(1.48)

r −a where qn1 = b̂ n qn∗ = b̂ n bn −an , Birt (L, l, p) stands for the right-tailed binomial distribution, n n i. e., it is the probability that at least l successes occur in L Bernoulli trials with parameter p, and

n − k mb )q (1 − q)n−mb −k . mb

b −k ̃n−k ) ( −K Mnk (x, q) = (x b̂ nmb +1 â n−m n

+

(1.49)

Proof. For technical simplicity, which is explained by the fact that we start counting with an increase in the degree of b̂ n , let us analyze the second representation from (1.19). It is easy to see from (1.19) that the non-zero terms denoted by vnk (xâ n ) and vnk (xb̂ n ) in the sums involved in formula (1.18) are written successfully and occur starting with some integer numbers ma (x) and mb (x), respectively. We are going to determine these numbers. First, note that the greatest term corresponding to the index i = n − ktn in the sum vnk (y) from (1.19) is of the order x exp{σ √nT(1 −

t t )} exp{−rT(1 − )} − K exp{−rT} → ∞ T T

as n → ∞.

Thus there exists a number n(x) such that the set of non-zero terms in both sums of the form (1.19) participating in (1.18) is non-empty if n ≥ n(x). Furthermore, a term is involved in the first sum if ̃n−k ) > 0. (x b̂ nmb +1 â nn−k−mb − K +

We rewrite this condition in the (equivalent) form b ̃n−k xb̂ nmb +1 â n−k−m >K n

and transform it for k = ktn : ̃n−kn , (mb + 1) log b̂ n + (n − ktn − mb ) log â n + log x > log K t

1.5 Convergence of Greek symbols |

43

or ̃n−kn − (n − k n ) log â n − log x − log b̂ n . mb log(b̂ n /â n ) > log K t t Note that ̃n−kn = log K − (n − k n ) log(1 + log K t t log â n = −

σ √T rT − log(1 + ), √n n

σ √T rT log b̂ n = − log(1 + ), √n n ̂b √ 2σ T log n = , √n â n

rT ), n and

whence mb >

log(K/x) 2σ √ Tn

log(1 + rT ) 1 n + (n − ktn − 1) + . 2 2σ √ Tn

Therefore one can put mb = 0 ∨ (⌊

log(K/x) 2σ √ Tn

log(1 + rT ) 1 n + (n − ktn − 1) + ⌋ + 1) 2 2σ √ Tn

(1.50)

for the first sum, where, as usual, ⌊z⌋ is the biggest integer not exceeding z, and similarly ma = 0 ∨ (⌊

log(K/x) 2σ √ Tn

log(1 + rT ) 1 n + (n − ktn + 1) + ⌋ + 1) 2 2σ √ Tn

(1.51)

for the second sum. It is obvious that for n ≥ n(x), only one of the following two cases may occur: either kn ma > mb or ma = mb = 0. In any case, Δnt (x) contains non-zero terms of both sums for i ranging from ma to n − ktn . Consider the case ma > mb . The terms of the first sum are non-zero for i ranging from mb to ma . It turns out that there exists only one value of i with this property and hence it equals mb . Indeed, ma − mb = ⌊

log( Kx ) 2σ √ Tn

log( Kx ) 1 1 + (n − ktn + 1)⌋ − ⌊ + (n − ktn − 1)⌋ = 1. T 2 2 2σ √ n

Thus kn Δnt (x)

n−k n

=

n

n

n−i−ktn ̂ i ̂ bn (bn

t )(q∗ )i (p∗ )n−i−kt x â ∑i=mt (n−k n n n i a

ktn

+ Mn 1ma =m ̸ b

x(b̂ n − â n )

− â n )

44 | 1 Financial markets. From discrete to continuous time n−ktn

n − ktn i n−i−ktn n−i−ktn ̂ i kn )(qn∗ ) (p∗n ) â n bn + Mnt 1ma =m ̸ b i

= ∑ ( i=ma n−ktn

n − ktn i n−k n −i kn )(b̂ n qn∗ ) (â n (p∗n )) t + Mnt 1ma =m ̸ b i

= ∑ ( i=ma n−ktn

i n−k n −i n − ktn kn )(qn1 ) (p1n ) t + Mnt 1ma =m ̸ b i

= ∑ ( i=ma

kn

= Birt (n − ktn , ma (x), qn1 ) + Mnt 1ma =m ̸ b,

(1.52)

where qn1 = b̂ n qn∗ and p1n = â n p∗n . This is indeed a probability distribution, since 1 + bn rn − an 1 + an bn − rn ⋅ + ⋅ b̂ n qn∗ + â n p∗n = 1 + rn bn − an 1 + rn bn − an r + r b − an − an bn + bn + an bn − rn − an rn = n n n = 1. (1 + rn )(bn − an ) The following analogue of the Esseen inequality is needed for the proof of the main convergence result (see [136, p. 111]). Theorem 1.51. Let {Ynj , 1 ≤ j ≤ n, n ≥ 1} be independent random variables. Assume 2

3

that 𝔼Ynj = 0, 𝔼(Ynj ) = (σnj )2 > 0, and 𝔼|Ynj | < ∞, 1 ≤ j ≤ n. Put Bn = ∑nj=1 (σnj )2 , 3

Fn (x) = ℙ(B−1/2 ∑nj=1 Ynj < x), and Ln = B−3/2 ∑nj=1 𝔼|Ynj | . n n Then

󵄨 󵄨 sup 󵄨󵄨󵄨Fn (x) − Φ(x)󵄨󵄨󵄨 ≤ cLn , x

(1.53)

where c is a positive constant not depending on n and Φ(x) is the standard normal distribution function.

1.5.3 Convergence of Δkn to Δ(x, T − t) Theorem 1.50 allows us to consider separately the convergence of components of the discrete Greek functional Δkn . Throughout below we assume that 0 < t < T, since the results for t = 0 and t = T are analogous and proofs are simpler. Let us recall the local de Moivre–Laplace theorem (see [32, p. 67]), which will be used below. Theorem 1.52 (Local de Moivre–Laplace theorem). Let μn be the number of successes that occur in n Bernoulli trials. If n → ∞, the probability p of success is a constant, and is bounded uniformly with respect to m and n (−∞ < a ≤ xm ≤ b < +∞), xm = m−np √npq

1.5 Convergence of Greek symbols | 45

then ℙ(μn = m) =

2 xm 1 e− 2 (1 + αn (m)), √2πnp(1 − p)

where |αn | < c/√n for xm ∈ [a, b], c > 0 being a constant. First we prove the following result. kn

Theorem 1.53. The convergence Mnt (x, qn∗ )1ma (x)=m ̸ b (x) → 0 as n → ∞ is uniform with respect to x > 0 belonging to an arbitrary bounded set. Proof. Note that the inequality ma > mb implies that ma > 0 and thus the maximum in the definition of ma can be dropped by the reasoning used in the proof of Theorem 1.50. Let us analyze the asymptotic behavior of ma and mb . Note that ktn < nt ; therefore T in our case ma >

log( Kx ) 2σ √ Tn

+

log(1 + rT ) t 1 1 n + n(1 − ) + →∞ 2 2 T 2σ √ Tn

as n → ∞.

(1.54)

Then mb = ma − 1 → ∞ as n → ∞. Moreover, by the same reasons, n − ktn − mb ≈ −

log(K/x) 2σ √ Tn

1 t + n(1 − ) → ∞ 2 T

as n → ∞, and mb ≈ ma ≈

log(K/x) 2σ √ Tn

1 t + n(1 − ). 2 T

To apply the local de Moivre–Laplace theorem we only need to prove that the sequence α(mb , n) :=

mb − (n − ktn )qn∗ √(n − ktn )p∗n qn∗

is uniformly bounded with respect to n. However, p∗n ≈ qn∗ ≈ 1/2, according to Remark 1.44, and therefore the numerator log( Kx ) 1 t , mb − (n − ktn )p∗n ≈ mb − (1 − )n ≈ 2 T 2σ √ Tn while the denominator 1 t √(n − ktn )p∗n qn∗ ≈ √n√1 − , 2 T

46 | 1 Financial markets. From discrete to continuous time and we get α(mb , n) ≈

log( Kx )

σ √T(1 − Tt )

as n → ∞.

,

Thus the sequence α(mb , n), n ≥ 1, is bounded for every x > 0. Therefore all the conditions for the local de Moivre–Laplace theorem are valid and hence, for sufficiently large n, 2 n − ktn ∗ mb ∗ n−mb −ktn 1 1 )qn pn = )) e−xn,t /2 (1 + O( √n mb √2πnp∗n qn∗

(

2 2 ≈ √ n−1/2 e−xn,t /2 (1 + O(n−1/2 )), π

where |xn,t | ≤ Ct for any t ∈ (0, T). Therefore for sufficiently large n kn Mnt (x, qn∗ )

2

−xn,t 1 2 n−m −k n ̃n ) e + o( ), = √ n−1/2 (x b̂ nmb +1 â n b t − K √n π n √n − k

(1.55)

t

where |xn,t | ≤ Ct , n ≥ 1. Now the only thing is to rewrite and bound for all sufficiently large n the value n mb +1 n−mb −kt ̂ ̃n from (1.55): xbn â n −K σ √ Tn

mb +1

e n−m −k 0 ≤ xb̂ nmb +1 â n b t − K ≤ x( ) 1 + rn n

=x ≈ =

n−mb −ktn

exp{σ √ Tn (2mb − n + ktn + 1)} n

(1 + rn )n−kt +1

x

e

√T

e−σ n ( ) 1 + rn

rT(1− Tt

x

er(T−t)

)

K

t t T log( x ) exp{σ √ ( + n(1 − ) − n(1 − ))} T n σ√ T T

K ⋅ = Ke−rT+rt , x

n

as n → ∞.

∗ Thus Mk(n) n (x, qn ) → 0 as n → ∞ uniformly with respect to any bounded set of x. t

Now we are ready to state and prove the main result concerning the convergence of the Greek symbol “Delta.” Theorem 1.54. Consider a pre-limit symmetric binomial model with the limit being the Black–Scholes model. Then the delta-hedge for a European call option in discrete time weakly converges to the delta functional of a European call option in continuous time as the number of periods in the discrete time model tends to infinity, that is, for any t ∈ [0, T] ℙ(n,∗) , ℙ(n) , d

Δkn (Xn (ktn )) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ Δ(X(t), T − t),

n → ∞.

1.5 Convergence of Greek symbols |

47

Proof. It is clear from what we discussed above that one only needs to check that Δkn (xn ) → Δ(x, T − t),

n→∞

under the condition that xn → x > 0. We apply representation (1.48) of Theorem 1.50. By Theorem 1.53, kn

Mnt (x, qn∗ )1ma (x)=m ̸ b (x) → 0 as n → ∞ in this representation. Now the convergence of Δkn (xn ) to Δ(x, T −t) will follow from the convergence of Birt (n−ktn , ma (xn ), qn1 ) to Δ(x, T−t). Here Birt (n−ktn , ma (xn ), qn1 ) is defined in (1.52), where xn substitutes for x. For the sake of a simpler notation we write ma instead of ma (xn ). Consider the probability that at least ma successes occur in a sequence of n Bernoulli trials with parameters n − k and qn1 , that is, n−k i n−i−k n−k Birt (n − k, ma , qn1 ) = ∑ ( )(qn1 ) (p1n ) . i i=m a

In other words, n−k

i n−i−k n−k )(qn1 ) (p1n ) = 1 − ℙ(νn < ma ), i

∑ (

i=ma

where n−k

νn = ∑ Xin , i=1

0

Xin = {

1

with probability p1n , with probability qn1 ,

qn1 + p1n = 1.

Thus it remains to establish that ℙ(νn < ma ) → 1 − Φ(d+ (x, T − t)) = Φ(

log(K/x) − (r + 21 σ 2 )(T − t) σ √T − t

)

as n → ∞. In order to do this, the next step of the proof is to apply Theorem 1.51 to the sum under consideration. Since 𝔼Xjn = qn1 , we pass to the centered terms, that is, put Yjn = Xjn − qn1 , j = 1, n. Then −qn1

with probability p1n ,

Yjn = {

p1n

with probability qn1 ,

𝔼Yjn = 0,

j = 1, n.

Now we evaluate the second and third moments: 2

2

2

2

𝔼(Yjn ) = p1n (qn1 ) + (p1n ) qn1 = (1 − p1n )(p1n (1 − p1n ) + (p1n ) ) = qn1 p1n ,

48 | 1 Financial markets. From discrete to continuous time 3 3 󵄨 󵄨3 𝔼󵄨󵄨󵄨Yjn 󵄨󵄨󵄨 = (qn1 ) p1n + (p1n ) (qn1 ) 2

3

3

2

4

3

= p1n − 3(p1n ) + 3(p1n ) − (p1n ) + (p1n ) − (p1n )

4

= −p1n (2(p1n ) − 4(p1n ) + 3p1n − 1). 3

It is clear that 𝔼|Yjn | < ∞ and n−k

Bn = ∑ p1n qn1 = (n − k)p1n qn1 . j=1

Therefore all the assumptions of Theorem 1.51 hold and inequality (1.53) can be applied to the sum under consideration. Consider Ln and take into account that k = ktn and the fact that p1n → 21 and qn1 → 21 as n → ∞: Ln =

B−3/2 n

3

3

=−

2

(n − k)p1n (2(p1n ) − 4(p1n ) + 3p1n − 1) √((n − k)p1n qn1 )3 2

=

2

1 1 1 1 ∑n−k j=1 pn (2(pn ) − 4(pn ) + 3pn − 1) 󵄨 󵄨3 ∑ 𝔼󵄨󵄨󵄨Yjn 󵄨󵄨󵄨 = − √((n − k)p1n qn1 )3 j=1

n−k

(1 − p1n )(2(p1n ) − 2p1n + 1) √(n − k)p1n (qn1 )3

=

3

=−

2

2(p1n ) − 4(p1n ) + 3p1n − 1 √(n − k)p1n (qn1 )3

2

2(p1n ) − 2p1n + 1 √(n − k)p1n qn1

→ 0,

that is, Ln → 0 as n → ∞, whence we conclude that 󵄨 󵄨 sup󵄨󵄨󵄨Fn (y) − Φ(y)󵄨󵄨󵄨 → 0, y∈ℝ

n → ∞.

(1.56)

Recall that n−k

n 1 ∑n−k j=1 Xj − (n − k)qn

j=1

√(n − k)p1n qn1

Fn (y) = ℙ(B−1/2 ∑ Yjn < y) = ℙ( n

< y)

by the definition of Yjn . It is obvious that Fn (αma ) = ℙ(νn < ma ), where αma :=

ma − (n − ktn )qn1 √(n − ktn )p1n qn1

.

Finally, it remains to prove that Fn (αma ) → Φ(

log(K/x) − (r + 21 σ 2 )(T − t) σ √T − t

).

1.5 Convergence of Greek symbols |

49

In order to prove the latter relation, we transform αma to a more convenient form. First we transform the probabilities more precisely: qn1 =

eσ

√ Tn

1+

T σ2 1 1 qn∗ = (1 + (r + )√ ) + o( ), √n 2 2 n

rT n

whence a similar asymptotic expansion for p1n follows. Therefore, αma =

ma (xn ) − (n − ktn )qn1 √(n −

ktn )p1n qn1

=

log(K/xn ) − (r +

σ2 )(T 2

σ √T − t

− t)

+ o(

1 ). √n

(1.57)

Now we put d(x) =

log(K/x) − (r + 21 σ 2 )(T − t) σ √T − t

and estimate 󵄨󵄨 󵄨󵄨 1 󵄨󵄨 󵄨 )) − Φ(d(x))󵄨󵄨󵄨 󵄨󵄨Fn (d(xn ) + o( 󵄨󵄨 󵄨󵄨 √n 󵄨󵄨 󵄨󵄨 1 󵄨 󵄨 󵄨 󵄨 ≤ sup 󵄨󵄨󵄨Fn (y) − Φ(y)󵄨󵄨󵄨 + 󵄨󵄨󵄨Φ(d(x)) − Φ(d(xn ) + o( ))󵄨󵄨󵄨 → 0 󵄨󵄨 √n 󵄨󵄨 y∈ℝ as n → ∞, which completes the proof of the theorem. Remark 1.55. We can get the same result considering the following modification of the symmetric binomial model: an = e(r−

σ2 2

)δ−σ √δ

− 1,

bn = e(r−

σ2 2

)δ+σ √δ

− 1,

where δ = Tn . In this case, to simplify the calculations, we can consider the discounting factor e−rδ on each period and a common discounting factor of the form e−r(T−t) instead t of (1 + rδ)−(n−kn ) . We put ̂n = a

σ2 an + 1 √ = e− 2 δ−σ δ , rδ e

σ2 ̂ = bn + 1 = e− 2 δ+σ√δ b n rδ e

and consider n−k

vnk (y) = e−r(T−t) ∑ ((an + 1)i (bn + 1)n−i−k y − K) i=0

n−k i n−i−k ×( )(p∗n ) (1 − p∗n ) , i The final result will be the same, but a little faster.

t∈(

+

kT (k + 1)T , ]. n n

(1.58)

50 | 1 Financial markets. From discrete to continuous time

1.6 General schemes of diffusion approximation In Section 1.4.3 we considered the functional CLT for the multiplicative scheme of a financial market. This theorem works in the case where the limit process is simply a geometric Brownian motion. The present section focuses on the more involved case, where we have a diffusion process in the limit. In this connection, we introduce a diffusion approximation for the recurrent schemes of financial markets. Our goal is to adapt the theorems of the diffusion approximation from [110] and [83] to the multiplicative financial models which, as explained in detail earlier, are natural for the pre-limit market. To be more precise, in Section 1.6.1 we present the general results from [110] concerning the conditions of the weak convergence for the discrete-time processes to the general diffusion process, i. e., we formulate the general functional limit theorem for the diffusion approximation. In Section 1.6.2 we adapt these results to the additive and, even more importantly, to multiplicative schemes. 1.6.1 General functional limit theorem for diffusion approximation Recall some notions from the classical semimartingale theory (see, for example, [110]). Let the interval 𝕋 = [0, T] and Ωℱ = (Ω, ℱ , 𝔽 = {ℱt , t ∈ 𝕋}, ℙ) be a complete filtered probability space whose filtration 𝔽 satisfies the standard assumptions of the rightcontinuity and completeness from Definition A.16; see Section 1.3.1. As before, let D(𝕋) stand for the set of all real-valued càdlàg functions on 𝕋 (see Section A.3). In what follows we consider only càdlàg processes, i. e., processes having with probability 1 càdlàg trajectories. Introductions to the concepts of martingale and local martingale are contained in Section A.3.1. Definition 1.56. A real-valued process X = {X(t), ℱt , t ∈ 𝕋} considered on the probability space Ωℱ is called a semimartingale if it admits the decomposition of the form X(t) = X(0) + M(t) + A(t),

(1.59)

where M is a local martingale with M(0) = 0 and A is a process of locally bounded variation (i. e., of bounded variation on any interval). A semimartingale {X(t), ℱt , t ∈ 𝕋} is called a special semimartingale if it admits the decomposition (1.59), in which process A is a process of locally integrable variation, that is, of variation integrable on any interval. For more detail concerning semimartingales see Definition A.41. Remark 1.57. It is established in Chapter 2 of [110] that X is a special semimartingale if and only if it admits decomposition (1.59) with a predictable process A of locally bounded variation and such predictable process is unique. We shall apply both equivalent forms of the definition of a special semimartingale.

1.6 General schemes of diffusion approximation |

51

Denote by ⟨M⟩ = {⟨M⟩(t), t ∈ 𝕋} the quadratic characteristic of the locally squareintegrable martingale {M(t), ℱt , t ∈ 𝕋}. It is a predictable non-decreasing process for which the process M 2 (t) − ⟨M⟩(t) is a local martingale. Also, denote the jump of any càdlàg process X as follows: ΔX(t) = X(t) − X(t−). Theorem 1.58 ([110]). If X = {X(t), ℱt , t ∈ 𝕋} is such a semimartingale that for some a > 0 and all t ∈ 𝕋 we have |ΔX(t)| ≤ a, then X is a special semimartingale. Let X = {X(t), ℱt , t ∈ 𝕋} be a semimartingale. For each a > 0 we denote the sum of its “big” jumps to moment t as t

a

󵄨 󵄨 X (t) = ∑ ΔX(s)1(󵄨󵄨󵄨ΔX(s)󵄨󵄨󵄨 > a) = ∫ ∫ xdμ(x), 0a

where μ is the measure of jumps of the process X. Evidently, this sum is finite a. s. for any càdlàg process X. Furthermore, denote the difference Y a (t) = X(t) − X a (t),

t ∈ 𝕋.

The jumps of the process Y a are bounded, |ΔY a (t)| ≤ a, so Y a is a special semimartingale according to Theorem 1.58. It means that there exist a local martingale M a and such a predictable process Ba (X) of locally bounded variation that Y a (t) = X(0) + Ba (X)(t) + M a (t),

t ∈ 𝕋,

Ba (0) = M a (0) = 0.

Thus X(t) = X(0) + Ba (X)(t) + M a (t) + X a (t),

t ∈ 𝕋.

In turn, a local martingale {M a (t), ℱt , t ∈ 𝕋} admits a decomposition M a (t) = M ac (t) + M ad (t) into the continuous and purely discontinuous parts, where the continuous component M ac does not depend on a and the purely discontinuous component M ad can be presented as ad

t

M (t) = ∫ ∫ xd(μ − ν)(x), 0 |x|≤a

where ν is the compensator (a dual predictable projection) of the measure μ. Let us denote M c = M ac and C(X)(t) = ⟨M c ⟩(t). The processes (Ba (X), C(X), ν) compose the triplet of predictable characteristics for the semimartingale X.

52 | 1 Financial markets. From discrete to continuous time Now we introduce the general pre-limit and limit processes participating in the diffusion approximation. Concerning the limit process X, let X = {X(t), ℱt , t ∈ 𝕋} be a continuous semimartingale. In this case it is obvious that ν ≡ 0 and Ba (X)(t) coincide for any a > 0. We denote as B(X)(t) the common value of all Ba (X)(t). Suppose that t

B(X)(t) = ∫ b(s, X)ds

(1.60)

0

and t

C(X)(t) = ∫ c2 (s, X)ds

(1.61)

0

for some predictable measurable functions b(t, x(⋅)), c(t, x(⋅)) : 𝕋 × D(𝕋) → ℝ. Moreover, we suppose that c(t, x(⋅)) > 0. In this case we can apply the generalized Lévy theorem (see, e. g., [69]) concluding that there exists a Wiener process W = (W(t), ℱt , t ∈ 𝕋) adapted to the filtration 𝔽 = {ℱt , t ∈ 𝕋} such that M c admits the t representation M c (t) = ∫0 c(s, X)dW(s). Therefore, X is the solution of the stochastic differential equation t

t

X(t) = X(0) + ∫ b(s, X)ds + ∫ c(s, X)dW(s). 0

(1.62)

0

Assume that the coefficients of equation (1.62) satisfy the following condition: there exists a function L : 𝕋 → ℝ+ such that for any t ∈ 𝕋 and any X ∈ D(𝕋) 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨b(t, X)󵄨󵄨󵄨 ≤ L(t)(1 + sup󵄨󵄨󵄨X(s)󵄨󵄨󵄨),

(1.63)

c2 (t, X) ≤ L(t)(1 + sup X 2 (s)).

(1.64)

s≤t

s≤t

Additionally, we assume that T

∫ L(t) dt < ∞.

(1.65)

0

Now, in order to construct the sequence of pre-limit stochastic objects, suppose that we have the sequence of the filtered probability spaces (Ω(n) , ℱ (n) , 𝔽(n) = {ℱt(n) , t ∈ 𝕋}, ℙ(n) ),

n ≥ 1,

and a sequence of semimartingales Xn = {Xn (t), ℱt(n) , t ∈ 𝕋} on the corresponding probability spaces, with trajectories in D(𝕋) a. s. and with the triplets of the predictable

1.6 General schemes of diffusion approximation |

53

characteristics (Bn,a , C n , νn ). Suppose that for any ε > 0 and a ∈ (0, 1] the following conditions hold: 󵄨 󵄨 lim ℙ(n) (sup󵄨󵄨󵄨ΔXn (t)󵄨󵄨󵄨 ≥ ε) = 0,

(1.66)

t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim ℙ(n) (sup󵄨󵄨󵄨Bn,a (t) − ∫ b(s, Xn )ds󵄨󵄨󵄨 ≥ ε) = 0, n 󵄨󵄨 t∈𝕋 󵄨󵄨󵄨 󵄨 0

(1.67)

n

t∈𝕋

t 󵄨󵄨 󵄨󵄨 󵄨󵄨 n,a 󵄨󵄨 󵄨 lim ℙ (sup󵄨󵄨⟨M ⟩(t) − ∫ c2 (s, Xn )ds󵄨󵄨󵄨 ≥ ε) = 0. n 󵄨 󵄨󵄨 t∈𝕋 󵄨󵄨 󵄨 0 (n)

(1.68)

Since this will not cause misunderstanding, we denote as ℙ and ℙ(n) , n ≥ 1, the measures that correspond to the processes X and Xn , n ≥ 1, respectively. ℙ(n) , ℙ, d

Theorem 1.59 ([110]). Let the conditions (1.63)–(1.68) hold. If, in addition, Xn (0) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ X(0) and functions b, c determine uniquely the measure ℙ that corresponds to the limit process X, then we have the weak convergence of probability measures ℙ(n) , ℙ

(Xn (t), t ∈ 𝕋) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ 𝕋). 1.6.2 Functional limit theorem for diffusion approximation of the sums and the products of random variables 1.6.2.1 The diffusion approximation for an additive scheme To adapt the well-known functional limit theorems towards the financial models, suppose now that we consider the semimartingale X = {X(t), ℱt , t ∈ 𝕋} from the previous section but in a simplified situation. More precisely, we suppose that the measurable functions b and c have forms b = b(t, x) and c = c(t, x) : 𝕋 × ℝ → ℝ. We assume also that the coefficients b and c satisfy the conditions for existence and uniqueness of the weak solution of the stochastic differential equation dX(t) = b(t, X(t))dt + c(t, X(t))dW(t),

t ∈ 𝕋,

X(0) = x0 .

(1.69)

Let c(t, x) ≥ 0, t ∈ 𝕋, x0 ∈ ℝ, and let the limit process X be a solution of this equation with some 𝔽-Wiener process W. Remark 1.60. The conditions for existence and uniqueness of the weak solution in the case of the homogeneous coefficients b and c were formulated in [96] and the most general conditions were obtained in [52] and [53–55]. For the inhomogeneous case, we just refer to the classical book [161] containing results on the existence and uniqueness of the solution of a martingale problem. A short discussion of these topics is contained in Section A.3.11.

54 | 1 Financial markets. From discrete to continuous time Now we simplify the pre-limit processes introducing the stepwise functions. Let n ≥ 1. Consider the sequence of the probability spaces (Ω(n) , ℱ (n) , 𝔽n = {ℱt(n) , t ∈ 𝕋}, ℙ(n) ) with a respective filtration and the sequence of stepwise semimartingales Xn = {Xn (t), ℱt(n) , t ∈ 𝕋} defined on a respective probability space, adapted to a respective filtration and admitting a representation Xn (t) = Xn (

kT ) n

kT (k + 1)T ≤t< . n n

for

(1.70)

So, the trajectories of the pre-limit processes Xn have the jumps at the points kT/n, k = 0, . . . , n, and are constant in the interior intervals. Denote ℱnk = σ{Xn (t), t ≤ kT } n and introduce the notation for the increments Qkn := ΔXn (

kT kT (k − 1)T ) := Xn ( ) − Xn ( ), n n n

k = 1, . . . , n.

Then the random variables Qkn are ℱnk -measurable, k = 1, . . . , n, and in what follows we identify ℱt(n) with ℱnk for kT ≤ t < (k+1)T . Let, as before, the notation ⌊x⌋ stand for the n n integer part of a number x. It follows from the definition of the triplet of predictable characteristics that in this case Bn,a (t) =

∑

1≤k≤⌊ nt ⌋ T

𝔼n (Qkn 1|Qk |≤a | ℱnk−1 ). n

Since Xn is a jump process, we have C n = 0. Hence ⟨M

n,a

t

2

⟩(t) = ∫ ∫ x2 dνn − ∑ ( ∫ xνn ({s}, dx)) 0 0, a ∈ (0, 1], 󵄨 󵄨 lim ℙ(n) ( sup 󵄨󵄨󵄨Qkn 󵄨󵄨󵄨 ≥ ε) = 0, n

(1.73)

t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim ℙ(n) (sup󵄨󵄨󵄨 ∑ 𝔼n (Qkn 1|Qk |≤a | ℱnk−1 ) − ∫ b(s, Xn (s))ds󵄨󵄨󵄨 ≥ ε) = 0, n n 󵄨󵄨 t∈𝕋 󵄨󵄨󵄨 󵄨 ⌋ 1≤k≤⌊ nt 0 T

(1.74)

t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim ℙ(n) (sup󵄨󵄨󵄨 ∑ Varn (Qkn 1|Qk |≤a | ℱnk−1 ) − ∫c2 (s, Xn (s))ds󵄨󵄨󵄨 ≥ ε) = 0. n n 󵄨󵄨 󵄨 t∈𝕋 󵄨󵄨 󵄨 ⌋ 1≤k≤⌊ nt 0 T

(1.75)

1≤k≤n

and

Finally, let the functions b, c uniquely determine the measure ℙ. Then we have weak convergence of the probability measures ℙ(n) , ℙ

(Xn (t), t ∈ 𝕋) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ 𝕋). 1.6.2.2 A discrete approximation scheme for the multiplicative processes in the financial market Consider the sequence of discrete-time financial markets consisting of two assets, one risk-free (bond) and one risky (stock). We suppose that the risk-free asset admits the representation Bn (t) = B0n

∏ (1 + rnk ),

⌋ 1≤k≤⌊ nt T

where {rnk > −1, n ≥ 1, 1 ≤ k ≤ n} are real numbers. Let the risky asset admit the representation Sn (t) = Sn0

∏ (1 + Rkn ),

1≤k≤⌊ nt ⌋ T

where {Rkn > −1, 1 ≤ k ≤ n} are random variables on the probability space (Ω(n) , ℱ (n) , ℙ(n) ), n ≥ 1. We introduce the sequence {𝔽(n) , n ≥ 1} of filtrations consisting of σ-fields ℱn0 = {0, Ω}, ℱnk = σ{Rin , 1 ≤ i ≤ k}. We are in a position to present the conditions of weak convergence of this multiplicative scheme to the limit model of the form t

B(t) = B(0) exp{∫ r(s)ds}, 0

t

1 S(t) = exp{X(t) − ∫ c2 (s, X(s))ds}, 2 0

56 | 1 Financial markets. From discrete to continuous time where the process X is the unique weak solution of equation (1.69). Let us introduce the processes Xn (t) =

∑

1≤k≤⌊ nt ⌋ T

Rkn

and Yn (t) =

1 2 ∑ (Rkn − (Rkn ) ). 2 nt

1≤k≤⌊ T ⌋

We denote as ℚ and ℚ(n) , n ≥ 1, the measures that correspond to the processes S and Sn , n ≥ 1, respectively. Theorem 1.62. 1) Let condition (A6) hold: (A6) (i) B0n → B(0) and sup0≤k≤n |rnk | → 0 , n → ∞; t

(ii) ∑1≤k≤⌊ nt ⌋ (rnk − 21 (rnk )2 ) → ∫0 r(s)ds , n → ∞; T

(iii) lim supn→∞ ∑1≤k≤n (rnk )2 < ∞. Then the pointwise convergence holds: Bn (t) → B(t), n → ∞.

2) Let conditions (A7) and (A8) hold: (A7)

(i) Sn0 → exp{X(0)} and sup1≤k≤n |Rkn | 󳨀 → 0, n → ∞; (ii) for any a ∈ (0, 1] ℙ

2

lim lim sup ℙ(n) ( ∑ 𝔼n ((Rkn ) 1|Rk |≤a | ℱnk−1 ) ≥ C) = 0;

C→∞

n→∞

1≤k≤n

n

(iii) for any a ∈ (0, 1] 󵄨 󵄨 lim lim sup ℙ(n) ( ∑ 󵄨󵄨󵄨𝔼n (Rkn 1|Rk |≤a | ℱnk−1 )󵄨󵄨󵄨 ≥ C) = 0; n n→∞

C→∞

1≤k≤n

(iv) for any ε > 0, a ∈ (0, 1] t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k k−1 󵄨 lim ℙ (sup󵄨󵄨 ∑ 𝔼n (Rn 1|Rk |≤a | ℱn ) − ∫ b(s, Xn (s))ds󵄨󵄨󵄨 ≥ ε) = 0; n n 󵄨󵄨 󵄨 t∈𝕋 󵄨󵄨 󵄨 ⌋ 1≤k≤⌊ nt 0 T (n)

(v) for any ε > 0, a ∈ (0, 1] t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 lim ℙ(n) (sup󵄨󵄨󵄨 ∑ 𝔼n ((Rkn ) 1|Rk |≤a | ℱnk−1 )−∫ c2 (s, Xn (s))ds󵄨󵄨󵄨 ≥ ε) = 0. n n 󵄨󵄨 t∈𝕋 󵄨󵄨󵄨 󵄨 ⌋ 1≤k≤⌊ nt 0 T

1.6 General schemes of diffusion approximation |

57

(A8) Functions b and c uniquely determine the measure ℚ. Then we have weak convergence of the probability measures ℚ(n) , ℚ

(Sn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐⇒ (S(t), t ∈ [0, T]). Proof. Regarding the convergence of Bn , let 0 < a < 1 be fixed. Due to condition (A6)(i) we can consider such n0 that for n ≥ n0 we have sup1≤k≤n |rnk | < a. For such n we present log(Bn (t)) as log(Bn (t)) = log(B0n ) + = log(B0n +

∑

⌋ 1≤k≤⌊ nt T

log(1 + rnk ))

1 2 ∑ (rnk − (rnk ) ) 2 nt

1≤k≤⌊ T ⌋

+ α(a, rnk , 0 ≤ k ≤ n)

2

∑ (rnk ) ,

1≤k≤⌊ nt ⌋ T

a n where |α(a, rnk , 0 ≤ k ≤ n)| does not exceed 3(1−a) 3 . Then the convergence of B follows immediately from conditions (A6)(i) and (ii). Consider the weak convergence of Sn . Due to condition (A7)(i), we can fix any 0 < a < 1 and it is enough to establish the corresponding convergence for

Sn,a (t) = Sn0

∏ (1 + Rkn,a ),

⌋ 1≤k≤⌊ nt T

where Rkn,a = Rkn 1|Rk |≤a . We have n

log Sn,a (t) = log Sn0 + = log Sn0 +

∑

1≤k≤⌊ nt ⌋ T

log(1 + Rkn,a )

1 2 ∑ (Rkn,a − (Rkn,a ) ) 2 nt

1≤k≤⌊ T ⌋

+ α(a, Rkn,a , 0 ≤ k ≤ n)

2

∑ (Rkn,a ) ,

1≤k≤⌊ nt ⌋ T

a where |α(a, Rkn,a , 0 ≤ k ≤ n)| does not exceed 3(1−a) 3 a. s. It follows from condition (A7)(ii) and the Lenglart–Rebolledo inequality (Theorem A.33, Lemma A.35, and Corollary A.37; see also [110], Chapter 1, p. 66) that for any a ∈ (0, 1] 2

lim lim sup ℙ(n) ( ∑ (Rkn,a ) ≥ C)

C→∞

n→∞

0≤k≤n

2

1

1

≤ lim lim sup(C − 2 + ℙ(n) ( ∑ 𝔼n ((Rkn ) 1|Rk |≤a | ℱnk−1 ) ≥ C 2 )) = 0. C→∞

n→∞

1≤k≤n

n

(1.76)

58 | 1 Financial markets. From discrete to continuous time So, we can fix an arbitrary δ > 0 and apply condition (A7)(ii) and (1.76) in order to find such C > 0 and n(δ, C) that for n ≥ n(δ, C) 2

ℙ(n) ( ∑ (Rkn,a ) ≥ C) < δ. 0≤k≤n

Therefore, with probability ℙ(n) , exceeding 1 − δ, 󵄨󵄨 󵄨 k 󵄨󵄨α(a, Rn,a , 0 ≤ k ≤ n)󵄨󵄨󵄨

2

∑ (Rkn,a )
0 2

lim ℙ(n) ( ∑ (𝔼n (Rkn 1|Rk |≤a | ℱnk−1 )) ≥ ε) = 0.

n→∞

1≤k≤n

n

Therefore, it follows from condition (A7)(v) that (1.75) holds, and we obtain that Xn weakly converges in measure to X. Now we can apply Theorem A.106 and deduce from the weak convergence above and condition (A7)(iii) that the couple of processes {Xn , [Xn ]} weakly converges in measure to {X, [X]}, where [⋅] means the quadratic variation and X is a weak solution to the stochastic differential equation (1.62). As t to [X], it equals ∫0 c2 (s, X(s))ds. Therefore, ∑1≤k≤[ n⋅ ] (Rkn )2 weakly converges in meaT

t

sure to ∫0 c2 (s, X(s))ds and we conclude that Yn weakly converges in measure to t

X − 21 ∫0 c2 (s, X(s))ds, whence finally the proof follows.

1.7 A recurrent scheme for the diffusion approximation when the limit process is a geometric Ornstein–Uhlenbeck process In this section we study a natural model for the stochastic interest rate as well as the starting point for stochastic volatility, namely, the Ornstein–Uhlenbeck process (Vasicek model). It is a unique solution of the Langevin equation of the general form dX(t) = (μ − νX(t))dt + σdW(t),

X(0) = x0 ∈ ℝ,

t ∈ 𝕋,

(1.77)

where μ, ν ∈ ℝ and σ > 0. The properties of the unique solution of the Langevin equation (1.77) make it a suitable tool for modeling interest rates, stochastic volatility, and

1.7 Geometric Ornstein–Uhlenbeck process | 59

some other objects in financial markets. The Ornstein–Uhlenbeck process is Markov and admits the explicit representation t

X(t) = x0 e−νt +

μ (1 − e−νt ) + σ ∫ e−θ(t−s) dW(s). ν 0

It is a Gaussian process with the following characteristics: 𝔼X(t) = x0 e−νt +

μ (1 − e−νt ), ν

Cov(X(t), X(s)) =

σ 2 −ν|t−s| (e − e−ν(t+s) ). 2ν

The Ornstein–Uhlenbeck process was used by Vasicek in [166] in deriving an equilibrium model of discount bond prices. This Gaussian process has been used extensively by other scientists in valuing bond options, futures, futures options, and other types of contingent claims. The examples are provided in [60, 68, 84, 90]. The motivation for considering the geometric Ornstein–Uhlenbeck process lies in its price recovery effect that is supplied by its mean reverting property. For example, the paper [162] contains one example of modeling a stochastic interest rate by the Vasicek model. However, there are some reasons in favor of the Vasicek model even if we consider the stock prices. One of the reasons is that in the standard Black–Scholes model the variance of T the total profit ∫0 dS(t) equals σ 2 T → ∞ as T → ∞ while in real markets it often tends S(t) to a finite value that is true for the Vasicek model. So, here we consider the geometric Ornstein–Uhlenbeck process as the limit model, and in Section 2.4.1 we shall consider the Ornstein–Uhlenbeck process as the model for stochastic volatility. Furthermore, we construct the recurrent scheme for the pre-limit market that is rather natural, in our opinion, since it is constructed based on the binomial scheme with the help of the scheme similar to the Euler approximation. For this we proceed with three consequent steps: first, we consider the Euler approximation scheme for the solution of a stochastic differential equation; second, we replace the increment of a Wiener process with binomial random summands; and third, we take into account the adjusting term that appears when we pass from the multiplicative financial schemes to the additive mathematical ones. To some extent, these ideas were realized in [139]; however, our approach is more explicit, direct, and general. Then we apply the results to the market for which the limit price process is modeled by the geometric Ornstein–Uhlenbeck process. The recurrent scheme for the diffusion approximation for the case when the limiting process is represented by the geometric Ornstein–Uhlenbeck process is constructed in Section 1.7, and in Section 1.7.2 we discuss the applicability of the geometric Ornstein–Uhlenbeck process in the sense that the corresponding financial model is arbitrage-free and complete. The conditions of convergence for the option prices including the joint convergence of the stock prices and the Radon–Nikodym derivatives are established in Section 1.7.3. Note that in Section 2.2 we slightly generalize the approximation scheme, assuming that the summands which replace the increments of a Wiener process can have a more general form and are not obligatory binomial. Some

60 | 1 Financial markets. From discrete to continuous time additional conditions are needed in this case. We also establish there the rate of convergence of option prices. Concerning different approximation models, another type of approximation of the financial market driven by the geometric Ornstein–Uhlenbeck process, and even by the geometric Ornstein–Uhlenbeck–Lévy process, was studied in [145].

1.7.1 Geometric Ornstein–Uhlenbeck process and construction of discrete scheme Let 𝕋 = [0, T], Ωℱ = (Ω, ℱ , 𝔽 = {ℱt , t ∈ 𝕋}, ℙ) be a complete filtered probability space satisfying the standard assumptions from Definition A.16 and consider the adapted Ornstein–Uhlenbeck process with constant parameters on this space. We are in a position to construct a discrete scheme that weakly converges in measure to the geometric Ornstein–Uhlenbeck (Vasicek) process which is given for tech2 nical convenience by the formula S(t) = exp{X(t) − σ2 t}, where X is a solution of equation (1.77) with ν = 1. So, the form of the Langevin equation is now dX(t) = (μ − X(t))dt + σdW(t),

X(0) = x0 ∈ ℝ,

t ∈ 𝕋,

(1.78)

where μ ∈ ℝ and σ > 0. Consider the following discrete approximation scheme for (1.78). Assume we have a sequence of probability spaces (Ω(n) , ℱ (n) , ℙ(n) ),

and let {qnk , n ≥ 1, 0 ≤ k ≤ n}

n ≥ 1,

be the sequence of i. i. d. random variables in the corresponding probability space, each with two possible values ±√ Tn , and ℙ(n) (qnk = ±√ Tn ) = 21 . Let n > T. We introduce the recurrent scheme: xn0 ∈ ℝ,

Rkn := xnk − xnk−1 =

(μ − xnk−1 )T + σqnk , n

1 ≤ k ≤ n.

(1.79)

Let ℱn0 = {0, Ω} and ℱnk = σ{Rin , 1 ≤ i ≤ k}. Denote Xn (t) =

∑

⌊ tn ⌋

1≤k≤⌊ nt ⌋ T

Rkn = xn T 1t≥ T , n

and let ℚ(n) be the measure corresponding to the process Sn (t) = exp{xn0 }

∏ (1 + Rkn ),

1≤k≤⌊ tn ⌋ T

t∈𝕋

whereas ℚ is the measure that corresponds to the process S(t) = exp{X(t) −

σ2 t}. 2

1.7 Geometric Ornstein–Uhlenbeck process | 61

Theorem 1.63. Let xn0 → x0 , n → ∞. Then we have weak convergence of the probability measures ℚ(n) , ℚ

(Sn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐⇒ (S(t), t ∈ [0, T]). Proof. According to Theorem 1.62, we need to check conditions (A7) and (A8). However, (A8) is evident, so we need to check only (A7). At first, we mention that the random variables xnk can be presented as xnk = xn0 (1 −

k

k

k−i

k T T T ) + μ(1 − (1 − ) ) + σ ∑ qni (1 − ) n n n i=1

,

(1.80)

whence there exists a constant C > 0 such that sup0≤k≤n |xnk | ≤ C√n a. s. Therefore, C sup0≤k≤n |Rkn | ≤ √n a. s. and this means that condition (A7)(i) holds. Furthermore, in order to establish condition (A7)(ii), we consider any fixed a ∈ (0, 1] and such n0 that C ≤ a. Then for any n ≥ n0 we have √n 0

2

∑ 𝔼n ((Rkn ) 1|Rk |≤a | ℱnk−1 ) ≤ C 2 , n

0≤k≤n

whence condition (A7)(ii) follows. To establish condition (A7)(iii), we should note that in our case for any ε > 0, a ∈ (0, 1] and n ≥ n0 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k k−1 󵄨 lim ℙ (sup󵄨󵄨 ∑ 𝔼n (Rn 1|Rk |≤a | ℱn ) − ∫ b(s, Xn (s))ds󵄨󵄨󵄨 ≥ ε) n n 󵄨 󵄨󵄨 t∈𝕋 󵄨󵄨 󵄨 ⌋ 1≤k≤⌊ nt 0 T (n)

t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = lim ℙ(n) (sup󵄨󵄨󵄨 ∑ 𝔼n (Rkn | ℱnk−1 ) − ∫(μ − Xn (s))ds󵄨󵄨󵄨 ≥ ε) n 󵄨󵄨 t∈𝕋 󵄨󵄨󵄨 󵄨 ⌋ 1≤k≤⌊ nt 0 T

󵄨󵄨 (μ − xnk−1 )T (μ − xnk )T 󵄨 − = lim ℙ(n) (sup󵄨󵄨󵄨 ∑ ∑ n n n t∈𝕋 󵄨󵄨 1≤k≤⌊ nt ⌋−1 1≤k≤⌊ nt ⌋ T

T

[ nt ]T T

󵄨󵄨 󵄨 − (μ − xn )(t − )󵄨󵄨󵄨 ≥ ε) 󵄨󵄨 n 󵄨󵄨 (μ − x0 )T ⌊ nt ⌋T 󵄨󵄨󵄨 ⌊ nt ⌋ 󵄨 n − (μ − xn T )(t − T )󵄨󵄨󵄨 ≥ ε) = 0. = lim ℙ(n) (sup󵄨󵄨󵄨 n 󵄨󵄨 n n t∈𝕋 󵄨󵄨 ⌊ nt ⌋ T

Now let us check condition (A7)(iv). At first we shall prove that 󵄨 󵄨 lim lim sup ℙ(n) (max 󵄨󵄨󵄨xnk 󵄨󵄨󵄨 ≥ C) = 0.

C→∞

n

1≤k≤n

Due to representation (1.80), it is enough to prove that 󵄨󵄨 k k−i 󵄨󵄨 󵄨󵄨 󵄨󵄨 T A := lim lim sup ℙ(n) (max 󵄨󵄨󵄨∑ qni (1 − ) 󵄨󵄨󵄨 ≥ C) = 0. 󵄨󵄨 C→∞ n 1≤k≤n󵄨󵄨 n 󵄨 󵄨i=1

62 | 1 Financial markets. From discrete to continuous time But the last assertion follows immediately from the Kolmogorov inequality for the sums of i. i. d. random variables: n

2

A ≤ lim C −2 ∑ 𝔼n (qni ) = lim C −2 T = 0. C→∞

C→∞

i=1

Now, we have for any ε > 0 and a ∈ (0, 1] t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k 2 k−1 󵄨 lim ℙ (sup󵄨󵄨 ∑ 𝔼n ((Rn ) 1|Rk |≤a | ℱn ) − ∫ c2 (s, Xn (s))ds󵄨󵄨󵄨 ≥ ε) n n 󵄨󵄨 󵄨 t∈𝕋 󵄨󵄨 󵄨 ⌋ 1≤k≤⌊ nt 0 T 󵄨 󵄨󵄨 󵄨󵄨 2 󵄨 = lim ℙ(n) (sup󵄨󵄨󵄨 ∑ 𝔼n ((Rkn ) | ℱnk−1 ) − σ 2 t 󵄨󵄨󵄨 ≥ ε) n 󵄨󵄨 󵄨 t∈𝕋 󵄨 1≤k≤⌊ nt ⌋ (n)

T

2 󵄨󵄨 󵄨󵄨 (μ − xnk )T nt T 󵄨 󵄨 ) + σ 2 ⌊ ⌋ − σ 2 t 󵄨󵄨󵄨 ≥ ε) = lim ℙ(n) (sup󵄨󵄨󵄨 ∑ ( n 󵄨󵄨 n T n t∈𝕋 󵄨󵄨 nt 1≤k≤⌊ ⌋ T

󵄨 󵄨 ≤ lim lim sup ℙ (max 󵄨󵄨󵄨xnk 󵄨󵄨󵄨 ≥ C) C→∞

(n)

n

1≤k≤n

+ lim ℙ(n) (sup(σ 2 t − σ 2 ⌊ n

t∈𝕋

nt T (|μ| + C)2 T 2 ⌋ + ) ≥ ε) T n n

= 0, and condition (A7)(iv) holds. Concerning condition (A7)(v), we have that for any ε > 0 and a ∈ (0, 1], 2

lim ℙ(n) ( ∑ (𝔼n (Rkn 1|Rk |≤a | ℱnk−1 )) ≥ ε) n

n

1≤k≤n

= lim ℙ(n) ( ∑ ( n

1≤k≤n

2

(μ − xnk )T ) ≥ ε) n

(|μ| + C)2 T 2 󵄨 󵄨 ≤ lim lim sup ℙ(n) (max 󵄨󵄨󵄨xnk 󵄨󵄨󵄨 ≥ C) + lim ℙ(n) ( ≥ ε) = 0. n C→∞ n 1≤k≤n n The theorem is proved. 1.7.2 Pre-limit and limit Ornstein–Uhlenbeck markets are arbitrage-free and complete At first, we start with the pre-limit discrete-time discounted Ornstein–Uhlenbeck market Yn (t) =

Sn (t) = exp{xn0 } Bn (t)

∏

1≤k≤⌊ nt ⌋ T

1 + Rkn 1 + rnk

,

1.7 Geometric Ornstein–Uhlenbeck process | 63

where the random variables {Rkn | 1 ≤ k ≤ n, n ≥ 1} are defined via (1.79), {rnk | 1 ≤ k ≤ n, n ≥ 1} stand for the non-random interest rates, Bn (t) =

∏ (1 + rnk ),

⌋ 1≤k≤⌊ nt T

and rnk satisfy two assumptions, namely, sup1≤k≤n rnk → 0 and Bn (t) → ert , n → ∞. 1 ) and let the sequence xn0 be bounded. Then there exists n0 Theorem 1.64. Let rnk = o( √n such that for any n > n0 the pre-limit market (Bn (t), Yn (t)) is arbitrage-free and complete.

Proof. We look for probability measures ℙ(n,∗) such that ℙ(n,∗) ∼ ℙ(n) and for which 𝔼∗n (Yn (t) | ℱns ) = Yn (s),

(1.81)

where s

i

ℱn = σ{Rn , 1 ≤ i ≤ ⌊

ns ns ⌋} = σ{qni , 1 ≤ i ≤ ⌊ ⌋}. T T

The relation (1.81) is equivalent to 𝔼∗n (Yn (

kT 󵄨󵄨󵄨 k−1 (k − 1)T ) 󵄨󵄨 ℱn ) = Yn ( ), 󵄨 n n

1 ≤ k ≤ n,

(1.82)

where ℱnk−1 = σ{qni , 1 ≤ i ≤ k −1}. According to Theorem A.26, all equivalent martingale measures ℙ(n,∗) have Radon–Nikodym derivatives of the form n dℙ(n,∗) = (1 + ΔMnk ), ∏ dℙ(n) k=1

where {Mnk , 1 ≤ k ≤ n} is some ℱnk - martingale, ΔMnk > −1. Evidently, in our framework, martingale Mn admits the representation k

i Mnk = ∑ ρi−1 n qn , i=1

i−1 where ρi−1 n is ℱn -adapted. Therefore, n dℙ(n,∗) k = (1 + ρk−1 ∏ n qn ). dℙ(n) k=1

(1.83)

Equality (1.82) is equivalent to 𝔼n ( dℙ Y ( kT ) | ℱnk−1 ) dℙ(n) n n (n,∗)

(n,∗) 𝔼n ( dℙ dℙ(n)

| ℱnk−1 )

= Yn (

(k − 1)T ), n

1 ≤ k ≤ n.

64 | 1 Financial markets. From discrete to continuous time Furthermore, we can provide the equivalent relations 𝔼n (

k dℙ(n,∗) 1 + Rn (n) k dℙ 1 + rn

𝔼n (

dℙ(n,∗) k R dℙ(n) n

(n,∗) 󵄨󵄨 k−1 󵄨󵄨 ℱ ) = 𝔼n ( dℙ 󵄨󵄨 n dℙ(n)

󵄨󵄨 k−1 󵄨󵄨 ℱ ), 󵄨󵄨 n

(n,∗) 󵄨󵄨 k−1 󵄨󵄨 ℱ ) = r k 𝔼n ( dℙ n 󵄨󵄨 n dℙ(n)

k k−1 −1 k k−1 k 𝔼n ((1 + ρk−1 n qn )((μ − xn )Tn + σqn ) | ℱn ) = rn ,

󵄨󵄨 k−1 󵄨󵄨 ℱ ), 󵄨󵄨 n

k k k−1 k (μ − xnk−1 )Tn−1 + σ𝔼n ((1 + ρk−1 n qn )qn | ℱn ) = rn .

(1.84)

Denote ynk = rnk − (μ − xnk−1 )Tn−1 . Then we immediately obtain from (1.84) that ρk−1 n =

nynk σT

k and ρk−1 n qn =

ynk

σqnk

.

To establish the arbitrage-free property and completeness, we only have to check the inequality

ynk σqnk

> −1. The latter inequality will follow from the relation T 󵄨󵄨 k 󵄨󵄨 󵄨󵄨yn 󵄨󵄨 < σ √ . n

Take into account the inequality 󵄨󵄨 k k−i 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ qi (1 − T ) 󵄨󵄨󵄨 ≤ √nT. n 󵄨󵄨 󵄨󵄨 n 󵄨󵄨i=0 󵄨󵄨 Additionally, it follows from (1.80) that xnk − μ = (xn0 − μ)(1 −

k

k−i

k T T ) + σ ∑ qni (1 − ) n n i=0

,

(1.85)

and we immediately conclude that k

T T T T rnk + ( (xn0 − μ) + σ √ )(1 − ) − σ √ n n n n ≤ rnk +

(xnk − μ)T n

(1.86) k

T T T T ≤ rnk + ( (xn0 − μ) − σ √ )(1 − ) + σ √ . n n n n 1 Evidently, if rnk = o( √n ) and xn0 is bounded, then for sufficiently large n k

T T T rnk + ( (xn0 − μ) + σ √ )(1 − ) > 0 n n n

1.7 Geometric Ornstein–Uhlenbeck process | 65

and k

T T T rnk + ( (xn0 − μ) − σ √ )(1 − ) < 0. n n n For such n the pre-limit market (Bn (t), Yn (t)) is arbitrage-free and complete. As to the limit process, the discounted price process has the form t

σ2 σ2 Y(t) = exp{X(t) − t − rt} = exp{∫(μ − X(s))ds + σW(t) + x0 − t − rt}. 2 2

(1.87)

0

Lemma 1.65. A financial market with bond B(t) = ert and a discounted risky asset defined by equality (1.87) is arbitrage-free and complete. Proof. We look for a measure ℙ∗ ∼ ℙ such that with respect to ℙ∗ process Y is a mar∗ tingale. Evidently, the Radon–Nikodym derivative equals dℙ | , where dℙ ℱT t

t

0

0

(μ − r − X(s))2 μ − r − X(s) dℙ∗ 󵄨󵄨󵄨󵄨 1 ds)}. dW(s) − ∫( 󵄨󵄨 = exp{− ∫ dℙ 󵄨󵄨ℱt σ 2 σ2 But this is correct only if the last relation corresponds to the martingale. According to [109] (V. 1, page 233, Example 3), the process t

t

0

0

1 φ(t)(β) = exp{∫ β(s)dW(s) − ∫ β2 (s)ds} 2 is a martingale on the interval [0, T] if β is a Gaussian process with sup 𝔼β2 (t) < ∞. t≤T

In our case β(t) = − dℙ dℙ

μ−r−X(t) σ

and, up to the constant terms, 𝔼β2 (t) ∼ 𝔼X 2 (t) ∼ et , whence

defines a new martingale measure on any interval [0, T]. Therefore, the limit market is arbitrage-free and complete. ∗

1.7.3 Convergence of the asset prices in the geometric Ornstein–Uhlenbeck model Suppose that we have an option on a stochastic risk-free market that is governed by the geometric Ornstein–Uhlenbeck process. If we want to establish the convergence of option prices, we need the joint convergence of the bond prices and the Radon–Nikodym derivatives. For this we apply the multi-dimensional functional limit theorem (Theorem A.107) (see [83] for more detail concerning multi-dimensional functional limit theorems).

66 | 1 Financial markets. From discrete to continuous time , n ≥ 1, 1 ≤ k ≤ n, and let xn0 → x0 as n → ∞. Then the Theorem 1.66. Let rnk = rT n two-dimensional sequence of stochastic processes {

dℙ(n,∗) 󵄨󵄨󵄨󵄨 , Y (⋅)} 󵄨 dℙn 󵄨󵄨󵄨ℱ⌊ n⋅ ⌋ n T

| , Y(⋅)}. Here the pre-limit process of the Radon– weakly converges in measure to { dℙ dℙ ℱ⋅ Nikodym derivative is defined as ∗

dℙ(n,∗) 󵄨󵄨󵄨󵄨 󵄨 dℙn 󵄨󵄨󵄨ℱ

⌊ nt ⌋ T

=

k ∏ (1 + ρk−1 n qn ).

⌋ 1≤k≤⌊ nt T

Proof. We can use the same representations and reasoning that have used in the general case when we proved Theorem 1.63. Namely, we present the Radon–Nikodym derivative as dℙ(n,∗) 󵄨󵄨󵄨󵄨 󵄨 dℙn 󵄨󵄨󵄨ℱ

⌊ nt ⌋ T

= exp{ ∑

1≤k≤⌊ nt ⌋ T

k log(1 + ρk−1 n qn )}

1 k−1 2 T k = exp{ ∑ (ρk−1 )} + oP (n, t) n qn − (ρn ) 2 n nt 1≤k≤⌊ T ⌋

= exp{ ∑

⌋ 1≤k≤⌊ nt T

(1.88) 2

r − μ + xnk−1 k 1 r − μ + xnk−1 T qn − ( ) } + oP (n, t), σ 2 σ n

where we do not present the exact form of the remainder term oP (n, t), but it can be bounded similarly to α(a, R(n,a) , 0 ≤ k ≤ n) ∑1≤k≤[ nt ] (R(n,a) )2 from the proof of Theok k T rem 1.61 and sup oP (n, t) → 0

0≤t≤T

in probability as n → ∞. Furthermore, Yn (t) = exp{xn0 +

∑

⌋ 1≤k≤⌊ nt T

(μ − xnk−1 )T +σ n

∑

1≤k≤⌊ nt ⌋ T

qnk −

σ2 t − rt} 2

+ oP (n, t). So, we have to establish the joint weak convergence of the pre-limit vector processes { ∑

1≤k≤⌊ nt ⌋ T

∑

1≤k≤⌊ nt ⌋ T

r − μ + xnk−1 k qn , σ

(μ − xnk−1 )T , n

∑ (

1≤k≤⌊ nt ⌋ T

∑

1≤k≤⌊ nt ⌋ T

qnk , 2

r − μ + xnk−1 T ) } σ n

1.7 Geometric Ornstein–Uhlenbeck process | 67

to the limit vector process r − μ + X(s) (r − μ + X(s))2 ds, ∫(μ − X(s))ds}. {∫ dW(s), W⋅ , ∫ σ σ2 ⋅

⋅

⋅

0

0

0

Weak convergence of finite-dimensional distributions follows from Theorem 1.63. To establish tightness, consider, for example, the first sum. Other sums can be considered similarly. It follows from (1.80) that 󵄨󵄨 k k−i 󵄨󵄨4 󵄨󵄨 󵄨󵄨 T 4 󵄨 󵄨4 󵄨 󵄨 𝔼󵄨󵄨󵄨Xnk 󵄨󵄨󵄨 ≤ 8(󵄨󵄨󵄨Xn0 󵄨󵄨󵄨 + |μ|) + 8σ 4 𝔼 max 󵄨󵄨󵄨∑ qni (1 − ) 󵄨󵄨󵄨 . 󵄨󵄨 n 1≤k≤n󵄨󵄨 󵄨 󵄨i=1 Furthermore, it follows from the Burkholder–Gundy inequality that 󵄨󵄨 k k−i 󵄨󵄨4 2k−2i 2 n 󵄨󵄨 󵄨󵄨 T T 2 󵄨 󵄨 i i ) 𝔼 max 󵄨󵄨󵄨∑ qn (1 − ) 󵄨󵄨󵄨 ≤ C(∑󵄨󵄨󵄨qn 󵄨󵄨󵄨 (1 − ) 󵄨󵄨 n n 1≤k≤n󵄨󵄨 i=1 󵄨 󵄨i=1 2k−2i

T2 n T = C 2 (∑(1 − ) n n i=1

2

) ≤ CT 2 .

Therefore, 𝔼|Xnk |4 < C1 for some constant C1 . Let us continue. For any 0 ≤ t1 < t2 ≤ T, again from the Burkholder–Gundy inequality, 4 󵄨󵄨 r − μ + Xnk−1 k 󵄨󵄨󵄨󵄨 󵄨 qn 󵄨󵄨 𝔼󵄨󵄨󵄨 ∑ 󵄨󵄨 󵄨󵄨 nt1 σ nt ⌊ ⌋ 0,

t ∈ 𝕋,

(1.89)

where b > 0, σ > 0. The integral form of the process X has the following form: t

t

X(t) = x0 + ∫(b − X(s))ds + σ ∫ √X(s)dW(s). 0

(1.90)

0

According to the paper [37], the condition σ 2 ≤ 2b

(1.91)

is necessary and sufficient for the process X to get positive values and to not hit zero. Furthermore, we will assume that condition (1.91) is satisfied. To establish respective functional limit theorems, we need a modification of the CIR process with bounded coefficients. This process will be called a “truncated” CIR process, unlike the original “non-truncated” one, and it is introduced as follows. Let C > 0. Consider the following stochastic differential equation with the same coefficients b and σ as in equation (1.89): dX C (t) = (b − X C (t) ∧ C)dt + σ √(X C (t) ∨ 0) ∧ CdW(t),

X(0) = x0 > 0,

t ∈ 𝕋. (1.92)

Lemma 1.69. For any C > 0, equation (1.92) has a unique strong solution. Proof. Since the coefficient σ(x) = σ √(x ∨ 0) ∧ C and b(x) = b − (x ∧ C) satisfy the condition of Theorem A.122 and also the growth condition (A.41), a global strong solution X C (t) exists uniquely for every given initial value x0 .

70 | 1 Financial markets. From discrete to continuous time Remark 1.70. Denote σ−ϵ = inf {t : X C (t) = −ϵ}, where 0 < ϵ < b. Suppose that ℙ(σ−ϵ < ∞) > 0. Then for any r < σ−ϵ such that X C (t) < 0, if t ∈ (r, σ−ϵ ), we have with positive probability dX C (t) = (b − X C (t) ∧ C)dt > 0 on the interval (r, σ−ϵ ), and hence t → X C (t) increases in this interval. This is obviously impossible. Therefore, X C (t) is non-negative and can be written as X C (0) = x0 > 0,

dX C (t) = (b − X C (t) ∧ C)dt + σ √X C (t) ∧ CdW(t),

t ∈ 𝕋.

(1.93)

The integral form of the process X C is as follows: t

t

0

0

X C (t) = x0 + ∫(b − X C (s) ∧ C)ds + σ ∫ √X C (s) ∧ CdW(s). Lemma 1.71. Let 2b ≥ σ 2 , i. e., condition (1.91) holds. Then the trajectories of the process X C are positive with probability 1 on the set 𝕋. Proof. In order to prove that the process X C is positive, we will provide a proof similar to the one given in [112, pp. 308–309] for the CIR process (1.89), with some modifications. Note that the coefficient g(x) := σ √x ∧ C of equation (1.93) is continuous and g 2 (x) > 0 for all x ∈ (0, ∞). Furthermore, the coefficient f (x) := b − x ∧ C is continuous on ℝ. Fix α and β such that 0 < α < x0 < β. Due to the non-singularity of g on [α, β], there exists a unique solution F(x) of the ordinary differential equation 1 f (x)F ′ (x) + g 2 (x)F ′′ (x) = −1, 2

α 0 has the form τ

C

p1 (X (τ)) = p1 (x0 ) + ∫ p′1 (X C (u))g(X C (u))dW(u), 0

and 𝔼p1 (X C (τ)) = p1 (x0 ). When t → ∞, we get p1 (x0 ) = 𝔼p1 (X C (τα ∧ τβ )) = p1 (α)ℙ(τα < τβ ) + p1 (β)ℙ(τβ < τα ), and hence p1 (β) − p1 (x0 ) p1 (β) − p1 (α)

ℙ(τα < τβ ) =

and

ℙ(τβ < τα ) =

p1 (x0 ) − p1 (α) . p1 (β) − p1 (α)

Assume that C > 1 ∨ β. First, consider x < 1. Then x

y

x

1

1

1

2(b − z) 2(y − 1) − 2b p1 (x) = ∫ exp{− ∫ dz}dy = ∫ y σ2 exp{ }dy, 2 σ z σ2 and under the condition σ 2 ≤ 2b we obtain the relation lim p1 (x) = −∞. x↓0

Now, oppositely, let x increase and tend to infinity. Denote C

C1 = ∫ exp{ 1

Then for x > C C

y

p1 (x) = ∫ exp{− ∫ 1

1

2(b − z) dz}dy σ2 z

2(y − 1) − 2b2 }y σ dy. σ2

(1.97)

1.8 Cox–Ingersoll–Ross model | 73 x

C

y

C

1

C

2(b − C) 2(b − z) dz − ∫ dz}dy + ∫ exp{− ∫ σ2 z σ2 C C

= ∫ exp{ 1

2(y − 1) − 2b2 2(C − 1) − 2b }y σ dy + C σ2 exp { } 2 σ σ2

x

× ∫ exp{− C

= C1 + C

2(b − C) (y − C)}dy σ2 C

− 2b2 +1 σ

2(C − 1) σ2 2(C − b) exp{ } × (exp{ (x − C)} − 1), 2(C − b) σ2 σ2 C

and then limx↑∞ p1 (x) = ∞. Define τ0 = lim τα α↓0

and τ∞ = lim τβ , β↑∞

and put τ = τ0 ∧ τ∞ . From (1.97) we immediately obtain ℙ( inf X C (t) ≤ α) ≥ ℙ(τα < τβ ) = 0≤t 0, ℙ(inf0≤t 0, XnC,k = XnC,k−1 +

(b − (XnC,k−1 ∧ C))T + σqnk √XnC,k−1 ∧ C, n

(1.99)

C,k C,k−1 QC,k n := Xn − Xn

=

(b − (XnC,k−1 ∧ C))T + σqnk √XnC,k−1 ∧ C, n

1 ≤ k ≤ n.

(1.100)

1.8.2.1 Correctness of construction of approximations The following lemma confirms the correctness of the construction of these approximations. Lemma 1.73. 1) Let condition 2b ≥ σ 2 hold and let n > 2T. Then all values given by equalities (1.98) and (1.99) are positive, so the approximations (1.98) and (1.99) are defined correctly. 2) There is a relation ℙ{Xnk ≠ XnC,k , 0 ≤ k ≤ n} → 0 when C → ∞. Proof. 1) We fix n > 2T and apply induction in k. For k = 1 Xn1 = x0 +

(b − x0 )T + σqn1 √x0 . n

(1.101)

76 | 1 Financial markets. From discrete to continuous time Let us show that x0 +

(b − x0 )T + σqn1 √x0 > 0. n

(1.102)

We denote α := √x0 and note that (1.102) holds if the quadratic inequality T T bT ) − σ√ α + >0 n n n

α2 (1 −

holds. However, this quadratic inequality holds because its discriminant D=

σ 2 T 4bT T 2bT 4bT 2 − (1 − ) ≤ − + 2 2T. So, Xn1 > 0. Assume now that Xnk > 0. It can be shown with the help of absolutely the same transformations that the random variable (b−X k )T Xnk+1 = Xnk + nn + σqnk √Xnk > 0 under the same conditions σ 2 ≤ 2b and n > 2T. The fact that the random variables given by (1.99) are positive can be proved by the same arguments, taking into account the scheme of their construction. 2) Note that Xnk can be represented in two different forms, and we shall use both of them: k

Xnk = x0 + ∑ i=1

= Xnk−1 +

k (b − Xni−1 )T + σ ∑ qni √Xni−1 n i=1

(b − Xnk−1 )T + σqnk √Xnk−1 . n

(1.103)

Compute for some 1 ≤ i ≤ n 2

𝔼(Xni ) = 𝔼(Xni−1 (1 − = 𝔼(Xni−1 (1 − 2

=(

2

T bT )+ + σ √Xni−1 qni ) n n 2

T bT σ2 T )+ ) + 𝔼Xni−1 n n n

bT σ 2 T 2bT T ) +[ + (1 − )]𝔼Xni−1 n n n n 2

+ (1 −

T 2 ) 𝔼(Xni−1 ) . n

(1.104)

Assume that 𝔼(Xnj )2 ≤ β2 , 1 ≤ j ≤ i − 1 for some β > 0. Then 𝔼Xnj ≤ β, 1 ≤ j ≤ i − 1. Our aim is to prove that the quadratic inequality of the form 2

(1 −

2

T σ 2 T 2bT T bT ) β2 + [ + (1 − )]β + ( ) < β2 n n n n n

1.8 Cox–Ingersoll–Ross model | 77

is true for some β. In an equivalent form, it can be presented as 2

(

σ 2 T 2bT T bT 2T T 2 2 − 2 )β − [ + (1 − )]β − ( ) > 0, n n n n n n

that is, (2 −

T b2 T T 2 )β − [σ 2 + 2b(1 − )]β − > 0. n n n

(1.105)

Calculating the roots of the latter quadratic polynomial, we get 2

β± =

σ 2 + 2b(1 − Tn ) ± √(σ 2 + 2b(1 − Tn ))2 + 4(2 − Tn ) bnT 2(2 − Tn )

.

Taking into account that n > 2T, we conclude that (1.105) obviously holds for any β satisfying inequality β>

σ 2 + 2b + √σ 4 + 4bσ 2 + 4b2 . 3

Given the initial value, for all 1 ≤ i ≤ n, 𝔼Xni ≤

σ 2 + 2b + √σ 4 + 4bσ 2 + 4b2 ∨ x0 =: γ. 3

Applying the Burkholder inequality to the martingale term, we continue: k

2

0 ≤ 𝔼 max (Xnk ) ≤ 2(x0 + bT)2 + 2σ 2 𝔼 max (∑ qni √Xni−1 ) 0≤k≤n

0≤k≤n

n

≤ 2(x0 + bT)2 + 8σ 2 𝔼(∑ qni √Xni−1 ) ≤ 2(x0 + bT)2 + 8σ 2 γT.

2

2

i=1

i=1

Therefore ℙ{Xnk ≠ XnC,k , 0 ≤ k ≤ n} = ℙ{ max Xnk ≥ C} 0≤k≤n

2

≤ C 𝔼 max (Xnk ) ≤ 2C −2 (x0 + bT)2 + 8σ 2 C −2 γT, −2

0≤k≤n

whence formula (1.101) follows, which means that the proof is complete.

78 | 1 Financial markets. From discrete to continuous time 1.8.2.2 Stepwise processes and weak convergence of measures Consider the sequence of stepwise processes which correspond to our discrete schemes. Namely, let Xn (t) = Xnk

for

(k + 1)T kT ≤t< n n

XnC (t) = XnC,k

for

kT (k + 1)T ≤t< . n n

and

Thus, trajectories of the processes Xn and XnC have jumps at the points {kT/n , 0 ≤ k ≤ n} and are constant on the interior intervals. Consider the sequence of filtrations ℱnk = σ{Xn (t), t ≤ kT }. The processes XnC are adapted to these filtrations. Therefore we can n consider the same filtration for both discrete approximation schemes, and moreover, we can identify ℱnt with ℱnk for kT ≤ t < (k+1)T . n n Remark 1.74. Now we can rewrite relation (1.101) as follows: ℙ{Xn (t) ≠ XnC (t), t ∈ [0, T]} → 0, when C → ∞. Denote by ℚ and ℚ(n) , n ≥ 1, the measures that correspond to the processes X and Xn , n ≥ 1, respectively, and denote by ℚ(C) and ℚ(n,C) , n ≥ 1, the measures that correspond to the processes X C and XnC , n ≥ 1, respectively. Recall that the notation ℚ(n,C) , ℚ(C)

󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ stands for the weak convergence of measures that correspond to the stochastic processes XnC , n ≥ 1, to X C . Now Theorem 1.61, applied in order to prove the weak convergence of measures ℚ(n,C) to the measure ℚ(C) , can be formulated as follows (recall that the random variables QC,k n were defined by equality (1.100)). Theorem 1.75. Assume the following conditions are satisfied: (i) For any ϵ > 0, 󵄨 󵄨󵄨 lim ℙ( sup 󵄨󵄨󵄨QC,k n 󵄨󵄨 ≥ ϵ) = 0. n

1≤k≤n

(ii) For any ϵ > 0, a ∈ (0, 1], 󵄨󵄨 󵄨󵄨 lim ℙ(sup󵄨󵄨󵄨 ∑ 𝔼(QC,k | ℱnk−1 ) n 1|QC,k n |≤a n t∈𝕋 󵄨󵄨󵄨 nt 1≤k≤⌊ T ⌋∨1 t

󵄨󵄨 󵄨󵄨 − ∫(b − XnC (s) ∧ C)ds󵄨󵄨󵄨 ≥ ϵ) = 0. 󵄨󵄨 󵄨 0

1.8 Cox–Ingersoll–Ross model | 79

(iii) For any ϵ > 0, a ∈ (0, 1], 󵄨󵄨 󵄨󵄨 2 lim ℙ(sup󵄨󵄨󵄨 ∑ (𝔼((QC,k | ℱnk−1 ) n ) 1|QC,k n |≤a n t∈𝕋 󵄨󵄨󵄨 nt 1≤k≤⌊ T ⌋∨1 t 󵄨󵄨 󵄨󵄨 k−1 2 2 C 󵄨󵄨 ≥ ϵ) = 0. − (𝔼(QC,k | ℱ 1 )) ) − σ (X (s) ∧ C)ds ∫ C,k n n n 󵄨󵄨 |Qn |≤a 󵄨󵄨 0

Then there is weak convergence of probability measures: ℚ(n,C) , ℚ(C)

(XnC (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X C (t), t ∈ [0, T]),

n → ∞.

Applying Theorem 1.75, we can prove the following result. Theorem 1.76. For any fixed C > 0, the weak convergence ℚ(n,C) , ℚ(C)

(XnC (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X C (t), t ∈ [0, T]),

n→∞

indeed holds true. Proof. According to Theorem 1.75, we need to check conditions (i)–(iii). Let us start noting that relation (1.99) implies the upper bound TC 󵄨 󵄨󵄨 (b + C)T max 󵄨󵄨󵄨QC,k + σ√ . n 󵄨󵄨 ≤ n n

0≤k≤n

C2 Hence, there exists a constant C2 > 0 such that max0≤k≤n |QC,k n | ≤ √n . This means that condition (i) holds. Furthermore, in order to establish condition (ii), we consider any fixed a > 0 and C2 ≤ a, that is, n ≥ ( Ca2 )2 . For such n, n ≥ 1 such that √n k−1 𝔼(QnC,k 1|QC,k |≤a | ℱnk−1 ) = 𝔼(QC,k n | ℱn ) n

(b − (XnC,k−1 ∧ C))T + σ𝔼qnk √XnC,k−1 ∧ C n (b − (XnC,k−1 ∧ C))T = . n

=

Consequently, for any ϵ > 0 we have t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k−1 C 󵄨󵄨 ≥ ϵ) ℙ(sup󵄨󵄨󵄨 ∑ 𝔼(QC,k 1 | ℱ ) − (b − (X (s) ∧ C))ds ∫ n n n 󵄨󵄨 |QC,k n |≤a t∈𝕋 󵄨󵄨󵄨 󵄨󵄨 1≤k≤⌊ nt ⌋∨1 0 T

󵄨󵄨 (b − (XnC,k−1 ∧ C))T T 󵄨 = ℙ(sup󵄨󵄨󵄨 ∑ − (b − (XnC,k ∧ C)) ∑ n n t∈𝕋 󵄨󵄨 1≤k≤⌊ nt ⌋∨1 0≤k≤(⌊ nt ⌋−1)∨0 T

T

(1.106)

80 | 1 Financial markets. From discrete to continuous time ⌋T 󵄨󵄨󵄨 ⌊ nt T )󵄨󵄨󵄨 ≥ ϵ) 󵄨󵄨 n nt 󵄨󵄨 ⌊ ⌋T 󵄨󵄨󵄨 C,⌊ nt ⌋ 󵄨 = ℙ(sup󵄨󵄨󵄨(b − (Xn T ∧ C))(t − T )󵄨󵄨󵄨 ≥ ϵ) → 0, 󵄨󵄨 n t∈𝕋 󵄨󵄨 C,⌊ nt ⌋ T

− (b − (Xn

∧ C))(t −

n → ∞.

Note that this probability equals zero starting from some n, because 0 ≤ (t −

⌊ nt ⌋T T n

)
0, 󵄨󵄨 󵄨󵄨 2 ℙ(sup󵄨󵄨󵄨 ∑ (𝔼((QC,k | ℱnk−1 ) n ) 1|QC,k n |≤a 󵄨 t∈𝕋 󵄨󵄨 nt 1≤k≤⌊ T ⌋∨1 −

(𝔼(QC,k n 1|QC,k n |≤a

|

k−1 2 ℱn )) )

t

󵄨󵄨 󵄨󵄨 − σ ∫(XnC (s) ∧ C)ds󵄨󵄨󵄨 ≥ ϵ) 󵄨󵄨 󵄨 0 2

2 󵄨󵄨 (b − (XnC,k−1 ∧ C))T σ 2 T C,k−1 󵄨 ) + (Xn ∧ C) = ℙ(sup󵄨󵄨󵄨 ∑ (( n n t∈𝕋 󵄨󵄨 1≤k≤⌊ nt ⌋∨1 T

−(

(b −

(XnC,k−1

− σ 2 (XnC (⌊

n

∧ C))T

∑

0≤k≤(⌊ nt ⌋−1)∨0 T

(

σ 2 T C,k (Xn ∧ C)) n

⌊ nt ⌋T 󵄨󵄨󵄨 nt ⌋) ∧ C)(t − T )󵄨󵄨󵄨 ≥ ϵ) 󵄨󵄨 T n

= ℙ(sup(σ 2 (XnC (⌊ t∈𝕋

2

) )−

⌊ nt ⌋T nt ⌋) ∧ C)(t − T )) ≥ ϵ) → 0, T n

So, relation (iii) is established and the theorem is proved. Theorem 1.77. The weak convergence ℚ(n) , ℚ

(Xn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, T]), holds.

n→∞

n → ∞.

1.8 Cox–Ingersoll–Ross model | 81

Proof. According to Theorems A.87 and 1.76, it is enough to prove that 󵄨 󵄨 lim lim sup ℙ{ sup 󵄨󵄨󵄨Xn (t) − XnC (t)󵄨󵄨󵄨 ≥ ϵ} = 0.

C→∞

n→∞

0≤t≤T

However, due to Remark 1.74, 󵄨 󵄨 lim lim sup ℙ{ sup 󵄨󵄨󵄨Xn (t) − XnC (t)󵄨󵄨󵄨 ≥ ϵ}

C→∞

n→∞

0≤t≤T

≤ lim lim sup ℙ{Xn (t) ≠ XnC (t), t ∈ [0, T]} = 0. C→∞

n→∞

The theorem is proved.

1.8.3 Multiplicative scheme for the Cox–Ingersoll–Ross process In this subsection we construct the multiplicative discrete approximation scheme for the geometric CIR process eX(t) , t ∈ [0, T], where X(t) is a CIR process given by equation (1.90). We construct the following multiplicative process based on the discrete approximation scheme (1.98)–(1.99), introducing limit and pre-limit processes as follows: SnC (t) = exp{x0 }

∏

1≤k≤⌊ tn ⌋∨1 T

SC (t) = exp {X C (t) −

(1 + QC,k n ),

t ∈ 𝕋,

t

σ2 ∫(X C (s) ∧ C)ds}, 2

t ∈ 𝕋,

0

Sn (t) = exp{x0 }

∏

1≤k≤⌊ tn ⌋∨1 T

(1 + Qkn ),

t ∈ 𝕋,

t

S(t) = exp {X(t) −

σ2 ∫ X(s)ds}, 2

t ∈ 𝕋,

0

S̃n (t) = exp{x0 }

∏

⌋∨1 1≤k≤⌊ tn T

[(1 + Qkn ) exp{

σ2 k X }], 2n n

t ∈ 𝕋,

and ̃ = exp{X(t)}, S(t)

t ∈ 𝕋.

̃ and 𝔾 ̃ (n) , n ≥ 1, the measures that correspond to the Denote by 𝔾C , 𝔾(n,C) , 𝔾, 𝔾(n) , 𝔾, ̃ and S̃ , n ≥ 1, respectively. processes SC , SnC , S, Sn , S, n In order to prove the weak convergence of measures, we apply Theorem 1.62, parts (A7) and (A8). It can be reformulated as follows.

82 | 1 Financial markets. From discrete to continuous time Theorem 1.78. Let the following conditions hold: (i) sup1≤k≤n |QC,k n | 󳨀→ 0, n → ∞; (ii) for any a ∈ (0, 1], ℙ

2

k−1 lim lim sup ℙ( ∑ 𝔼((QC,k n ) 1|QC,k |≤a | ℱn ) ≥ D) = 0;

D→∞

n→∞

n

1≤k≤n

(iii) for any a ∈ (0, 1], 󵄨 󵄨 lim lim sup ℙ( ∑ 󵄨󵄨󵄨𝔼(QC,k | ℱnk−1 )󵄨󵄨󵄨 ≥ D) = 0; n 1|QC,k n |≤a n→∞

D→∞

1≤k≤n

(iv) for any ϵ > 0, a ∈ (0, 1], 󵄨󵄨 󵄨󵄨 | ℱnk−1 ) 𝔼(QC,k lim ℙ(sup󵄨󵄨󵄨 ∑ n 1|QC,k n |≤a n t∈𝕋 󵄨󵄨󵄨 nt 1≤k≤⌊ T ⌋∨1 t

− ∫(b − 0

XnC (s)

󵄨󵄨 󵄨󵄨 ∧ C)ds󵄨󵄨󵄨 ≥ ϵ) = 0; 󵄨󵄨 󵄨

(v) for any ϵ > 0, a ∈ (0, 1], t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 C k−1 2 󵄨󵄨 ≥ ϵ) = 0. (X (s) ∧ C)ds | ℱ ) − σ lim ℙ(sup󵄨󵄨󵄨 ∑ 𝔼((QC,k ) 1 ∫ C,k n n n 󵄨󵄨 |≤a |Q n n t∈𝕋 󵄨󵄨󵄨 󵄨󵄨 ⌋∨1 1≤k≤⌊ nt 0 T

Then the weak convergence of the probability measures 𝔾(n,C) , 𝔾C

(SnC (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (SC (t), t ∈ [0, T]),

n→∞

holds. Applying Theorem 1.78, we can prove the following result. Theorem 1.79. The weak convergence 𝔾(n,C) , 𝔾C

(SnC (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (SC (t), t ∈ [0, T]),

n→∞

indeed holds true. Proof. According to Theorem 1.78 we need to check if conditions (i)–(v) hold. It was established in the proof of Theorem 1.76 that conditions (i) and (iv) are satisfied. Let us prove that condition (ii) holds. Indeed, it was established in the proof of Theorem 1.76 that for any fixed C > 0 there exists some C2 > 0 not depending on n such that 󵄨 󵄨 C sup 󵄨󵄨󵄨QnC,k 󵄨󵄨󵄨 ≤ 2 . √n 0≤k≤n

1.8 Cox–Ingersoll–Ross model | 83

So, for all a ∈ (0, 1], starting from some number n 2

k−1 ∑ 𝔼((QC,k n ) 1|QC,k |≤a | ℱn ) n

1≤k≤n

2

C22 ≤ C22 , n 1≤k≤n

k−1 = ∑ 𝔼((QC,k n ) | ℱn ) ≤ ∑ 1≤k≤n

whence condition (ii) holds. Now, (1.106) implies that for all a ∈ (0, 1], starting from some number n C3 󵄨󵄨 󵄨 󵄨 C,k k−1 󵄨󵄨 , | ℱnk−1 )󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼(QC,k 󵄨󵄨𝔼(Qn 1|QC,k n | ℱn )󵄨󵄨 ≤ n |≤a n whence condition (iii) holds. Let us check condition (v). For any ϵ > 0 and a ∈ (0, 1], t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 2 k−1 2 C 󵄨󵄨 ≥ ϵ) lim ℙ(sup󵄨󵄨󵄨 ∑ 𝔼((QC,k ) 1 | ℱ ) − σ (X (s) ∧ C)ds ∫ C,k n n n 󵄨󵄨 |Qn |≤a n 󵄨 t∈𝕋 󵄨󵄨 󵄨󵄨 1≤k≤⌊ nt ⌋∨1 0 T 2 󵄨󵄨 (b − (XnC,k−1 ∧ C))T σ 2 T C,k−1 󵄨 ) + (Xn ∧ C)) = lim ℙ(sup󵄨󵄨󵄨 ∑ (( n n n t∈𝕋 󵄨󵄨 1≤k≤⌊ nt ⌋∨1 T

−

∑

0≤k≤(⌊ nt ⌋−1)∨0 T

− σ 2 (XnC (⌊

σ 2 T C,k (Xn ∧ C)) n

⌊ nt ⌋T 󵄨󵄨󵄨 nt ⌋ ∨ 1) ∧ C)(t − T )󵄨󵄨󵄨 ≥ ϵ) 󵄨󵄨 T n

≤ lim ℙ(sup( n

(

t∈𝕋

⌊ nt ⌋T nt (|b| + C)2 Tt + σ 2 (XnC (⌊ ⌋ ∨ 1) ∧ C)(t − T )) ≥ ϵ) n T n

= 0. The theorem is proved. Theorem 1.80. The weak convergence 𝔾(n) , 𝔾

(Sn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐⇒ (S(t), t ∈ [0, T]),

n→∞

holds true. Proof. The proof immediately follows from Theorems A.87 and 1.79 and Remark 1.74. Indeed, 󵄨 󵄨 lim lim sup ℙ{ sup 󵄨󵄨󵄨Xn (t) − XnC (t)󵄨󵄨󵄨 ≥ ϵ}

C→∞

n→∞

0≤t≤T

≤ lim lim sup ℙ{Xn (t) ≠ XnC (t), t ∈ [0, T]} = 0. C→∞

n→∞

84 | 1 Financial markets. From discrete to continuous time Remark 1.81. It can be proved in a similar way that the weak convergence ̃ (n) , 𝔾 ̃ 𝔾

̃ t ∈ [0, T]), (S̃n (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐⇒ (S(t),

n→∞

holds true.

1.9 General conditions of weak convergence of discrete-time multiplicative schemes to asset prices with memory All the previous considerations were related to the case when the ultimate stochastic process and the corresponding market model are Markov and semimartingale ones, and in some sense, they have no memory. More precisely, to analyze the functional limit theorems from Sections 1.4–1.8, we see that geometric Brownian motion and Ornstein–Uhlenbeck and CIR limit processes are Markov and semimartingale processes. However, the presence of memory in financial markets has already been so convincingly recorded that for many years models have been studied that could well model this memory, and the question of approximation of non-Markov asset prices and other components of financial market processes by discrete-time random sequences is also studied. As regards purely theoretical results on functional limit theorems in which the limit process is not Markov, we cite the papers [42], where the limit process is stationary, and [71], where the limit process is semistable and Gaussian. In turn, with regard to memory modeling, considering the processes with short- or long-range dependence, it is easiest to use fractional Brownian motion (fBm), which is a Gaussian self-similar process with stationary correlated increments. There are two approaches to the problem: to model prices themselves using processes with memory, in particular, to consider the models involving fBm, or to concentrate the model’s memory in stochastic volatility. The first approach has the peculiarity that an ultimate market with memory allows arbitrage, while pre-limit markets can be arbitrage-free. The existence of arbitrage was first established in the paper [141] and discussed in detail in the book [126]. However, such an approach has the right to exist, if only because regardless of possible financial applications, it is reasonable to prove functional limit theorems in which the limit process is an fBm or some related process. For the first time, a discrete approximation of fBm by a binomial market model was considered in the paper [156], and a fairly thorough analysis of the number of so called arbitrage points in such a market was made in the paper [36]. However, even fractional Black–Scholes models can be approximated by various discrete-time sequences, and the purpose of this section is to formulate and illustrate by examples the functional limit theorem and its multiplicative version, in which both the pre-limit sequence of processes and the limiting process are quite general, but simple to consider. Moreover, the fractional binomial market considered by [156] is a special case of our model.

1.9 Weak convergence to asset price with memory | 85

Thus, the main objectives of this section are as follows. To start with, we consider an additive stochastic sequence that is based on the sequence of i. i. d. random variables and has the coefficients that allow for this stochastic sequence to be dependent on the past. For such a sequence, we formulate the conditions of weak convergence to some limit process in terms of coefficients and the characteristic function of any basic random variable. These conditions are stated in Theorem 1.82. This theorem is of course a special case of general functional limit theorems, but it has the advantages that it is formulated in terms of coefficients, that the coefficients are such that they immediately show the dependence on the past, and that the limit process in it is not required to have any special properties with respect to distribution, self-similarity, etc. However, then, in order to apply our general theorem to more practical situations, in Theorem 1.83 we adapt the general conditions to the case where the limit process is Gaussian. Then we go to the multiplicative scheme in order to get the a. s. positive limit process that can model the asset price on the financial market. So, we assume that all multipliers in the pre-limit multiplicative scheme are positive, and this imposes additional restrictions on the coefficients. In addition, we consider only Bernoulli basic random variables. The next goal is to apply these general results to the case where the limit processes in the additive scheme are fBm and Riemann–Liouville fBm. In the case of the limit fBm we consider the pre-limit processes that are constructed regarding Cholesky decomposition of the covariance function of fBm. In both cases we were lucky in the sense that such coefficients are suitable also for the multiplicative scheme. Our proofs require deep study of the properties of the Cholesky decomposition for the covariance matrix of fBm. It turns out that all elements of the upper-triangular matrix in this decomposition are positive, and moreover, the rows of this matrix are increasing. We also suppose that the columns of this matrix are decreasing. This conjecture is confirmed by numerical results; however, its proof remains an interesting open problem. Note that stochastic volatility with memory is considered in Section 2.5. In Section 1.9.1 we establish sufficient conditions for the weak convergence of continuous-time random walks of rather general form to some limit in the space D([0, T]). The case of Gaussian limit is studied in more detail. Multiplicative version of this result is also obtained. Sections 1.9.2 and 1.9.4 are devoted to two particular examples of the general scheme investigated in Section 1.9.1. In Section 1.9.2 we consider a discrete process that converges to fBm. This example is based on Cholesky decomposition of the covariance matrix of fBm. In Section 1.9.3 we investigate possible perturbations of the coefficients in the scheme studied in Section 1.9.2. Section 1.9.4 is devoted to another example, where the limit process is a so-called Riemann–Liouville fBm.

86 | 1 Financial markets. From discrete to continuous time 1.9.1 General conditions of weak convergence As usual, let T > 0, 𝕋 = [0, T], (Ω, ℱ , 𝔽 = {ℱt , t ∈ 𝕋}, ℙ) be a stochastic basis, i. e., a complete probability space (Ω, ℱ , ℙ) with a filtration 𝔽 satisfying standard assumptions. 1.9.1.1 Convergence of sums For any n ≥ 1, consider the uniform partition {tnk = kT/n, 0 ≤ k ≤ n} of the interval [0, T]. Let {ξi , i ≥ 1} be a sequence of i. i. d. random variables with 𝔼ξi = 0 and 𝔼ξi2 = 1. Assume that for each n ≥ 1 we are given a triangular array of real numbers {cnj,k , 1 ≤ j ≤ k ≤ n}. Define a stochastic process k

Xn (tnk ) = ∑ cnj,k ξj , j=1

k = 0, 1, . . . , n;

let Xn (t) = Xn (tnk−1 ) for t ∈ [tnk−1 , tnk ), k = 1, . . . , n. Because of the dependence of the coefficients cnj,k on k, the increments of Xn may depend on the past, and the dependence may be strong. Let us first establish general conditions of weak convergence of the sequence {Xn , n ≥ 1} in terms of coefficients cnj,k and the characteristic function φ(λ) = 𝔼eiλξ1 of the underlying noise. Having stepwise pre-limit processes, we use the Skorokhod topology in the space D([0, T]) of càdlàg functions. As before, notation ℙ(n) refers to the measures corresponding to the pre-limit processes while ℙ is the measure corresponding to the limit process. Theorem 1.82. Assume that the following assumptions hold: (B1) There exists a stochastic process {X(t), t ∈ [0, T]} such that ℙ(n) , ℙ, fdd

(Xn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, T]) as n → ∞, that is, for any l ≥ 1, 0 = t0 < t1 < t2 < ⋅ ⋅ ⋅ < tl ≤ T, and λ1 , . . . , λl ∈ ℝ, l

⌊ntr /T⌋

∏

∏

r=1 p=⌊ntr−1 /T⌋+1

l

l

m=r

r=1

φ( ∑ λm cnp,⌊ntm /T⌋ ) → 𝔼 exp{i ∑ λr X(tr )}

(1.107)

as n → ∞. (B2) There exist positive constants K and α such that for all integers n ≥ 1 and 0 ≤ j < k ≤ n, j

2

k

2

∑(cnr,k − cnr,j ) + ∑ (cnr,k ) ≤ K(

r=1

r=j+1

1+α

k−j ) n

.

(1.108)

1.9 Weak convergence to asset price with memory | 87

Then the weak convergence of measures holds: ℙ(n) , ℙ

(Xn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, T])

as n → ∞

in the Skorokhod topology on the space D([0, T]). Proof. First, note that l

l

m=1

m=1

⌊ntm /T⌋

𝔼 exp{i ∑ λm Xn (tm )} = 𝔼 exp{i ∑ λm ∑ ξp cnp,⌊ntm /T⌋ } p=1

l

m

⌊ntr /T⌋

= 𝔼 exp{i ∑ λm ∑ m=1 l

= 𝔼 exp{i ∑

∑

r=1 p=⌊ntr−1 /T⌋+1 l

⌊ntr /T⌋

∑

r=1 p=⌊ntr−1 /T⌋+1

l

⌊ntr /T⌋

=∏

∏

r=1 p=⌊ntr−1 /T⌋+1

ξp cnp,⌊ntm /T⌋ }

ξp ∑ λm cnp,⌊ntm /T⌋ } m=r

l

φ( ∑ λm cnp,⌊ntm /T⌋ ). m=r

(1.109)

Therefore, the weak convergence of finite-dimensional distributions is equivalent to condition (1.107). In order to prove the weak convergence of measures, it suffices to establish the tightness of the sequence {Xn , n ≥ 1}. To start with, let us mention that for 0 ≤ t1 < t2 ≤ T, 2

𝔼(Xn (t2 ) − Xn (t1 )) ⌊nt1 /T⌋

= 𝔼( ∑

r=1

⌊nt1 /T⌋

(cnr,⌊nt2 /T⌋

−

2 r,⌊nt1 /T⌋ cn )ξr ) 2

= ∑ (cnr,⌊nt2 /T⌋ − cnr,⌊nt1 /T⌋ ) + r=1

⌊nt2 /T⌋

+ 𝔼(

⌊nt2 /T⌋

∑

r=⌊nt1 /T⌋+1

∑

r=⌊nt1 /T⌋+1 2

2 r,⌊nt2 /T⌋ cn ξr )

(cnr,⌊nt2 /T⌋ ) =: An (t1 , t2 ),

(1.110)

where, as usual, the sum over the empty set equals zero. Furthermore, let us prove that there exists C > 0 such that An (t1 , t2 )An (t2 , t3 ) ≤ C(t3 − t1 )2+2α

(1.111)

for all n ≥ 1 and for all 0 ≤ t1 < t2 < t3 ≤ T. We consider two cases. Case 1: Let t3 − t1 < T/n. In this case we have ⌊nt3 /T⌋ − ⌊nt1 /T⌋ < n(t3 − t1 )/T + 1 < 2, which means that ⌊nt3 /T⌋ − ⌊nt1 /T⌋ ≤ 1. This implies that at least one of the following equalities is true: ⌊nt1 /T⌋ = ⌊nt2 /T⌋ or ⌊nt2 /T⌋ = ⌊nt3 /T⌋. If ⌊nt1 /T⌋ = ⌊nt2 /T⌋, then An (t1 , t2 ) = 0 and inequality (1.111) holds for any C > 0. Similarly, it holds in the case ⌊nt2 /T⌋ = ⌊nt3 /T⌋, because An (t2 , t3 ) = 0.

88 | 1 Financial markets. From discrete to continuous time Case 2: Let t3 − t1 ≥ T/n. In this case ⌊nt2 /T⌋ − ⌊nt1 /T⌋ t2 − t1 1 2(t3 − t1 ) ≤ + ≤ . n T n T Then condition (1.108) implies that An (t1 , t2 ) ≤ K(

1+α

⌊nt2 /T⌋ − ⌊nt1 /T⌋ ) n

1+α

2 ≤ K( ) T

(t3 − t1 )1+α .

The same bound holds for An (t2 , t3 ). Therefore, we have 2+2α

2 An (t1 , t2 )An (t2 , t3 ) ≤ K 2 ( ) T

(t3 − t1 )2+2α ,

that is, (1.111) holds with C = K 2 (2/T)2+2α . Thus, (1.111) is proved in both cases. Now using inequalities (1.110) and (1.111), we may write 󵄨 󵄨󵄨 󵄨 𝔼( 󵄨󵄨󵄨Xn (t2 ) − Xn (t1 )󵄨󵄨󵄨󵄨󵄨󵄨Xn (t3 ) − Xn (t2 )󵄨󵄨󵄨 ) 󵄨 󵄨2 󵄨 󵄨2 1/2 ≤ (𝔼󵄨󵄨󵄨Xn (t2 ) − Xn (t1 )󵄨󵄨󵄨 𝔼󵄨󵄨󵄨Xn (t3 ) − Xn (t2 )󵄨󵄨󵄨 ) 1/2

= (An (t1 , t2 )An (t2 , t3 ))

≤ C 1/2 (t3 − t1 )1+α .

Consequently, the sequence of processes is tight; see Theorem A.97 and Remark A.98. Hence, the statement follows. If X is a Gaussian process, we can formulate sufficient conditions for the convergence of finite-dimensional distributions in terms of the covariance function. As usual, ∑0j=1 = 0. Theorem 1.83. Assume that there exists a stochastic process X = {X(t), t ∈ [0, 1]} such that the following conditions hold: (C1) X is Gaussian and centered; (C2) for all t, s ∈ [0, 1], ⌊nt⌋∧⌊ns⌋

∑ j=1

cnj,⌊nt⌋ cnj,⌊ns⌋ → Cov(X(t), X(s))

as n → ∞;

(C3) max1≤j≤k≤n |cnj,k | → 0 as n → ∞. Then ℙ(n) , ℙ, fdd

(Xn (t), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, 1]) as n → ∞.

1.9 Weak convergence to asset price with memory | 89

Proof. For any l > 1, let us consider the characteristic function of l-dimensional distribution: l

𝔼 exp{i ∑ λm Xn (tm )},

0 = t0 < t1 < t2 < ⋅ ⋅ ⋅ < tl ≤ 1,

m=1

λ1 , . . . , λl ∈ ℝ.

According to (1.109), l

⌊ntl ⌋

m=1

p=1

Zn := ∑ λm Xn (tm ) = ∑ αn,p ξp , where for 1 ≤ r ≤ l, ⌊ntr−1 ⌋ + 1 ≤ p ≤ ⌊ntr ⌋, n ≥ 1, we define l

αn,p := ∑ λm cnp,⌊ntm ⌋ . m=r

For every n ≥ 1, the random variables ηn,p := αn,p ξp , p ≥ 1, are independent. We will apply Lindeberg’s CLT (see Theorem A.111) for the scheme of series {ηn,1 , . . . , ηn,⌊ntl ⌋ }. Let us calculate the variance: ⌊ntl ⌋

l

⌊ntl ⌋

⌊ntr ⌋

2 σn2 := Var( ∑ ηn,p ) = ∑ αn,p =∑ p=1

l

=∑

l

2

( ∑ λm cnp,⌊ntm ⌋ )

r=1 p=⌊ntr−1 ⌋+1 m=r

p=1

l

⌊ntr ⌋

∑

l

p,⌊ntq ⌋

∑ ∑ λm λq cnp,⌊ntm ⌋ cn

∑

r=1 p=⌊ntr−1 ⌋+1 m=r q=r l

m∧q

l

= ∑ ∑ λm λq ∑ m=1 q=1 l

⌊ntr ⌋

∑

r=1 p=⌊ntr−1 ⌋+1

l

= ∑ ∑ λm λq

⌊ntm ⌋∧⌊ntq ⌋

∑

m=1 q=1

p=1

p,⌊ntq ⌋

cnp,⌊ntm ⌋ cn

p,⌊ntq ⌋

cnp,⌊ntm ⌋ cn

.

Hence, by assumption (C2), we have l

l

σn2 → ∑ ∑ λm λq Cov(X(tm ), X(tq )), m=1 q=1

n → ∞.

Now we are ready to verify Lindeberg’s condition. We have for any ε > 0 Ln (ε) :=

l l 1 1 2 𝔼( ξp2 1{|ξp |≥ εσn } ). ∑ 𝔼( η2n,p 1{|ηn,p |≥εσn } ) = 2 ∑ αn,p 2 |αn,p | σn p=1 σn p=1

⌊nt ⌋

⌊nt ⌋

We can estimate |αn,p | for ⌊ntr−1 ⌋ + 1 ≤ p ≤ ⌊ntr ⌋ as follows: 󵄨󵄨 l 󵄨󵄨 l 󵄨󵄨 󵄨 j,k 󵄨 p,⌊ntm ⌋ 󵄨󵄨󵄨 󵄨 |αn,p | = 󵄨󵄨 ∑ λm cn 󵄨󵄨 ≤ max 󵄨󵄨󵄨cn 󵄨󵄨󵄨 ∑ |λm |. 󵄨󵄨m=r 󵄨󵄨 1≤j≤k≤n m=1 󵄨 󵄨

(1.112)

90 | 1 Financial markets. From discrete to continuous time Therefore, σn σn =: an . ≥ |αn,p | max1≤j≤k≤n |cnj,k | ∑l |λm | m=1 Note that due to (C3) and (1.112), an → +∞, n → ∞. Hence, Ln (ε) ≤

l l 1 1 2 2 α ( α2 )𝔼( ξ12 1{|ξ1 |≥εan } ), 𝔼( ξ 1 ) = ∑ ∑ {|ξ |≥εa } n,p p p n σn2 p=1 σn2 p=1 n,p

⌊nt ⌋

⌊nt ⌋

2 because ξp , p ≥ 1, are i. i. d. random variables. Since σn2 = ∑p=1l αn,p , we see that ⌊nt ⌋

Ln (ε) ≤ 𝔼( ξ12 1{|ξ1 |≥εan } ) → 0,

n → ∞,

by the dominated convergence theorem.

ℙ(n) , ℙ, d

According to Lindeberg’s CLT, Zn 󳨐󳨐󳨐󳨐󳨐󳨐⇒ 𝒩 (0, σ 2 ), where l

l

σ 2 = lim σn2 = ∑ ∑ λm λq Cov(X(tm ), X(tq )); n→∞

m=1 q=1

see (1.112). Thus, l

𝔼 exp{i ∑ λm Xn (tm )} = 𝔼 exp{iZn } → exp{− m=1

= exp{−

σ2 } 2

1 l l ∑ ∑ λ λ Cov(X(tm ), X(tq ))} 2 m=1 q=1 m q l

= 𝔼 exp{i ∑ λm X(tm )}. m=1

1.9.1.2 Multiplicative scheme Consider now a multiplicative counterpart of the process considered above. Namely, 0 let {bj,k n , 1 ≤ j ≤ k ≤ n} be a triangular array of real numbers. Assuming ∏k=1 = 1, define ⌊nt/T⌋

Sn (t) = ∏ (1 + Znk ), k=1

t ∈ [0, T],

where k

Znk = ∑ bj,k n ξj , j=1

k = 1, . . . , n.

1.9 Weak convergence to asset price with memory | 91

To ensure that the values of Sn are positive, we assume that {ξr , r ≥ 1} are i. i. d. Rademacher random variables, i. e., ℙ(ξr = 1) = ℙ(ξr = −1) = 1/2, and that bj,k n satisfy the following assumption: k

󵄨 󵄨󵄨 ∑󵄨󵄨󵄨bj,k n 󵄨󵄨 < 1,

k = 1, . . . , n.

j=1

Our aim is to investigate the weak convergence of Sn to some positive process S. It is more convenient to work with logarithms, i. e., to consider ⌊nt/T⌋

log Sn (t) = ∑ log(1 + Znk ), k=1

t ∈ [0, T].

We will need a uniform version of the above boundedness assumption: (D1) supn≥1,k=1,...,n ∑kj=1 |bj,k n | < 1. We will also need the following assumption: 2 (D2) ∑nk=1 ∑kj=1 (bj,k n ) → 0, n → ∞. Theorem 1.84. Assume (D1) and (D2). Let also assumptions (B1) and (B2) hold for k

cnj,k = ∑ bj,i n,

1 ≤ j ≤ k ≤ n,

i=j

with some process X. Then ℙ(n) , ℙ

(Sn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐⇒ (S(t) = eX(t) , t ∈ [0, T])

as n → ∞

in the Skorokhod topology on the space D([0, T]). Proof. Let supn≥1,k=1,...,n ∑kj=1 |bj,k n | = a ∈ (0, 1). By the Taylor formula, for x ∈ (−a, a), there exists a value C(a) such that log(1 + x) = x − θ(x)x 2 , where 0 < θ(x) ≤ C(a). Therefore, ⌊nt/T⌋

⌊nt/T⌋

⌊nt/T⌋

k=1

k=1

2

log Sn (t) = ∑ log(1 + Znk ) = ∑ Znk − ∑ θ(Znk )(Znk ) =:

k=1 Xn1 (t)

+

Xn2 (t).

Write ⌊nt/T⌋ k

⌊nt/T⌋

⌊nt/T⌋

⌊nt/T⌋

j=1

k=j

j=1

j,k j,k Xn1 (t) = ∑ ∑ bj,k n ξj = ∑ ξj ∑ bn = ∑ cn ξj . k=1 j=1

92 | 1 Financial markets. From discrete to continuous time By Theorem 1.82, ℙ(n) , ℙ

(Xn1 (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, T])

as n → ∞

in the Skorokhod topology on the space D([0, T]). Furthermore, n

n

k

2 󵄨 󵄨󵄨2 󵄨 󵄨 𝔼 sup 󵄨󵄨󵄨Xn2 (t)󵄨󵄨󵄨 ≤ C(a) ∑ 𝔼(Znk ) = C(a) ∑ ∑󵄨󵄨󵄨bj,k n 󵄨󵄨 → 0, t∈[0,T]

k=1 j=1

k=1

n → ∞.

Therefore, supt∈[0,T] |Xn2 (t)| → 0, n → ∞, so ℙ

ℙ(n) , ℙ

(Xn2 (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐⇒ 0

as n → ∞

in the Skorokhod topology in D([0, T]). By Slutsky’s theorem (see, e. g., [73, p. 318]), we get ℙ(n) , ℙ

(log Sn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, T]),

as n → ∞

in the Skorokhod topology in D([0, T]), whence the claim follows. Remark 1.85. The statement of Theorem 1.84 remains valid if we replace Rademacher random variables by any other sequence {ξr , r ≥ 1} of i. i. d. random variables such that 𝔼ξr = 0, 𝔼ξr2 = 1, and |ξr | ≤ 1 for all r ≥ 1. The latter condition along with assumption (D1) ensures that Znk > −1 for all 1 ≤ k ≤ n, and consequently, the values of Sn are positive. 1.9.2 Fractional Brownian motion as a limit process and pre-limit coefficients taken from Cholesky decomposition of its covariance function Let H ∈ ( 21 , 1) and for technical simplicity let T = 1. Let BH = {BH (t), t ∈ [0, 1]} be an fBm, i. e., a centered Gaussian process with covariance function 1 R(s, t) = (s2H + t 2H − |t − s|2H ). 2

(1.113)

Some properties of fBm are considered in Section A.3.3. For n ≥ 1 we define the triangular array {dj,k , 1 ≤ j ≤ k ≤ n} by the following relation: p∧r

∑ dj,p dj,r = R(p, r). j=1

(1.114)

It is known that such sequence {dj,k , 1 ≤ j ≤ k ≤ n} exists and it is unique, since (1.114) is the Cholesky decomposition of a positive definite matrix (the covariance matrix of fBm).

1.9 Weak convergence to asset price with memory | 93

Define cnj,k =

dj,k nH

(1.115)

.

Theorem 1.86. Let {ξi , i ≥ 1} be a sequence of i. i. d. random variables with 𝔼ξi = 0 and 𝔼ξi2 = 1. Assume that {cnj,k , 1 ≤ j ≤ k ≤ n} are defined by (1.115) and ⌊nt⌋

Xn (t) = ∑ cnj,⌊nt⌋ ξj ,

t ∈ [0, 1].

j=1

Then ℙ(n) , ℙ

(Xn (t), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐⇒ (BH (t), t ∈ [0, 1])

as n → ∞

in the Skorokhod topology on the space D([0, 1]). In order to prove Theorem 1.86, we need to verify the conditions of Theorem 1.82 in the particular case X = BH . Since fBm is a Gaussian process, we will use Theorem 1.83 in order to prove the convergence of finite-dimensional distributions. Let us start with assumption (B2). Lemma 1.87. Let {cnj,k , 1 ≤ j ≤ k ≤ n} be defined by (1.115). Then for all 1 ≤ j < k ≤ n, j

k

2

2

∑(cnr,k − cnr,j ) + ∑ (cnr,k ) =

r=1

r=j+1

(k − j)2H . n2H

Proof. By applying successively (1.115), (1.114), and (1.113), we get j

2

k

∑(cnr,k − cnr,j ) + ∑ (cnr,k )

r=1

2

r=j+1

= =

1

n2H 1

j

k

(∑(dr,k − dr,j )2 + ∑ (dr,k )2 ) r=1

r=j+1

j

k

j

n

r=1

r=1

r=1

(∑(dr,j )2 + ∑(dr,k )2 − 2 ∑ dr,j dr,k ) 2H 1

(R(j, j) + R(k, k) − 2R(j, k)) n2H (k − j)2H = . n2H =

Thus assumption (B2) holds. In order to check condition (B1), we will use Theorem 1.83. First, we will establish some further properties of the coefficients dj,k . Let us

94 | 1 Financial markets. From discrete to continuous time consider the discrete-time stochastic process Gk := BH (k) − BH (k − 1), k ≥ 1. It is a stationary process with covariance γk = 𝔼( G1 Gk+1 ) = 𝔼( BH (1)(BH (k + 1) − BH (k)) ) 1 = (|k + 1|2H + |k − 1|2H − 2|k|2H ). 2 trix

(1.116)

In other words, (G1 , G2 , . . . , Gn )⊤ is a centered Gaussian vector with covariance maγ0 γ1 ( γ2 ( Γn = ( . ( .. γn−2 (γn−1

γ1 γ0 γ1 .. . γn−3 γn−2

γ2 γ1 γ0 .. . γn−4 γn−3

... ... ... .. . ... ...

γn−2 γn−3 γn−4 .. . γ0 γ1

γn−1 γn−2 γn−3 ) ) . .. ) . ) γ1 γ0 )

(1.117)

Lemma 1.88. The Cholesky decomposition of Γn , given by ℓ1,1 ℓ1,2 ( Γn = (ℓ1,3 .. . ℓ ( 1,n

0 ℓ2,2 ℓ2,3 .. . ℓ2,n

0 0 ℓ3,3 .. . ℓ3,n

... ... ... .. . ...

ℓ1,1 0 0 0 (0 0 ) )( .. .. . . ℓn,n ) ( 0

ℓ1,2 ℓ2,2 0 .. . 0

ℓ1,3 ℓ2,3 ℓ3,3 .. . 0

... ... ... .. . ...

ℓ1,n ℓ2,n ℓ3,n ) ), .. . ℓn,n )

has the following properties: ℓi,j ≥ 0,

1 ≤ i ≤ j ≤ n,

√2 − 22H−1 ≤ ℓi,i ≤ 1 = ℓ1,1 ,

1 ≤ i ≤ n.

(1.118) (1.119)

Remark 1.89. Similarly to (1.114), we can write the Cholesky decomposition of Γn coordinatewise as follows: p∧r

∑ ℓj,p ℓj,r = γp−r , j=1

p, r ∈ {1, . . . , n}.

(1.120)

Proof. In order to prove (1.118), we will apply Theorem A.130. To this end we need to verify the following conditions on the sequence {γ0 , γ1 , . . . , γn }: (a) monotonicity and positivity: γ0 ≥ γ1 ≥ γ2 ≥ ⋅ ⋅ ⋅ ≥ γn ≥ 0;

(1.121)

γ0 − γ1 ≥ γ1 − γ2 ≥ ⋅ ⋅ ⋅ ≥ γn−1 − γn ≥ 0.

(1.122)

(b) convexity:

1.9 Weak convergence to asset price with memory | 95

Note that by (1.116), γ0 = 1

and γ1 = 22H−1 − 1,

(1.123)

and hence, γ1 ≤ γ0 (recall that 1/2 < H < 1). For k ≥ 1, we have 11

γk = H(2H − 1) ∫∫(z − y + k)2H−2 dz dy 00

(1.124)

11

≥ H(2H − 1) ∫∫(z − y + k + 1)2H−2 dz dy = γk+1 , 00

whence (1.121) follows. Now let us prove (1.122). By (1.123), γ0 − γ1 = 2 − 22H−1 . Applying (1.124), we may write 11

γ1 − γ2 = H(2H − 1) ∫∫((z − y + 1)2H−2 − (z − y + 2)2H−2 ) dz dy 00

111

= H(2H − 1)(2 − 2H) ∫∫∫(z − y + 1 + u)2H−3 du dz dy 000

111

≤ H(2H − 1)(2 − 2H) ∫∫∫(z + u)2H−3 du dz dy 1

000

= H(2H − 1) ∫(z 2H−2 − (z + 1)2H−2 ) dz = H(2 − 2

0 2H−1

) ≤ 2 − 22H−1 = γ0 − γ1 ,

(1.125)

where we have used that the function y 󳨃→ (z − y + 1 + u)2H−3 is increasing for any z, u ∈ (0, 1). Similarly, for k ≥ 2, we have 111

γk − γk+1 = H(2H − 1)(2 − 2H) ∫∫∫(z − y + k + u)2H−3 du dz dy 000

111

≤ H(2H − 1)(2 − 2H) ∫∫∫(z − y + k − 1 + u)2H−3 du dz dy 000

= γk−1 − γk . Hence, (1.122) is proved. Then by applying Theorem A.130 we get (1.118). Furthermore, by Corollary A.131, we have the following lower bound: ℓi,i ≥ √γ0 − γ1 + γj−1 (γj−2 − γj−1 ),

2 ≥ i ≥ n,

96 | 1 Financial markets. From discrete to continuous time whence ℓi,i ≥ √γ0 − γ1 = √2 − 22H−1 ,

2 ≥ i ≥ n.

Finally, equality (1.120) implies that p

2 = γ0 = 1, ∑ ℓj,p j=1

1 ≤ p ≤ n.

Therefore, ℓ1,1 = 1 and ℓp,p ≤ 1 for all 1 ≤ p ≤ n. It is not hard to see that the sequences {dj,k , 1 ≤ j ≤ k ≤ n} and {ℓj,k , 1 ≤ j ≤ k ≤ n} are related as follows: dj,k − dj,k−1 , ℓj,k = { dj,j ,

j < k,

(1.126)

j = k.

Indeed, by (1.114) we have γp−r = Cov(BH (p) − BH (p − 1), BH (r) − BH (r − 1)) p∧r

p∧(r−1)

(p−1)∧r

j=1

j=1

j=1

= ∑ dj,p dj,r − ∑ dj,p dj,r−1 − ∑ dj,p−1 dj,r (p−1)∧(r−1)

+

∑ j=1

dj,p−1 dj,r−1

p∧r

= ∑ (dj,p − dj,p−1 )(dj,r − dj,r−1 ) j=1

(here dp,p−1 := 0), and comparing this expression with (1.120), we see that (1.126) holds. From (1.126) and Lemma 1.88, we immediately obtain the following result. Lemma 1.90. The coefficients di,j , 1 ≤ i ≤ j ≤ n, are increasing with respect to j and are positive: 0 < √2 − 22H−1 ≤ di,i ≤ di,i+1 ≤ ⋅ ⋅ ⋅ ≤ di,n ,

1 ≤ i ≤ n;

(1.127)

moreover, di,i ≤ 1 = d1,1 ,

1 ≤ i ≤ n.

(1.128)

Lemma 1.91. For all r ≥ 1, max dj,r ≤ CH r 2H−1 , 1≤j≤r

where CH =

H⋅22−2H . √2−22H−1

(1.129)

1.9 Weak convergence to asset price with memory | 97

Proof. By (1.114), for 1 < j < r we have j

∑ di,j di,r = R(j, r). i=1

Therefore, taking into account inequality (1.127), we get j−1

dj,j dj,r = R(j, r) − ∑ di,j di,r i=1

j−1

≤ R(j, r) − ∑ di,j−1 di,r = R(j, r) − R(j − 1, r) i=1

1 = (j2H − (r − j)2H − (j − 1)2H + (r − j + 1)2H ) 2 j

= H ∫ (x 2H−1 + (r − x)2H−1 ) dx. j−1

Note that the maximal value of the function f (x) = x 2H−1 + (r − x)2H−1 , x ∈ [0, r], is attained at the point x = 2r and equals f ( 2r ) = 2 ⋅ ( 2r )2H−1 = 22−2H r 2H−1 . Therefore dj,j dj,r ≤ H ⋅ 22−2H r 2H−1 . Using (1.127), we get dj,r ≤

H ⋅ 22−2H r 2H−1 H ⋅ 22−2H r 2H−1 ≤ . dj,j √2 − 22H−1

Similarly, in the case j = 1 we have 1

1 d1,r = R(1, r) = (1 + r 2H − (r − 1)2H ) = H ∫(x2H−1 + (r − x)2H−1 ) dx 2 ≤ H ⋅ 22−2H r 2H−1 ≤ CH r 2H−1 .

0

(1.130)

Finally, if j = r, then using (1.127), (1.128), and (1.130), we obtain dr,r ≤ d1,1 ≤ d1,r ≤ CH r 2H−1 . Remark 1.92. Lemma 1.91 claims that max1≤j≤r dj,r = O(r 2H−1 ) as r → ∞. Note that the asymptotic rate of convergence O(r 2H−1 ) is exact, since 1 max dj,r ≥ d1,r = (1 + r 2H − (r − 1)2H ) = Hr 2H−1 + O(r 2H−2 ), 1≤j≤r 2

r → ∞.

Moreover, we suppose that max1≤j≤r dj,r = d1,r ; however, the proof of this equality remains an open problem.

98 | 1 Financial markets. From discrete to continuous time Lemma 1.93. Let Xn be the process defined in Theorem 1.86. Then ℙ(n) , ℙ, fdd

(Xn (t), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (BH (t), t ∈ [0, 1]) as n → ∞. Proof. Let us check the conditions of Theorem 1.83. Evidently, assumption (C1) holds, because fBm is a centered Gaussian process. The verification of (C2) is straightforward. Applying (1.114), we get ⌊nt⌋∧⌊ns⌋

∑ j=1

cnj,⌊nt⌋ cnj,⌊ns⌋ =

1

⌊nt⌋∧⌊ns⌋

n2H

= R(

∑ j=1

dj,⌊nt⌋ dj,⌊ns⌋ =

⌊nt⌋ ⌊ns⌋ , ) → R(t, s), n n

1 R(⌊nt⌋, ⌊ns⌋) n2H n → ∞.

(1.131)

Finally, using Lemma 1.91, we can estimate C C 1 󵄨 󵄨 max 󵄨󵄨󵄨cnj,k 󵄨󵄨󵄨 = H max dj,k ≤ HH max k 2H−1 = HH max k 2H−1 n 1≤j≤k≤n n 1≤k≤n n 1≤k≤n

1≤j≤k≤n

= CH nH−1 → 0,

n → ∞.

(1.132)

Hence, assumption (C3) also holds. Thus, the result follows from Theorem 1.83.

1.9.2.1 Multiplicative scheme Now let us verify the conditions of Theorem 1.84 for the sequence dj,k −dj,k−1

n bj,k n = { dj,j , H n

H

,

j < k, j = k.

(1.133)

Evidently, k

∑ bj,i n = i=j

dj,k nH

= cnj,k ,

(1.134)

where the coefficients cnj,k are defined by (1.115). Hence, the conditions of Theorem 1.82 for cnj,k are satisfied. In the following Lemmas 1.94 and 1.95 we check assumptions (D2) and (D1), respectively. Lemma 1.94. Let {bj,k , 1 ≤ j ≤ k ≤ n} be defined by (1.133). Then n

k

2

∑ ∑(bj,k n ) →0

k=1 j=1

as n → ∞.

1.9 Weak convergence to asset price with memory | 99

Proof. It follows from (1.133), (1.114), and (1.113) that for all 1 ≤ k ≤ n, k

2

∑(bj,k n ) = j=1

=

1

k−1

( ∑ (dj,k − dj,k−1 )2 + (dk,k )2 ) 2H

n

j=1

1

n2H 1

k

k−1

k−1

j=1

j=1

j=1

(∑(dj,k )2 + ∑ (dj,k−1 )2 − 2 ∑ dj,k dj,k−1 )

(R(k, k) + R(k − 1, k − 1) − 2R(k, k − 1)) n2H 1 = 2H (k 2H + (k − 1)2H − k 2H − (k − 1)2H + 1) n 1 = 2H . n =

(1.135)

Therefore n

k

2

1−2H →0 ∑ ∑(bj,k n ) =n

k=1 j=1

as n → ∞.

Lemma 1.95. Let {bj,k , 1 ≤ j ≤ k ≤ n} be defined by (1.133). Then sup

k

󵄨 󵄨󵄨 ∑󵄨󵄨󵄨bj,k n 󵄨󵄨 < 1.

n≥2,1≤k≤n j=1

Proof. Using the Cauchy–Schwarz inequality and equality (1.135), we obtain that for n ≥ 2 and 1 ≤ k ≤ n, k k 1 √k 󵄨 󵄨󵄨 √ j,k 2 k (bn ) = H ≤ n 2 −H < 1. ≤ ∑ ∑󵄨󵄨󵄨bj,k 󵄨 n 󵄨 n j=1 j=1

(1.136)

Thus, we have proved that all assumptions of Theorem 1.84 are satisfied. As a consequence, we obtain the following result. Theorem 1.96. Assume that {bj,k n , 1 ≤ j ≤ k ≤ n} is a triangular array of real numbers defined by (1.133) and {ξj , j ≥ 1} are i. i. d. Rademacher random variables. Then the sequence of stochastic processes ⌊nt⌋

k

k=1

j=1

ℙ(n) , ℙ

H

B (Sn (t) = ∏ (1 + ∑ bj,k n ξj ), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐⇒ (S(t) = e

(t)

, t ∈ [0, 1])

as n → ∞ in the Skorokhod topology on the space D([0, 1]). Remark 1.97. Theorems 1.86 and 1.96 suggest one possible way to approximate fBm by a discrete model. Another scheme was proposed by [156]. It worth noting that his approximation is also a particular case of the general scheme described in Section 1.9.1,

100 | 1 Financial markets. From discrete to continuous time but with the following coefficients: j n

k cnj,k = √n ∫ z( , s) ds, n j−1 n

where t

2HΓ( 32 − H)

1 3 1 1 z(t, s) = √ (H − )s 2 −H ∫ uH− 2 (u − s)H− 2 du. 1 2 Γ(H + 2 )Γ(2 − 2H) s

Note that assumptions (B1) and (B2) for such cnj,k are verified in the proof of [156, Theorem 1]; in particular, the tightness condition (B2) is established in [156, Equation (8)]. The coefficients bj,k n of the corresponding multiplicative scheme are equal to j

bj,k n

k k−1 j,k j,k−1 n { {cn − cn = √n ∫j−1 (z( n , s) − z( n , s)) ds, j ≤ k − 1, n ={ {ck,k = √n ∫ nk z( k , s) ds, j = k. k−1 n { n n

Then, by the Cauchy–Schwarz inequality, we have k

2 ∑(bj,k n ) j=1

j n

k−1

2

k n

2

k k k−1 , s)) ds) + n( ∫ z( , s) ds) = n ∑ ( ∫ (z( , s) − z( n n n j=1 j−1 n

k−1

k−1 n

j n

2

k n

2

k k−1 k , s)) ds + ∫ z( , s) ds ≤ ∑ ∫ (z( , s) − z( n n n j=1 j−1 n

k−1 n

= ∫ z( 0

k−1 n

2

k n

2

k−1 k , s) ds + ∫ z( , s) ds n n 0

k−1 n

k k−1 − 2 ∫ z( , s)z( , s) ds. n n

(1.137)

0

The function z(t, s) is the kernel of the following Molchan–Golosov representation of fBm as an integral with respect to Wiener process W = {W(t), t ≥ 0}: t

BH (t) = ∫ z(t, s) dW(s); 0

see Appendix A.3.3.

1.9 Weak convergence to asset price with memory | 101

Therefore the covariance function of BH equals t∧s

R(t, s) = ∫ z(t, u)z(s, u) du, 0

and we obtain from (1.137) that k

2

∑(bj,k n ) ≤ R( j=1

k k k−1 k 1 k−1 k−1 , ) + R( , ) − 2R( , ) = 2H . n n n n n n n

(1.138)

Conditions (D2) and (D1) are derived from the bound (1.138) similarly to the derivation of Lemmas 1.94 and 1.95 from equality (1.135).

1.9.3 Possible perturbations of the coefficients in Cholesky decomposition We now discuss the question how it is possible to perturb the coefficients (1.114) and (1.115) in the pre-limit sequence so that the convergence to fBm is preserved. In this sense, we estimate the rate of convergence of the perturbations to zero, sufficient to preserve the convergence. Theorem 1.98. 1. Let the coefficients {cnj,k , 1 ≤ j ≤ k ≤ n} and the random variables {ξi , i ≥ 1} be the same as in Theorem 1.86. Consider the perturbed coefficients c̃nj,k = cnj,k + εnj , where a sequence {εnj , 1 ≤ j ≤ n} satisfies the following conditions: (i) there exist positive constants C and α such that k

2

∑ (εnr ) ≤ C(

r=j+1

1+α

k−j ) n

for all 0 ≤ j < k ≤ n; (ii) ∑nj=1 (εnj )2 → 0 as n → ∞. Then the processes ⌊nt⌋

ℙ(n) , ℙ

(Xn (t) = ∑ c̃nj,⌊nt⌋ ξj , t ∈ [0, 1]) 󳨐󳨐󳨐󳨐⇒ (BH (t), t ∈ [0, 1]) as n → ∞ j=1

in the Skorokhod topology on the space D([0, 1]).

102 | 1 Financial markets. From discrete to continuous time 2. Assume, additionally, that the following assumption holds: (iii) There exists n0 > 0 such that for all n ≥ n0 , j1/2 󵄨󵄨 j 󵄨󵄨 󵄨󵄨εn 󵄨󵄨 < 1 − H , n and {ξj , j ≥ 1} are i. i. d. Rademacher random variables. Let c̃nj,k − c̃nj,k−1 , b̃ j,k = { n c̃nj,j ,

j < k, j = k.

Then the sequence of stochastic processes ⌊nt⌋

k

k=1

j=1

H

ℙ(n) , ℙ

B (Sn (t) = ∏ (1 + ∑ b̃ j,k n ξj ), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐⇒ (S(t) = e

(t)

, t ∈ [0, 1])

as n → ∞ in the Skorokhod topology on the space D([0, 1]). Proof. 1. Let us prove that conditions (B2), (C2), and (C3) remain valid if we replace the coefficients cnj,k by c̃nj,k . Applying assumption (i) and Lemma 1.87, we get j

2

k

2

j

k

2

2

∑(c̃nr,k − c̃nr,j ) + ∑ (c̃nr,k ) = ∑(cnr,k − cnr,j ) + ∑ (cnr,k + εnr )

r=1

r=j+1

r=1

r=j+1

j

k

2

2

k

≤ 2 ∑(cnr,k − cnr,j ) + 2 ∑ (cnr,k ) + 2 ∑ (εnr ) r=1

≤ 2(

r=j+1

2H

k−j ) n

+ 2C(

2

r=j+1

k−j ) n

1+α

,

whence (B2) follows. Furthermore, for 0 ≤ s ≤ t ≤ 1, we have ⌊nt⌋∧⌊ns⌋

∑ j=1

⌊ns⌋

c̃nj,⌊nt⌋ c̃nj,⌊ns⌋ = ∑ (cnj,⌊nt⌋ + εnj )(cnj,⌊ns⌋ + εnj ) j=1

⌊ns⌋

⌊ns⌋

j=1

j=1

2

⌊ns⌋

⌊ns⌋

j=1

j=1

= ∑ cnj,⌊nt⌋ cnj,⌊ns⌋ + ∑ (εnj ) + ∑ εnj cnj,⌊ns⌋ + ∑ εnj cnj,⌊nt⌋ .

(1.139)

The first term in the right-hand side of (1.139) converges to R(t, s) by (1.131) as n → ∞. The second term is bounded by the sum ∑nj=1 (εnj )2 , which tends to zero according to assumption (ii). Using the Cauchy–Schwarz inequality and (1.113), we can bound the third term as follows: 󵄨󵄨⌊ns⌋ 󵄨󵄨 ⌊ns⌋ ⌊ns⌋ 󵄨󵄨 󵄨 󵄨󵄨 ∑ εj cj,⌊ns⌋ 󵄨󵄨󵄨 ≤ √ ∑ (εnj )2 ∑ (cnj,⌊ns⌋ )2 n n 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 j=1 j=1 󵄨 j=1

1.9 Weak convergence to asset price with memory | 103

n

j 2

≤ √∑(εn ) R( j=1

⌊ns⌋ ⌊ns⌋ , )→0 n n

, ⌊ns⌋ ) = ( ⌊ns⌋ )2H → s2H as n → ∞. Similarly, for the as n → ∞ by (ii), since R( ⌊ns⌋ n n n fourth term we have 󵄨󵄨 󵄨󵄨⌊ns⌋ ⌊ns⌋ ⌊ns⌋ 󵄨 󵄨󵄨 󵄨󵄨 ∑ εj cj,⌊nt⌋ 󵄨󵄨󵄨 ≤ √ ∑ (εnj )2 ∑ (cnj,⌊nt⌋ )2 n n 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 j=1 j=1 j=1 n

j 2

⌊nt⌋

j,⌊nt⌋ 2

) →0

≤ √∑(εn ) ∑ (cn j=1

j=1

as n → ∞. Thus, ∑⌊nt⌋∧⌊ns⌋ c̃nj,⌊nt⌋ c̃nj,⌊ns⌋ → R(t, s) as n → ∞, i. e., assumption (C2) is j=1 verified. Finally, assumption (C3) also holds true, since 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 max 󵄨󵄨󵄨c̃nj,k 󵄨󵄨󵄨 ≤ max 󵄨󵄨󵄨cnj,k 󵄨󵄨󵄨 + max󵄨󵄨󵄨εnj 󵄨󵄨󵄨, 1≤j≤n 1≤j≤k≤n

1≤j≤k≤n

max1≤j≤k≤n |cnj,k | → 0, n → ∞, by (1.132), and n

󵄨 󵄨 j 2 max󵄨󵄨󵄨εnj 󵄨󵄨󵄨 ≤ √∑(εn ) → 0, 1≤j≤n

j=1

n → ∞,

by (ii). 2. Now let us verify conditions (D1) and (D2) for the convergence of the corresponding multiplicative scheme. Note that c̃nj,k − c̃nj,k−1 , j < k, cnj,k − cnj,k−1 , j < k, bj,k j < k, n , b̃ j,k = { = { = { n j,j j,j j j,j j c̃n , j=k c̃n + εn , j=k bn + εn , j = k.

(1.140)

Therefore, k

k−1

k

j=1

j=1

j=1

2 j,k 2 k,k k 2 j,k 2 k 2 ∑(b̃ j,k n ) = ∑ (bn ) + (bn + εn ) ≤ 2 ∑(bn ) + 2(εn ) .

Consequently, using (ii) and Lemma 1.94 we get n

k

2

n

k

2

n

2

j,k k ∑ ∑(b̃ j,k n ) ≤ 2 ∑ ∑(bn ) + 2 ∑ (εn ) → 0

k=1 j=1

k=1 j=1

k=1

as n → ∞,

i. e., assumption (D2) holds. Applying successively (1.140), (7), and (iii), we get k k 1/2 󵄨󵄨 k 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨 j,k 󵄨󵄨 󵄨󵄨 k 󵄨󵄨 k ≤ b + ε ≤ ∑󵄨󵄨󵄨b̃ j,k ∑ 󵄨 󵄨 󵄨 󵄨 󵄨 n 󵄨 󵄨 n 󵄨 󵄨 n 󵄨 nH + 󵄨󵄨εn 󵄨󵄨 < 1, j=1 j=1

for all n ≥ n0 and all 1 ≤ k ≤ n. Hence, assumption (D1) is also satisfied, and the convergence of the multiplicative scheme follows from Theorem 1.84.

104 | 1 Financial markets. From discrete to continuous time Example 1.99. The following sequences of εnj satisfy conditions (i)–(iii) of Theorem 1.98: a) εnj = n1β , β > 21 , b) εnj =

jγ , nβ

γ ≥ 0, β − γ > 21 .

1.9.4 Riemann–Liouville fractional Brownian motion as a limit process Let H ∈ ( 21 , 1), T = 1, and let us define t

Z H (t) = ∫(t − u)H−1/2 dW(u),

t ∈ [0, 1],

(1.141)

0

where W = {W(u), u ∈ [0, 1]} is a Wiener process. The process X is known as Riemann– Liouville fBm or type II fBm; see, e. g., [41, 113]. Define bj,k n =

H− k

nH

1 2

(k − j + 1)H−3/2 ,

cnj,k = ∑ bj,i n = i=j

H−

nH

1 ≤ j ≤ k ≤ n,

1 k 2

∑(i − j + 1)H−3/2 = i=j

H−

nH

(1.142) 1 k−j+1 H−3/2 2

∑ l l=1

,

(1.143)

1 ≤ j ≤ k ≤ n. Theorem 1.100. Let {ξi , i ≥ 1} be a sequence of i. i. d. random variables with 𝔼ξi = 0 and 𝔼ξi2 = 1. Assume that {cnj,k , 1 ≤ j ≤ k ≤ n} are defined by (1.143) and ⌊nt⌋

Xn (t) = ∑ cnj,⌊nt⌋ ξj , j=1

t ∈ [0, 1].

Then ℙ(n) , ℙ

(Xn (t), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐⇒ (Z H , t ∈ [0, 1]),

n→∞

in the Skorokhod topology on the space D([0, 1]). First, let us prove the convergence of finite-dimensional distributions. Lemma 1.101. Under assumptions of Theorem 1.100, the finite-dimensional distributions of Xn converge to those of Z H . Proof. In order to prove the lemma, we will verify the conditions of Theorem 1.83. Let us start with proving that the covariance function of Xn converges to the covariance

1.9 Weak convergence to asset price with memory | 105

function of Z H as n → ∞. Indeed, for s ≤ t we have ⌊ns⌋

∑ cnj,⌊nt⌋ cnj,⌊ns⌋ =

(H − 21 )2

⌊ns⌋ ⌊nt⌋−j+1

n2H

∑

j=1

∑

j=1

l=1

lH−3/2

⌊ns⌋−j+1

∑

k=1

k H−3/2 .

By the Euler–Maclaurin formula, n

∑ lH−3/2 = l=1

nH−1/2 1 2

H−

+ O(1)

as n → ∞.

(1.144)

Therefore, for s > 0, ⌊ns⌋

∑ cnj,⌊nt⌋ cnj,⌊ns⌋ ∼ j=1

1

⌊ns⌋

H−1/2

∑ (⌊nt⌋ − j + 1)

n2H

j=1

2H ⌊ns⌋

=

⌊ns⌋ n2H

→s

2H

∑( j=1

H−1/2

(⌊ns⌋ − j + 1) H−1/2

⌊nt⌋ j − 1 − ) ⌊ns⌋ ⌊ns⌋

1

H−1/2

t ∫( − y) s

H−1/2

(1 − y)

0

t

H−1/2

(1 −

j−1 ) ⌊ns⌋

1 ⌊ns⌋

s

dy = ∫(t − u)H−1/2 (s − u)H−1/2 du 0

s

= 𝔼( ∫(t − u)H−1/2 dW(u) ∫(s − u)H−1/2 dW(u) ) 0

H

0

H

= Cov(Z (t), Z (s)). Hence, condition (C2) is satisfied. Note that by (1.143) and (1.144), 1 k−j+1 H − 21 n H−3/2 󵄨 󵄨 H−2 H−3/2 max max 󵄨󵄨󵄨cnj,k 󵄨󵄨󵄨 = l = ∑l ∑ 1≤j≤k≤n nH 1≤j≤k≤n l=1 nH l=1

= O(n

−1/2

(1.145)

n → ∞,

),

and condition (C3) also holds. Thus the assumptions of Theorem 1.83 are satisfied. This concludes the proof. Now let us verify the tightness condition (B2). Lemma 1.102. Let the numbers {cnj,k , 1 ≤ j ≤ k ≤ n} be defined by (1.143). Then there exists a constant C > 0 such that for all 1 ≤ j < k ≤ n, j

2

k

2

∑(cnr,k − cnr,j ) + ∑ (cnr,k ) ≤ C(

r=1

r=j+1

H+1/2

k−j ) n

.

(1.146)

106 | 1 Financial markets. From discrete to continuous time Proof. We estimate each of the sums in the left-hand side of (1.146). Using (1.143), we get j

(H − 21 )2

2

∑(cnr,k − cnr,j ) =

n2H

r=1

k−r+1

2

r=1 l=j−r+2

(H − 21 )2

=

j

∑( ∑ lH−3/2 )

n2H

j

i+k−j

i=1

l=i+1

∑( ∑ l

H−3/2

2

).

The inner sum can be bounded as follows: i+k−j

3

H

i+k−j

H

3

H

i+k−j l

3

H

3

∑ lH−3/2 ≤ i 2 − 4 ∑ l 2 − 4 ≤ i 2 − 4 ∑ ∫ x 2 − 4 dx l=i+1

l=i+1

H

l=i+1 l−1

i+k−j

3

H

3

H

i H

H

H 2

+

1

+

1 4

H

1

H 2

1

(k − j) 2 + 4

3

≤ i 2 −4

H

(i + k − j) 2 + 4 − i 2 + 4

3

= i 2 − 4 ∫ x 2 − 4 dx = i 2 − 4

1 4

,

where we used the inequality xβ − yβ ≤ (x − y)β for x > y > 0 and 0 < β ≤ 1. Then j

∑(cnr,k r=1 Since i

H−3/2

≤x

H−3/2 j

∑i

−

2 cnr,j )

(H − 1 )2 (k − j)H+ 2 j H− 3 ≤ H 21 ∑i 2. n2H ( + )2 i=1 1

2

4

for x ∈ [i − 1, i], we see that

H−3/2

i=1

i

j

≤∑∫x

H−3/2

i=1 i−1

j

jH−1/2

dx = ∫ x H−3/2 dx =

H−

0

1 2

.

(1.147)

Consequently, we have j

∑(cnr,k r=1

−

2 cnr,j )

≤ ≤

H−

( H2 +

H−

( H2 +

1 H− 21 j k− 2 ( ) ( 1 2 n n ) 4 1 k− 2 ( 1 2 n ) 4

j

j

H+ 21

)

H+ 21

)

(1.148)

.

Now we estimate the second sum in (1.146). By the change of variables m = k −r +1, we get k

2

∑ (cnr,k ) =

r=j+1

=

(H − 21 )2 n2H

k

k−r+1

r=j+1

l=1

∑ ( ∑ lH−3/2 )

2 (H − 21 )2 k−j m H−3/2 ( l ) . ∑ ∑ n2H m=1 l=1

2

1.9 Weak convergence to asset price with memory | 107

We bound the inner sum using (1.147) and estimate m2H−1 ≤ (k − j)2H−1 . We obtain k

2

∑ (cnr,k ) ≤

r=j+1

k−j

1

∑ m2H−1 ≤

n2H

m=1

H− 21

k−j =( ) n

(k − j)2H n2H H+ 21

k−j ) ( n

(1.149)

H+ 21

k−j ) ≤( n

.

Combining (1.148) and (1.149), we conclude the proof. Now let us verify the conditions of Theorem 1.84. We start with condition (D1). Lemma 1.103. Let {bj,k n , 1 ≤ j ≤ k ≤ n} be a triangular array of non-negative numbers defined by (1.142). Then for all n ≥ 1 and for all k = 1, . . . , n, k

∑ bj,k n ≤ j=1

1 . √2

(1.150)

Proof. It follows from (1.142) and (1.147) that k

∑ bj,k n =

H−

j=1

If k ≥ 2, then we have

k H−1/2 nH

nH

1 k 2

∑(k − j + 1)H−3/2 = j=1

= ( nk )H √1 ≤ k

1 √2

k

1,1 ∑ bj,k n = bn =

nH

1 k 2

k H−1/2 . nH

∑ lH−3/2 ≤ l=1

(1.151)

and (1.150) is valid. In the remaining case k = 1

H−

nH

j=1

H−

1 2

≤H−

1 1 1 < < , 2 2 √2

and (1.150) also holds. Now let us verify condition (D2). Lemma 1.104. The numbers {bj,k n , 1 ≤ j ≤ k ≤ n} defined by (1.142) satisfy condition (D2). Proof. We have n

k

2

∑ ∑(bj,k n ) =

k=1 j=1

= ≤

(H − 21 )2 n2H

n

k

∑ ∑(k − j + 1)2H−3 =

k=1 j=1

(H − 21 )2 n n 2H−3 ∑∑l n2H l=1 k=l (H − 21 )2 n 2H−3 → ∑l n2H−1 l=1

since H ∈ (1/2, 1) and ∑nl=1 l2H−3 < ∞.

= 0

(H − 21 )2 n2H

n

k=1 l=1

(H − 21 )2 n 2H−3 (n ∑l n2H l=1 as n → ∞,

k

∑ ∑ l2H−3 − l + 1)

108 | 1 Financial markets. From discrete to continuous time Thus, we have proved that all assumptions of Theorem 1.84 are satisfied. As a consequence, we obtain the following result. Theorem 1.105. Assume that {bj,k n , 1 ≤ j ≤ k ≤ n} is a triangular array of real numbers defined by (1.142), {ξj , j ≥ 1} are i. i. d. Rademacher random variables, and the process Z H is given by (1.141). Then the sequence of stochastic processes ⌊nt⌋

k

k=1

j=1

{Sn (t) = ∏ (1 + ∑ bj,k n ξj ), t ∈ [0, 1]} converges in the Skorokhod topology on the space D([0, 1]) to {S(t) = eZ

H

(t)

, t ∈ [0, 1]}.

2 Rate of convergence of asset and option prices 2.1 The rate of convergence of option prices when the limit is a Black–Scholes model 2.1.1 Introduction Since the specialists in real finances consider the problem of the rate of convergence to be of high importance, we consider the mathematical tools that can help in determining and increasing the rate of convergence. There can be different approaches to its estimation. One of the possible approaches is appropriate when the pre-limit model is binomial and the limit model is Black–Scholes. In this case it is possible to adapt the results on approximating the binomial distribution by the normal distribution. It was considered in [30], where after some elegant modification of the prelimit model and the involvement of an additional parameter, the bound n1 is obtained for the rate of convergence of call option prices. It is a good result since the standard 1 ; however, such scheme has a limited range of applicability. rate of convergence is √n Among the papers devoted to the rate of convergence of option prices for binomial and trinomial models converging to the Black–Scholes model, we mention also the works [76, 167, 168] and the case where the Black–Scholes model is discretely monitored, studied in [26]. We certainly do not claim that our list of papers devoted to the “binomial-Gaussian” rate of convergence is complete. Another possible approach to estimate the rate of convergence of option prices consists in adapting the estimates of the rate of convergence in the general pre-limit martingale model in which there are no restrictions on distribution and applying, with some modifications and transformations, the results concerning the rate of convergence in the martingale CLT. Such results are contained, e. g., in [108], and the attempts to adapt them to financial schemes are contained, e. g., in [2, 26, 45, 72, 117, 119]. However, the rate of convergence of option prices obtained in such a framework may be unsatisfactory. For example, it was proved in [119] that approximation by the geometric Ornstein–Uhlenbeck model using 1 the general rate of convergence in the martingale CLT gives only n1/8 . At last, it is possible to analyze the rate of convergence of option prices and other functionals using the rate of convergence of the solutions of corresponding partial differential equations or their transformations (see, e. g., [24, 51, 100]). In the present section we consider three different cases: in Section 2.1.2 we treat the rate of convergence in the pre-limit binomial market with fixed interest rate (Section 2.1.2.1), obtaining the rate n−1/2 up to a constant; then we refine the rate of convergence for the similar binomial model in Section 2.1.2.2, obtaining the rate n−1 up to a constant and give some instructions about how further refining is possible. In Section 2.1.3 we estimate the rate of convergence for the case when the risky asset prices on the pre-limit market have uniformly distributed jumps, obtaining the rate n−1/2 up to a constant. In Section 2.1.4 we propose to analyze the rate of convergence in the general pre-limit martingale model without restrictions https://doi.org/10.1515/9783110654240-002

110 | 2 Rate of convergence of asset and option prices on the distribution of the jumps of asset prices. In the additive scheme, the limit process is a diffusion process with zero drift and non-random diffusion coefficient (that is, it is not far from a Wiener process). After some modifications and transformations, we apply the results concerning the rate of convergence in the martingale CLT from [108]. We obtain the rate n−1/8 up to a constant and the same rate of convergence for the multiplicative scheme. Some steps in similar directions were taken in [2, 26, 45]. For the rate of convergence in the martingale and semimartingale schemes, see also [21, 72, 75, 97]. 2.1.2 The rate of convergence in the binomial model At first we consider the binomial model and study the Gaussian approximation of binomial probabilities. This approximation developed to several powers supplies the pre-assigned rate of convergence of option prices. As an example, we consider the European call option. For options with smooth payoffs the situation is even better. 2.1.2.1 Preliminary rate of convergence for fixed interest rate Let T > 0. We consider the interval [0, T] and divide it uniformly into n ≥ 1 intervals. Taking into account Remark 1.55 and especially formula (1.58), we shall study the following modification of the symmetric binomial model, where interest rate r ≥ 0 is taken into account: an = e(r−

σ2 2

)δn −σ√δn

− 1,

bn = e(r−

σ2 2

)δn +σ√δn

rn = erδn − 1,

− 1,

δn =

T . n

Let us calculate probabilities that supply the non-arbitrage property of the market according to (1.14): p∗n

(n,∗)

=ℙ

= = =

b −r (Rn = an ) = n n = bn − an eσ√δn −

e

σ√δn −

σ2 2

δn

σ2 2

δn

−e

−1

e(r−

e(r−

σ2 2

σ2 2

)δn +σ√δn

)δn +σ√δn

− erδn

− e(r−

σ2 2

)δn −σ√δn

2

−σ√δn − σ2 δn

σ√δn −

σ2 δ 2 n

σ√δn −

1 3 3/2 σ δn 2

σ2 δ )2 2 n

+ 21 (σ√δn −

+ 61 (σ√δn −

2σ√δn + o(√δn )

+

1 3 3/2 σ δn 6

2σ√δn + o(√δn )

+ o(δn3/2 )

That is, we can rewrite p∗n as p∗n =

un − erδn . un − dn

=

σ2 δ )3 2 n

+ o(δn3/2 )

1 1 2 − σ δn + o(δn ). 2 6

(2.1)

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 111

Let the initial asset price be x. Then at time t = 0 the option price of a discounted call option with maturity date T in the discrete-time model equals πncall (x) = e−rT 𝔼∗n (Snn − K)

+

n + n n−k k = e−rT ∑ (x(bn + 1)k (an + 1)n−k − K) ( )(p∗n ) (1 − p∗n ) k k=1 n

k

n−k

= e−rT ∑ x((1 − p∗n )(bn + 1)) (p∗n (an + 1)) k=Kn

n ( ) k

n n n−k k − K ∑ ( )(p∗n ) (1 − p∗n ) e−rT k k=K n

n

n = x ∑ ( )(p1n )k (1 − p1n )n−k k k=K n

n n − e−rT ∑ ( )(p2n )k (1 − p2n )n−k , k k=K n

where Kn is defined as the smallest k for which x(bn + 1)k (an + 1)n−k ≥ K, and the new notations are discussed in detail below. In other words, we look for the smallest k for which k≥

log Kx − n log(an + 1) b +1

log an +1

(2.2)

,

n

or, finally, k≥

σ2 )δn 2

log Kx − n(r −

2σ√δn

+ nσ√δn

.

Similar but more complicated calculations are contained in the proof of Theorem 1.50. So, we obtain Kn = ⌊

log Kx − n(r −

σ2 )δn 2

2σ√δn

+ nσ√δn

K

=⌊

n 1 log x − T(r − + 2 2 σ √T/√n

=⌊

n √n (log + 2 2

K x

σ2 ) 2

− T(r − σ √T

⌋+1 σ2 )) 2

⌋ + 1.

Concerning the new notations, p2n = (1 − p∗n ),

⌋+1

1 − p2n = p∗n ,

(2.3)

112 | 2 Rate of convergence of asset and option prices p1n = (1 − p∗n )un e−rδn , 1 − p1n = p∗n dn e−rδn , so that p1n and 1 − p1n can be considered as the adjusted probabilities. Indeed, p∗n =

un −erδn un −dn

; therefore,

((1 − p∗n )un + p∗n dn )e−rδn = (un − p∗n (un − dn ))e−rδn

= (un − (un − erδn ))e−rδn = 1.

Now, rewrite πncall (x) as πncall (x) = xBirt (Kn , n, p1n ) − KBirt (Kn , n, p2n ), where Birt (⋅, ⋅, ⋅) stands for the right-tailed binomial distribution (see also Theorem 1.50), p2n = 1 − p∗n = p1n = (1 − p∗n )e−

σ2 2

1 + o(√δn ), 2

δn +σ√δn

1 1 = ( − σ 2 δn + o(δn )) 2 6

2

1 σ2 σ2 δn + σ √δn + (σ √δn − δn ) + o(δn )) 2 2 2 1 1 1 2 = + σ √δn − σ δn + o(δn ). 2 2 6 × (1 −

As mentioned in [85], page 114, and in [76], according to the de Moivre–Laplace theorem, Birt (⌊np + √np(1 − p)z⌋, n, p) = 1 − Φ(z) + O(

1 ), √n

as well as Birt (⌊np + √np(1 − p)z⌋ + 1, n, p) = 1 − Φ(z) + O(

1 ). √n

Also, Φ(z + O(

1 1 )) = Φ(z) + O( ), √n √n

1 where Φ(⋅) is the standard normal distribution function, and this term O( √n ) absorbs the distinction between Kn and Kn − 1 when we calculate the corresponding number z. Therefore, let us for technical simplicity consider a non-integer value

log ̃n = n + √n K 2 2

K x

− T(r − σ √T

σ2 ) 2

=:

n √n ̃ C. + 2 2

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 113

Considering Birt (Kn − 1, n, p2n ), we take p = p2n = 21 + o(√δn ), and we obtain immediately that √p(1 − p) = 21 + o(√δn ); therefore, up to a term of order o(√δn ) we can put ̃= z2 = C

log Kx − T(r − σ √T

σ2 ) 2

.

̃n in two different forms: To specify z1 , we present K ̃ ̃n = np1n + z1 √np1n (1 − p1n ) = n + √n C. K 2 2 Taking into account the last relation, together with (2.1), we obtain that up to a term of order o(√δn ), z1 = =

n 2

+

√n ̃ C 2

− np1n

√np1n (1 − p1n )

√n(− 21 σ√δn

= −σ √T +

+ 1 2

=

̃ √n( 21 − p1n ) + 21 C √p1n (1 − p1n ) ̃ + o(δ )) + 1 C

1 2 σ δn 6

n

+ O(δn )

2

1̃ 1 1 2 T 1̃ σ + C + o(√δn ) = −σ √T + C + O( ). √n 12 √n 2 2

Therefore, taking into account that 1 − Φ(z) = Φ(−z), we obtain πncall (x)

= xΦ( − Ke = xΦ(

− log Kx + T(r − σ √T

−rT

Φ(

+ σ √T)

− log Kx + T(r −

log Kx + T(r +

+ O(

σ2 ) 2

σ √T

σ √T

σ2 ) 2

σ2 ) 2

) − Ke

) + O(

−rT

Φ(

1 ) √n

log Kx + T(r − σ √T

1 ). √n

σ2 ) 2

) (2.4)

Taking into account the standard Black–Scholes formula (see (1.42)–(1.44)) and (2.4), we see that our calculations led us to the following result. Theorem 2.1. For an + 1 = e(r−

σ2 2

)δn −σ√δn

and bn + 1 = e(r−

πncall (x) = π call (x) + O(

σ2 2

)δn +σ√δn

we have

1 ). √n

2.1.2.2 Refining the rate of convergence. Varying interest rate 1 In some cases, the rate of convergence O( √n ) is not satisfactory and should be im-

1 proved. The specialists in real finances state that the rate O( n1 ) but not O( √n ) is satisfactory. So, let us improve our model and our approximation results in order to achieve

114 | 2 Rate of convergence of asset and option prices the rate of convergence O( n1 ). Note that for contingent claims with smooth payoff functions, ℍ = g(Snn ) with g ∈ C 2 (ℝ), it holds true that πncall (x) = π call (x) + O( n1 ) [76]. However, real payoff functions are not so smooth. To start with, recall that recently we σ2

σ2

considered the model with un = bn + 1 = e(r− 2 )δn +σ√δn and dn = an + 1 = e(r− 2 )δn −σ√δn . 2 The modifications consists in changing the coefficient (r − σ2 ). The results in this direction were obtained by J. B. Walsh in [167]. Here we discuss and modify the result from Chang and Palmer [30]. In order to optimize the rate of convergence, it is reasonable to use the following result from [164], page 119. In what follows ⌈z⌉ = z − ⌊z⌋ is a fractional part of number z. Lemma 2.2. Provided that pn → 21 as n → ∞ (so qn := 1 − pn → tail of the binomial distribution admits asymptotic expansion

1 2

as n → ∞), the right

n n Birt (jn , n, pn ) = ∑ ( )(pn )k (qn )n−k k k=j n

ξ2n

=

ξ2n ξ1n u2 qn − pn 1 2 2 ((1 − ξ2n )e− 2 − (1 − ξ1n )e− 2 ) ∫ e− 2 du + √2π 6√2πnpn qn 2

2

ξ1n

ξ2n ξ1n 1 1 ((1 − θ1n )e− 2 + (1 − θ2n )e− 2 ) + O( ), n √2πnpn qn 2

2

+ j −np − 1

nq + 1

where ξ1n = n√np nq 2 , ξ2n = √npn q2 , and θ1n and θ2n are respective fractional parts of the n n n n numbers nqn − ξ1n √npn qn and npn + ξ2n √npn qn . 2

2

Theorem 2.3. Let un = eλn σ δn +σ√δn and dn = eλn σ δn −σ√δn with λn being an arbitrary bounded function of n. Then the price of a European call option satisfies d22

d12

πncall (x) = π call (x) + (xe− 2 − Ke−rT− 2 ) where Δn = 1 − 2⌈

log Kx + n log dn 2σ√δn

Δn 1 + O( ), √n n

⌉.

The Black–Scholes values of the arguments d+ = d+ (T, x) and d− = d− (T, x) are taken from (1.43) and (1.44), respectively. Proof. As before, at time t = 0 the option price of a discounted call option with maturity date T calculated in the discrete-time model equals n n n n πncall (x) = x ∑ ( )(p1n )k (q1n )n−k − Ke−rT ∑ ( )(p2n )k (q2n )n−k , k k k=K k=K n

where, as always, x is an initial risky asset price.

n

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 115

Now we expand p2n using a Taylor expansion and for technical simplicity put y = yn = √δn . All o(⋅) and O(⋅) are considered as y = yn → 0, that is, as δn → 0. So, 2

p2n

erδn − dn e(r−λn σ )δn +σ√δn − 1 = = un − dn e2σ√δn − 1 1 1 2 = ((r − λn σ 2 )δn + σ √δn + σ 2 δn + (r − λn σ 2 ) δn2 2 2 + σ √δn3 (r − λn σ 2 ) +

σ 3 δn3/2 + o(δn )) 6

× (2σ √δn + 2σ 2 δn +

4 3 32 σ δn + o(δn )) 3

−1

1 σ2 = (σ + (r − λn σ 2 + σ 2 )y + (r − λn σ 2 + )σy2 2 6 1 1 1 2 + ( (r − λn σ 2 ) + σ 4 + σ 2 (r − λn σ 2 ))y3 + o(y3 )) 2 24 2 × (2σ + 2σ 2 y + =:

4 3 2 σ y + o(y2 )) 3

−1

A + By + Cy2 + Dy3 + o(y3 ) . 2A + B1 y + C1 y2 + o(y2 )

(2.5)

To adjust and to specify the descending order in the numerator and denominator in (2.5), we rewrite the probability as p2n =

A + By + Cy2 + O(y3 ) . 2A + B1 y + C1 y2 + O(y3 )

(2.6)

Now we apply the following formula for the twice continuously differentiable function: 1 1 f ′ (0) −f ′′ (0)f (0) + 2(f ′ (0))2 2 = − 2 y+ y + o(y2 ). f (y) f (0) f (0) 2f 3 (0) We obtain that the denominator of the fraction in the right-hand side of (2.6) equals B (B )2 − 2AC1 2 1 − 12 y + 1 y + O(y3 ). 2A 4A 8A3 Combining the latter value with (2.6), we obtain B (B )2 − 2AC1 2 1 − 12 y + 1 y + O(y3 )) 2A 4A 8A3 B2 − 2AC1 BB1 B 1 B C = +( − 1 )y + ( 1 − + )y2 + O(y3 ). 2 2A 4A 8A2 4A2 2A

p2n = (A + By + Cy2 + O(y3 ))(

116 | 2 Rate of convergence of asset and option prices Now, returning to the values of A, B, B1 and other coefficients, we conclude that r − λn σ 2 − 21 σ 2 B B 1 σ 2 2σ 2 − 1 = (r − λn σ 2 + − )= 2A 4A 2σ 2 2 2σ and B21 − 2AC1 BB1 C = 0. − + 8A2 4A2 2A It means that we can rewrite out probability as p2n =

1 3 1 1 + αy + O(y3 ) = + αδn2 + O(δn2 ), 2 2

where α=

r − (λn + 21 )σ 2 2σ

.

As the next step, we transform our ξin , i = 1, 2: ξ1n = to (2.2) and (2.3), Kn = ⌊

log Kx − n log(an + 1) b +1

log an +1 n

̃n ⌋ + 1. := ⌊K

⌋=⌊

Kn −np2n − 21 √np2n q2n

, where, according

K 2 n √n (log x − Tλn σ ) + ⌋+1 2 2 σ √T

Obviously, we can rewrite ξ1n as ξ1n = −

1 + 2np2n − 2Kn . 2√np2n q2n

Furthermore, 1 3 1 −2Kn + 1 + 2np2n = −2Kn + 1 + 2n( + αδn2 + O(δn2 )) 2 1

3

= −2Kn + 1 + n + 2nαδn2 + o(δn2 ) = (nδn = T) 1

= −2Kn + 1 + n + 2√Tnα + O(δn2 ). ̃n + n + 2√Tnα = √nd− , where d− = d− (T, x) Now our aim is to establish the relation −2K is taken (see (1.44)). Indeed, it is true because ̃n + n + 2√Tnα −2K = −2(

K 2 r − (λn + 21 )σ 2 n √n (log x − Tλn σ ) + ) + n + 2√Tn ⋅ 2 2 2σ σ √T

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 117

= =

log Kx

σ√δn log Kx σ√δn

= √n(

+ λn σ √Tn + 2√Tn ⋅ + √Tn ⋅

r − (λn + 21 )σ 2 2σ

r √ σ − Tn ⋅ σ 2

log Kx + rT − σ √T

σ2 T 2

) = √nd− .

̃n − {K ̃n } + 1, −2Kn = −2K ̃n + 2{K ̃n } − 2. Also, Kn = K Therefore, −2Kn + 2np2n + 1

1

̃n + 2{K ̃n } − 2 + 1 + n + 2√Tnα + O(δn2 ) = −2K 1

̃n } − 1 + d− √n + O(δn2 ) = 2{K 1

= −Δn + d− √n + O(δn2 ).

(2.7)

Also,

so

3 3 1 1 1 1 p2n (1 − p2n ) = ( + αδn2 + O(δn2 ))( − αδn2 + O(δn2 )) 2 2 3 1 = − α2 δn + O(δn2 ), 4 3 3 1 −1 = (1 − 4α2 δn + O(δn2 )) 2 = 1 + 2α2 δn + O(δn2 ). 2√p2n (1 − p2n )

We can conclude that

ξ1n = −d− +

Δn 1 + O( ), √n n

−ξ1n = d− −

Δn 1 + O( ). √n n

and with evidence,

Now, examine one by one the terms in Lemma 2.2 for the probabilities p2n . Let ξ2n

∞

u2

u2

∞

u2

I = ∫ e− 2 du = I1 − I2 = ∫ e− 2 du − ∫ e− 2 du. ξ1n

ξ1n

ξ2n

We can rewrite I1 as follows: −ξ1n

I1 = ∫ e −∞ x

u2

where f (x) = ∫d e− 2 du. −

2

− u2

d2

−ξ1

du = ∫ + ∫ = √2πΦ(d− ) + f (−ξ1 ), −∞

d−

118 | 2 Rate of convergence of asset and option prices By a Taylor expansion, for some η ∈ (−ξ1n , d− ), f (−ξ1n ) = e

−

2 d− 2

d e− (−ξ1n − d− ) − − 2

η2

2 d− 2

(−ξ1n − d− )2 +

f ′′′ (η) (−ξ1n − d− )3 , 6

η2

where f ′′′ (η) = −e− 2 + η2 e− 2 is a bounded function. Then 2 d−

f (−ξ1n ) = e− 2 (−

Δn d− Δ2n 1 − ) + o( ), √n 2n n

and so 2 d−

I1 = √2πΦ(d− ) + e− 2 (−

Δn d− Δ2n 1 − ) + o( ). √n 2n n u2

In the next step, we will evaluate I2 = ∫ξ e− 2 du. We have ∞ 2n

1 q2n + 2n ξ2n = →1 √n √p2n q2n

as n → ∞. Hence we can find n0 such that ξ2n ≥ 2 for n ≥ n0 . For n ≥ n0 ∞

1 |I2 | ≤ ∫ e−u du = e−ξ2n = e−√nO(1) = o( ). n ξ2n

Summarizing, ξ2n

2 u2 1 − d2− Δn 1 1 e + O( ). ∫ e− 2 du = Φ(d− ) − √2π √2π √n n

ξ1n

Let us consider the next terms: p2n − q2n =

2α√T 1 + O( 3/2 ), √n n

so q2n − p2n 4α√T 1 =− + O( 2 ). n n √np2n q2n Furthermore, it is easily understandable that the asymptotic behavior of ξin is as follows: −ξ1n → d− , ξ2n → ∞ as n → ∞. Therefore, e−

2 ξ2n 2

2 (1 − ξ2n ) − e−

2 ξ1n 2

2 d−

2 (1 − ξ1n ) → e− 2 (1 − d−2 ),

n → ∞,

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 119

with the rate of convergence O( n1 ), and, consequently, ξ2n ξ1n q2n − p2n 2 2 (e− 2 (1 − ξ2n ) − e− 2 (1 − ξ1n )) 6√2πnp2n q2n 2 d− 1 1 −4α√T (−e− 2 (1 − d−2 )) + o( ) = O( ). = √ n n 6n 2π 2

2

The next term can be evaluated very simply: ξ2n ξ1n 1 1 (e− 2 (1 − θ1n ) − e− 2 (1 − θ2n )) = O( ). √ n 12n 2π 2

2

Finally, 2 d−

n

1 n e− 2 Δn n−k + O( ). = Φ(d− ) − ∑ ( )pk2n q2n √2π √n k n k=kn n−k As to ∑nk=kn (nk )pk1n q1n , here corresponding lower and upper limits of integration equal

ξ̂1n = and p1n =

1 2

1

Kn − np1n − √np1n q1n

1 2

,

ξ̂2n =

nq1n +

1 2

√np1n q1n

,

3

̂ δn2 + O(δn2 ), +α ̂= α

r − (λn − 21 )σ 2 2σ

.

Replacing ξ1n and ξ2n by ξ̂1n and ξ̂2n and providing similar calculations, we get the same result but with d+ instead of d− . For example, (2.7) will take the form −2Kn + 2np2n + 1

(2.8) 1 2

̃n + 2{K ̃n } − 2 + 1 + n + 2√Tnα ̂ + O(δn ) = −2K 1

̃n } − 1 + d+ √n + O(δn2 ) = 2{K 1

= −Δn + d+ √n + O(δn2 ),

(2.9)

and as a result of similar calculations, n

2 d+

n e− 2 Δn 1 n−k = Φ(d+ ) − + O( ). ∑ ( )pk1n q1n √ √n k n 2π k=kn With this observation, the proof easily follows.

120 | 2 Rate of convergence of asset and option prices Remark 2.4. It is possible, of course, to get the next coefficients in the approximate expansion. In order to calculate them, it is necessary to continue, for instance, the calculation of the coefficients in (2.5). From an analytical point of view, this is not very interesting, so we do not present these calculations; however, from a computational point of view, it is very simple, and all the preliminary work has been done in the proof of the above theorem.

2.1.3 The rate of convergence of option prices in the model with uniformly distributed asset jump Consider another model of the market, which is, in our opinion, of practical interest. Namely, determine the rate of convergence of the fair prices of European options for a model of a financial market in which the jump of the stock price is uniformly distributed over a certain interval, by using the theorem on asymptotic decompositions of the distribution function of the sum of i. i. d. r. v. from the book [136]. Indeed, in numerous cases, we can only predict an interval of possible jumps of the stock price but not specific values of these jumps. Thus, to simplify the problem, we assume that the jump is uniformly distributed over a certain known interval with respect to an objective measure. The pre-limit market is incomplete despite the fact that the limit market is complete. We choose a single and specific martingale measure on an incomplete market and show that the rate of convergence of the corresponding fair price of the European call and put options to the Black–Scholes price with respect to this measure is not less than 1/√n. 2.1.3.1 Description of the parameters of the pre-limit “uniform” model As usual, we consider a model of a financial market in a scheme of series in which the entire time interval [0, T] in the nth series is split into steps Tn , 2T , . . . , nT , n ≥ 1. n n Consider a risk-free asset Bn operating on this market. Its price has the form Bkn := (1 + rn )k , k = 0, . . . , n, n ≥ 1, rn > −1, and, in addition, lim (1 + rn )n = erT ,

n→∞

(2.10)

where r is a finite constant. Moreover, for technical simplicity we assume that rn = rT + o( n1 ). n Assume that a positive risky asset Sn also operates on the same market. For this asset, we assume that its jump is represented by the sequence of random variables Rkn :=

Snk − Snk−1 Snk−1

,

k = 1, . . . , n,

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 121

and Rkn is uniformly distributed in the kth period of the nth series over the segment [αn , βn ] and, moreover, −1 < αn ≤ rn ≤ βn , k = 1, n lim α n→∞ n

(2.11)

}.

= lim βn = 0 n→∞

Let Rkn , k = 1, . . . , n be mutually independent random variables. Consider the case where the parameters have a special form. Namely, let the model be symmetric with rn =

√T

√T

and αn , βn such that 1 + αn = e−σ n , 1 + βn = eσ n for some σ > 0. Obviously, in this case conditions (2.10) and (2.11) are satisfied because rT n

lim (1 + rn )n = lim (1 +

n→∞

n→∞

n

rT ) = erT . n

Note that √nrn → 0, √nαn → −σ √T, √nβn → σ √T as n → ∞. Thus, for large n we can write −1 < αn < rn < βn , and, in addition, lim αn = lim βn = 0. We also refer the reader to Remark 1.18, where the symmetric binomial model was initially introduced, and to Remark 1.44, where non-arbitrage properties of the symmetric binomial model were established. Note again, however, that our model is not binomial. Still, the price process for the risky asset has the standard form k

Snk = Snk−1 (1 + Rkn ) = Sn0 ∏(1 + Rin ), i=1

k = 1, n,

where the initial value Sn0 = x > 0 is a given constant. For technical simplicity, we sometimes put x = 1. The sequence Sn , n ≥ 1, of the risky assets with the consequent prices Snk is studied on a respective sequence of the probability spaces (Ω(n) , ℱ (n) , ℙ(n) ). Filtration is taken in the form ℱnk := σ(Sn1 , . . . , Snk ), k = 1, . . . , n. Our goal is to determine martingale probability measure(s) ℙ(n,∗) for each n, i. e., the measure with respect to which the discounted price Xnk :=

Snk

(1 + rn )k

,

k = 0, . . . , n,

is a ℙ(n,∗) -martingale with respect to the filtration 𝔽(n) = {ℱnk , 1 ≤ k ≤ n}. In what follows, 𝔼n and Varn denote, respectively, the mathematical expectation and variance with respect to the measure ℙ(n) while 𝔼∗n and Var∗n denote, respectively, the mathematical expectation and variance with respect to the measure ℙ(n,∗) .

122 | 2 Rate of convergence of asset and option prices So, we look for the martingale measure(s) satisfying the following conditions: (i) 𝔼∗n (Xnk | Fnk−1 ) = Xnk−1 ∀ 1 ≤ k ≤ n, n ≥ 1; (ii) ℙ(n,∗) (Ω(n) ) = 1. These conditions are equivalent to the following ones: 𝔼(

dℙ(n,∗) k R ) = rn , dℙn n

𝔼(

dℙ(n,∗) ) = 1. dℙn

Let us try to choose the Radon–Nikodym derivative

dℙ(n,∗) dℙn

(2.12)

in such a way that with re-

, Rkn has a smooth density ϕ∗n (x). It is clear that the function ϕ∗n (x), satisfy-

spect to ℙ ing equalities (2.12), is not unique. Thus, the analyzed financial market is arbitrationfree but incomplete. We are looking for the distribution density of the Radon–Nikodym derivative as (n,∗)

ϕ∗n (x) =

cn x + dn , βn − αn

x ∈ (αn , βn ).

(2.13)

This means that the density of distribution of the Radon–Nikodym derivative with respect to the uniform distribution is equal to cn x + dn , x ∈ (αn , βn ). As a result of the solution of (2.12), for the required function of the form (2.13), we transform (2.13) to the form βn

∫

αn

cn x + dn dx = 1, βn − αn

and βn

∫x

αn

cn x + dn dx = rn . βn − αn

This system of linear equations obviously has the unique solution 12rn − 6(αn + βn ) , (βn − αn )2 6r (α + βn ) − 12αn βn . dn = 4 − n n (βn − αn )2 cn =

Moreover, cn → 3( σr2 − 1/2) and dn → 1 as n → ∞. Together with αn , βn tending to zero, it means that the inequality ϕ∗n (x) ≥ 0 is true for all sufficiently large n. Now we establish several auxiliary statements. Lemma 2.5. The symmetric model under consideration has asymptotic variance of the form σn2 := Var∗n (Rkn ) =

σ2 T 1 + O( 3/2 ). n n

(2.14)

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 123

Proof. Since the random variable Rkn with respect to the measure ℙ(n,∗) is distributed over [αn , βn ] with the density of distribution of the Radon–Nikodym derivative ϕ∗n (x) = cn x+dn , we have β −α n

n

βn

σn2 = Var∗n (Rkn ) = ∫(x − rn )2 ϕ∗n (x) dx αn

βn

= ∫(x2 − 2 αn

r 2 T 2 c x + dn rT x+ 2 ) n dx n βn − αn n

βn

=

1 rT r2 T 2 ∫(cn x3 − 2 cn x2 + cn 2 x βn − αn n n αn

r 2 T 2 dn 2rT )dx dn x + n n2 c β4 − αn4 2 rTcn βn3 − αn3 r 2 T 2 cn αn + βn = n n − + 4 βn − αn 3 n βn − αn 2 n2 + dn x 2 −

2rTdn αn + βn r 2 T 2 dn dn 2 (αn + αn βn + βn2 ) − + 3 n 2 n2 c d 2rTcn = n (αn + βn )(αn2 + βn2 ) + ( n − )(αn2 + αn βn + βn2 ) 4 3 3n 2rTdn r2 T 2 r2 T 2 + ( 2 cn − )(αn + βn ) + 2 dn . 2n 2n n +

(2.15)

Note that asymptotically αn + βn and αn2 + βn2 are of the same order: αn + βn = O( n1 ), αn2 + βn2 = O( n1 ). Therefore the right-hand side of (2.15) contains the term

βn2 )

=

σ2 T n

+

o( n1 ),

and the other terms are

O( n12 )

dn 2 (αn 3

+ αn βn +

as n → ∞. So, lemma is proved.

Lemma 2.6. The following equalities hold for the symmetric model: 1 𝔼∗n [log Snn ] = rT − σ 2 T + O(n−1/2 ), 2

Var∗n [log Snn ] = σ 2 T + O(n−1/2 )

as n → ∞.

Proof. It is necessary to determine the parameters of the distribution of log Snn . To this end, similarly to what we did when proving Theorem 1.41, let us apply the Taylor formula to n

log Snn = log ∏(1 + Rkn ). k=1

By virtue of the Taylor formula, log (1 + x) = x − 21 x 2 + ρ(x)x2 , where ρ(x) is such that |ρ(x)| ≤ δ(α, β) for −1 < α ≤ x ≤ β and δ(α, β) → 0 as α, β → 0. More precisely, if |αn | |βn | − 21 < αn < x < βn < 21 , then δ(αn , βn ) ≤ 3(1+θα ∨ 3(1+θβ ≤ 83 |αn | ∨ |βn |, 0 ≤ θ ≤ 1. )3 )3 n

n

124 | 2 Rate of convergence of asset and option prices Therefore, n 1 2 log Snn = ∑ (Rkn − (Rkn ) ) + Δn , 2 k=1

where Δn satisfies the inequality n

2

|Δn | ≤ δ(αn , βn ) ∑ (Rkn ) , k=1

(2.16)

and δ(αn , βn ) ≤

8 T T T 8 √T |α | ∨ |βn | ≤ (eσ n σ √ ) ∨ (σ √ ) ≤ 8σ √ , 3 n 3 n n n

provided that n > Tσ 2 . Since ℙ(n,∗) is a martingale measure, we have 𝔼∗n [Rkn ] = rn =

rT . n

To determine the parameters of the distribution Snn , we use Lemma 2.5 and bounds (2.16). This gives 1 1 𝔼∗n [log Snn ] = nrn − (σn2 n + nrn2 ) + δn,0 = rT − σ 2 T + δn,1 , 2 2 where δn,0 = 𝔼∗n [Δn ], δn,1 = δn + O(n−1/2 ), and, in view of (2.14), n

|δn,0 | ≤ 𝔼∗n |Δn | ≤ δ(αn , βn ) ∑ (Var∗n (Rkn ) + rn2 ) = O(n−1/2 ). k=1

Summarizing, δn,1 = O(n−1/2 ). Furthermore, we denote ukn = Rkn − 21 (Rkn )2 . Then 3

Var∗n ukn = Var∗n (Rkn ) − 𝔼∗n (Rkn ) − =

σ2 T 1 + O( 3/2 ) n n

1 1 ∗ k 4 2 2 2 𝔼 (R ) + 𝔼∗n Rkn 𝔼∗n (Rkn ) − (𝔼∗n (Rkn ) ) 4 n n 4 (2.17)

and lim n Var∗n ukn = σ 2 T.

n→∞

It follows immediately from all previous calculations that n

Var∗n [log Snn ] = ∑ Var∗n ukn + O(n−1/2 ) = σ 2 T + O(n−1/2 ). k=1

Thus, the distribution of log Snn has the following parameters: rT − 21 σ 2 T + O(n−1/2 ) and σ 2 T + O(n−1/2 ). The lemma is proved.

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 125

Remark 2.7. (i) In other words, for the parameters of the distribution Snn , there exist constants c3 , c4 > 0 such that the following inequalities are true: 󵄨󵄨 ∗ 󵄨 c n ∗ 󵄨󵄨𝔼n [log Sn ] − 𝔼n [log S(T)]󵄨󵄨󵄨 ≤ 3 √n and

󵄨󵄨 󵄨 c ∗ n ∗ 󵄨󵄨Varn [log Sn ] − Varn [log S(T)]󵄨󵄨󵄨 ≤ 4 , √n

where log S(T) = rT − 21 σ 2 T + σW(T). (ii) It immediately follows from (2.17) that the variance of the random variable Xn1 = log(1 + R1n ) − 𝔼∗n log(1 + R1n ) satisfies the same relation: Var∗n Xn1 =

σ2 T 1 + O( 3/2 ). n n

To formulate the theorem, we consider an arbitrary initial value Sn0 = x > 0 assuming it the same in all series. Theorem 2.8. Let conditions (2.10) and (2.11) be satisfied. Also, let the martingale measure be defined by the density of distribution of the Radon–Nikodym derivative of the form (2.13) and satisfy assumptions (2.12). Then the distribution of Snn with respect to ℙ(n,∗) weakly converges to a log-normal distribution with parameters 1 log x + rT − σ 2 T 2

and

σ √T,

i. e., to the distribution of a random variable 1 S(T) = x exp{σW(T) + (r − σ 2 )T}. 2 Proof. We use the CLT formulated as Theorem A.110. See also the proof of Theorem 1.41. Assume that the independent random variables {Ynk , n ≥ 1, 1 ≤ k ≤ n} satisfy the following conditions on (Ω(n) , ℱ (n) , ℙ(n) ) for each fixed n ≥ 1: (i) there exists a constant γn such that γn → 0 and |Ynk | ≤ γn , ℙn -a. s., (ii) ∑nk=1 𝔼n [Ynk ] → m, (iii) ∑nk=1 Varn [Ynk ] → σ 2 . Then the distribution of the sum n

Zn := ∑ Ynk , k=1

n ≥ 1,

weakly converges to the normal distribution with mean value m and variance σ 2 .

126 | 2 Rate of convergence of asset and option prices In the analyzed case, the limit distribution for n

log Snn = ∑ log(1 + Rkn ) + log x k=1

is normal according to the CLT in the form of Theorem A.110 and Lemma 2.6. At the same time, the distribution Snn is log-normal with the corresponding parameters. The theorem is proved. 2.1.3.2 The rate of convergence in the “uniform” model Theorem 2.9. For the model of a market in which the jump of risky asset prices is a random variable uniformly distributed over a symmetric interval with respect to an objective measure, there exists a martingale measure such that the rate of convergence of the fair prices of put and call options with respect to this measure to the fair price in the limit 1 model is not worse than √n , up to a constant. Proof. According to the Black–Scholes formula, the fair price of the discounted European put option Cput has the form π put (x) = e−rT 𝔼∗ (K − S(T)) , +

where ℙ∗ is the martingale measure, K is the strike price of the option, x is an initial asset price, and S(T) is the non-discounted asset price at time T; see (1.40). Now we write the fair prices of this option in the pre-limit and limit models and estimate their difference. For technical simplicity, assume that x = 1. Thus, in the limit model, we get, applying integration by parts, π

put

∞

(1) = e

−rT

𝔼 (K − S(T)) = e ∗

+

−rT

∫ (K − z)+ dF ∗ (z) 0

K

K

= e−rT ∫(K − z)dF ∗ (z) = e−rT ∫ F ∗ (z)dz, 0

(2.18)

0

where F is the distribution function of the random variable S(T), ∗

F ∗ (z) = ℙ∗ (S(T) < z) = ℙ∗ (log S(T) < log z)) = ℙ∗ ( = ℙ∗ ( = Φ(

log S(T) − 𝔼∗ (log S(T)) log z − 𝔼∗ (log S(T)) < ) √Var∗ (log S(T)) √Var∗ (log S(T))

log S(T) − rT + 21 σ 2 T

σ √T log z − rT + 21 σ 2 T σ √T


0 and f (y) = eyσ

√T

, the latter

1 1 ∫ F ∗ (z)dz = rT ∫ Φ(y(z))dz erT e y(K)

y(K)

−∞

−∞

1 2 1 √ = rT ∫ Φ(y)eyσ T+rT− 2 σ T σ √Tdy = k0 ∫ Φ(y)f (y)dy. e

Similarly, integral (2.19) admits the following change of variables: y = yn (z) =

log z − rT + 21 σ 2 T + υn √σ 2 T + υn

,

where υn = O(n−1/2 ), according to Remark 2.7. This yields K

K

1 1 ∫ F̃n∗ (yn (z))dz ∫ Fn∗ (z)dz = n (1 + rn )n ) (1 + rT n 0

yn (K)

0

= kn ∫ F̃n∗ (y)fn (y)dy. −∞

Here F̃n∗ (y) is the distribution function of the sum of centered and normalized i. i. d. r. v. F̃n∗ (y) = ℙ(n,∗) (

∑nj=1 Xnj √n Var∗n Xn1

< y),

where Xnj = log(1 + Rjn ) − 𝔼∗n log(1 + Rjn ), with Rjn being uniformly distributed in the jth period of the nth series over the segment [αn , βn ], with respect to the objective measure, satisfying assumptions (2.11), 1

2

kn = erT+υn − 2 σ T √σ 2 T + υn (1 +

rT ) n

−n

> 0,

128 | 2 Rate of convergence of asset and option prices 1

2

2

and fn (y) = ey σ T+υn , υn = O(n−1/2 ). Note that kn = e− 2 σ T σ √T(1 + υn ). Also, denote a = y(K) and an := yn (K). Thus, it is necessary to estimate the expression √

an a 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨󵄨k0 ∫ Φ(y)f (y)dy − kn ∫ F̃ ∗ (y)fn (y)dy󵄨󵄨󵄨 n 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨 −∞ −∞ a

a

󵄨 󵄨 ≤ |k0 − kn | ∫ f (y)dy + |kn | ∫ 󵄨󵄨󵄨fn (y) − f (y)󵄨󵄨󵄨dy −∞

−∞

a 󵄨󵄨 an 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 + |kn |󵄨󵄨 ∫ fn (y)dy󵄨󵄨 + |kn | ∫ f (y)󵄨󵄨󵄨Φ(y) − F̃n∗ (y)󵄨󵄨󵄨dy 󵄨󵄨 󵄨󵄨 󵄨a 󵄨 −∞

=: I1 + I2 + I3 + I4 .

a

Each term can be estimated separately. First, integral I5 = ∫−∞ eyσ whence a

I1 = |k0 − kn | ∫ f (y)dy = |k0 − kn |I5 ≤ −∞

√T

dy is finite,

c1 1 + o( ), √n √n

where c1 does not depend on n. Furthermore, a

󵄨󵄨 f (a) fn (a) 󵄨󵄨󵄨󵄨 c 󵄨 󵄨 󵄨 I2 = kn ∫ 󵄨󵄨󵄨fn (y) − f (y)󵄨󵄨󵄨dy = kn 󵄨󵄨󵄨− + 󵄨󵄨 ≤ 2 . 󵄨󵄨 σ √T √ 2 󵄨󵄨 √n σ T + υn −∞ Similarly, we can state that an

c 󵄨 󵄨 I3 = kn ∫ 󵄨󵄨󵄨fn (y)󵄨󵄨󵄨dy ≤ 3 . √n a

Finally, we have a

󵄨 󵄨 I4 = kn ∫ f (y)󵄨󵄨󵄨Φ(y) − F̃n∗ (y)󵄨󵄨󵄨dy −∞ a

1 󵄨 󵄨 ), ≤ c4 ∫ 󵄨󵄨󵄨Φ(y) − F̃n∗ (y)󵄨󵄨󵄨dy + o( √n

(2.20)

−∞

where c4 also does not depend on n. So, we should consider and bound the difference |Φ(y) − F̃n∗ (y)|, using Theorem A.112. In order to do this, we first check the condition 𝔼∗n |Xn1 |3 < ∞. Note that T T log(1 + R1n ) ∈ [log(1 + αn ), log(1 + βn )] = [−σ √ , σ √ ]. n n

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 129

So, 󵄨 󵄨3 󵄨 󵄨3 󵄨 󵄨3 𝔼∗n 󵄨󵄨󵄨Xn1 󵄨󵄨󵄨 ≤ 4𝔼∗n 󵄨󵄨󵄨log(1 + R1n )󵄨󵄨󵄨 + 4(𝔼∗n 󵄨󵄨󵄨log(1 + R1n )󵄨󵄨󵄨) C 󵄨 󵄨3 ≤ 8𝔼∗n 󵄨󵄨󵄨log(1 + R1n )󵄨󵄨󵄨 ≤ 3/2 , n

(2.21)

where constant C does not depend on n. Evidently, 𝔼∗n |Xn1 |k < ∞ for any k ≥ 1. 1

Denote Fn (y) = ℙ(n,∗) (Xn1 < y), fn (t) = 𝔼∗n eitXn . Applying Theorem A.112 with k = 3, we get 1 󵄨󵄨 ̃ ∗ 󵄨 |Q (y)| 1 2 3 󵄨󵄨Fn (y) − Φ(y)󵄨󵄨󵄨 ≤ n1/2 + C(Kn + Kn + Kn ), n

where

Kn1 = (Var∗n Xn1 )

n

−3/2 −1/2

−3

(1 + |y|)

(2.22)

|z|3 dFn (z),

∫ |z|≥√n Var∗n Xn1 (1+|y|)

Kn2 = (Var∗n Xn1 ) n−1 (1 + |y|) −2

−4

|z|≤√n Var∗n Xn1 (1+|y|)

n

−4 󵄨 󵄨 1 Kn3 = (sup 󵄨󵄨󵄨fn (t)󵄨󵄨󵄨 + ) n6 (1 + |y|) , 2n |t|≥ςn

Q1n (y) = − =−

|z|4 dFn (z),

∫

ςn =

γn3 1 −y2 /2 e H2 (y)( ) 3 √2π 3!(Var∗n Xn1 ) 2

Var∗n Xn1 , 12𝔼∗n |Xn1 |3

γn3 1 −y2 /2 2 , e (y − 1) 3 √2π 6(Var∗n Xn1 ) 2

where γn3 is the cumulant of order three of the distribution Xn1 and H2 (y) = y2 − 1 is the second Chebyshev–Hermite polynomial. Let us consider each term in the right-hand side of (2.22) separately. In order to bound

|Q1n (y)| , n1/2

note that

Fn (x) = ℙ(n,∗) (Xn1 < x) = ℙ(n,∗) (log(1 + R1n ) < 𝔼∗n log(1 + R1n ) + x) 1

= ℙ(n,∗) (R1n < e𝔼n log(1+Rn )+x − 1) = ℙ(n,∗) (R1n < eAn +x − 1), ∗

where An = 𝔼∗n log(1 + R1n ). We now determine the moment generating function 1

un (t) = 𝔼∗n etXn =

2An

=

βn

1 ∫ etx (cn (ex+An − 1)ex+An ) + dn ex+An )dx βn − αn αn

cn e (e(t+2)βn − e(t+2)αn ) (βn − αn )(t + 2) +

(dn − cn )eAn (e(t+1)βn − e(t+1)αn ). (βn − αn )(t + 1)

130 | 2 Rate of convergence of asset and option prices The cumulant is given by the formula γn3 = (log un (t))t |t=0 ′′′

=(

′ 3 ′ ′′ 󵄨 2u′n (t)(u′′ u′′′ n (t)un (t) − (un (t)) ) 󵄨󵄨󵄨 n (t)un (t) − un (t)un (t) − )󵄨󵄨 . 3 2 󵄨󵄨t=0 un (t) un (t)

Evidently, un (0) = 1 and u′n (0) = 𝔼∗n Xn1 = 0; therefore, γn3 can be simplified to 3

′ ′′ ′ γn3 = u′′′ n (t) − 3un (t)un (t) + 2(un (t)) |t=0 3

2

3

3

= 𝔼∗n (Xn1 ) − 3𝔼Xn1 𝔼(Xn1 ) + 2(𝔼∗n Xn1 ) = 𝔼∗n (Xn1 ) , whence, taking into account the order of decreasing of R1n with n and the equality from part (ii) of Remark 2.7, γn3

3 6(Var∗n Xn1 ) 2

=

𝔼∗n (Xn1 )3

3 6(Var∗n Xn1 ) 2

=

𝔼∗n (R1n )3 3 3/2 6 σnT3/2

+ O(n−1/2 ).

Moreover, βn

3

𝔼∗n (R1n ) = ∫ y3 αn

cn y + dn dy βn − αn

dn 2 c (α + βn2 )(αn + βn ) + n (αn4 + αn3 βn + αn2 βn2 + αn βn3 + βn4 ) 4 n 5 D −2 = 2 + o(n ), n =

where D is a constant not depending on n. Summarizing, we get that 2

−y /2 2 |y − 1| 󵄨󵄨 1 󵄨󵄨 Le , 󵄨󵄨Qn (y)󵄨󵄨 ≤ n1/2

where L > 0 is some constant. In order to bound Kn1 , recall that Var∗n Xn1 ∼ mark 2.7(ii). Furthermore,

σ2 T , n

(2.23) n → ∞, according to Re-

󵄨 󵄨3 |z|3 dFn (z) ≤ ∫ |z|3 dFn (z) = 𝔼∗n 󵄨󵄨󵄨Xn1 󵄨󵄨󵄨 = O(n−3/2 ),

∫

ℝ

|z|≥√n Var∗n Xn1 (1+|y|)

n → ∞, by (2.21). Therefore, Kn1 = (1 + |y|)

−3

⋅ O(n−1/2 ),

n → ∞.

(2.24)

Similarly, since ∫ |z|≤√n Var∗n Xn1 (1+|y|)

|z|4 dFn (z) ≤ √n Var∗n Xn1 (1 + |y|) ∫ |z|3 dFn (z), ℝ

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 131

we see that Kn2 ≤ (Var∗n Xn1 )

n

−3/2 −1/2

−3 −3 󵄨 󵄨3 (1 + |y|) 𝔼∗n 󵄨󵄨󵄨Xn1 󵄨󵄨󵄨 = (1 + |y|) O(n−1/2 )

(2.25)

as n → ∞. In order to bound Kn3 , let us consider the characteristic function 1

1

1

fn (t) = 𝔼∗n eitXn = 𝔼∗n eit log(1+Rn )−it𝔼 log(1+Rn ) 1

1

1

= 𝔼∗n eit log(1+Rn ) e−it𝔼 log(1+Rn ) := fn1 (t)e−it𝔼 log(1+Rn ) . 1

Evidently, |e−it𝔼 log(1+Rn ) | = 1, and fn1 (t) can be transformed as follows: fn1 (t)

=

𝔼∗n exp{it log(1

+

log(1+βn )

=

∫

dn dz βn − αn

log(1+βn )

∫

dn (ez − 1) dz βn − αn

exp{itz}

cn (ez − 1)ez dz = Jn1 (t) + Jn2 (t) + Jn3 (t). βn − αn

log(1+βn )

∫

cn y + dn dy βn − αn

exp{itz}

log(1+αn )

+

αn

exp{itz}

log(1+αn )

+

= ∫ exp{it log(1 + y)}

cn (ez − 1) + dn z e dz βn − αn

log(1+βn )

∫

βn

exp{itz}

log(1+αn )

=

R1n )}

log(1+αn )

Consider, for example, Re Jn1 (t)

dn = βn − αn =

log(1+βn )

cos tz dz

∫ log(1+αn )

dn (sin[t log(1 + βn )] − sin[t log(1 + αn )]) → 0 (βn − αn )t

as t → ∞ for any n > 1. The same is true for Im Jn1 (t). Integrals Jn2 (t) and Jn3 (t) can be considered similarly; therefore, we consider one of them, say Jn2 (t). It is possible to provide the following bounds for it, taking into account that dn → 1, and therefore |dn | < 2 for sufficiently large n: |dn | 󵄨󵄨 2 󵄨󵄨 󵄨󵄨Jn (t)󵄨󵄨 ≤ βn − αn

log(1+βn )

∫ log(1+αn )

󵄨󵄨 z 󵄨 󵄨󵄨e − 1󵄨󵄨󵄨 dz

132 | 2 Rate of convergence of asset and option prices

≤ ≤

|dn | βn − αn

z∈[log(1+αn ),log(1+βn )]

2 e

σ √ Tn

1 + βn 󵄨󵄨 z 󵄨 󵄨󵄨e − 1󵄨󵄨󵄨 log 1+α

sup

−e

−σ √ Tn

βn ⋅ 2σ √

n

T ≤ 4βn → 0 n

as n → ∞. It means that we can consider n0 > 1 such that for any n ≥ n0 󵄨 󵄨 󵄨 󵄨 lim sup󵄨󵄨󵄨fn (t)󵄨󵄨󵄨 ≤ lim sup󵄨󵄨󵄨fn1 (t)󵄨󵄨󵄨 < 1. t→∞

t→∞

Hence, for sufficiently large n 󵄨 󵄨 sup 󵄨󵄨󵄨fn (t)󵄨󵄨󵄨 < 1,

|t|≥ςn

since Var∗n Xn1 ςn = ≥ 12𝔼∗n |Xn1 |3

σ2 T n

1 + O( n3/2 )

12Cn−3/2

n → ∞,

→ ∞,

1 n by Remark 2.7(ii) and (2.21). Therefore, the factor (sup|t|≥ςn |fn (t)| + 2n ) decreases faster than n−p for any p > 0. So Kn3 = (1 + |y|)−4 o(n−1/2 ), n → ∞. Combining this with (2.22)–(2.25), we arrive at −y 󵄨󵄨 ̃ ∗ 󵄨 Le 󵄨󵄨Fn (y) − Φ(y)󵄨󵄨󵄨 ≤

2

|y2 − 1| −3 + (1 + |y|) ⋅ O(n−1/2 ), n

/2

n → ∞.

Thus, simplifying the notations for constants in (2.20) and taking into account 2 a a 1 that the integrals ∫−∞ e−y /2 |y2 − 1|dy and ∫−∞ (1+|y|) 3 dy are convergent, we obtain a

󵄨 󵄨 I4 = C ∫ 󵄨󵄨󵄨F̃n∗ (y) − Φ(y)󵄨󵄨󵄨dy −∞ a

≤C ∫ −∞

e−y

2

/2

a

O(n−1/2 ) |y2 − 1| dy + C ∫ dy = O(n−1/2 ). n (1 + |y|)3 −∞

Therefore, for x = 1 consequently, for any x > 0 󵄨󵄨 put 󵄨 put −1/2 󵄨󵄨π (x) − πn (x)󵄨󵄨󵄨 = O(n ). Referring to the put–call parity equation, we can state that K , (1 + rn )n K π call (x) = π put (x) + x − rT . e

πncall (x) = πnput (x) + x −

rn =

rT , n

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 133

Taking into account these relations and combining them with the fact that |erT − )n | = O(n−1 ) (see Lemma A.1), we conclude that (1 + rT n rT n 󵄨 K|e − (1 + rn ) | 󵄨󵄨 call 󵄨 󵄨 put put call 󵄨󵄨π (x) − πn (x)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨π (x) − πn (x)󵄨󵄨󵄨 + erT (1 + rn )n n 󵄨 c rT 󵄨󵄨󵄨 K 󵄨󵄨 ≤ 9 + rT 󵄨󵄨󵄨erT − (1 + ) 󵄨󵄨󵄨 = O(n−1/2 ). √n e 󵄨󵄨 n 󵄨󵄨

The theorem is proved. 2.1.4 The rate of convergence of option prices when a general martingale-type discrete-time scheme approximates the Black–Scholes model In this subsection we consider the rate of convergence of option prices in the case when the general pre-limit market approximates the Black–Scholes model. We are now in the framework of Section 1.6.2.1 and we will use the notation and results from this subsection. However, now the limit process is simplified, in comparison with equation (1.69), and has the form t

X(t) = ∫ c(s)dW(s),

t ∈ 𝕋 = [0, T],

0

where W is a Wiener process and c = c(s), s ∈ [0, T], is a measurable function such T that ∫0 c2 (s)ds < ∞. Concerning the pre-limit processes, we consider the sequence of probability spaces (Ω(n) , ℱ (n) , 𝔽(n) = {ℱt(n) , t ∈ 𝕋}, ℙ(n) ) with filtration and a sequence of stepwise semimartingales Xn = {Xn (t), ℱt(n) , t ∈ 𝕋} that are defined on the corresponding probability space and admit a representation Xn (t) = Xn (

kT (k + 1)T kT ) for ≤t< . n n n

So, the trajectories of the process Xn have the jumps at the points kT/n, k = 0, . . . , n, and are constant on the interior intervals. Denote k

ℱn = σ{Xn (t), t ≤

kT } n

and Qkn = ΔXn (

kT kT (k − 1)T ) = Xn ( ) − Xn ( ), n n n

k = 1, . . . , n.

Then the random variables Qkn are ℱnk -measurable, k = 1, . . . , n, and in what follows we identify ℱt(n) with ℱnk for kT ≤ t < (k+1)T . n n

134 | 2 Rate of convergence of asset and option prices 2.1.4.1 Additive scheme, the rate of convergence of distribution functions Theorems 1.61 and 1.62 give very mild conditions of weak convergence of the additive and multiplicative schemes to diffusion processes. However, they are not sufficient to establish the rate of convergence. In this connection, we shall use the following (more restrictive) result from [108]. Denote GT,n (x) = ℙ(n) (Xn (T) ≤ x) and let Φ(x) be the standard Gaussian distribution function. Theorem 2.10 ([108]). Assume that Xn (0) = 0 and that the following conditions are satisfied: (E1) (i) n

󵄨 󵄨 lim Jn1 := lim ∑ 𝔼n (󵄨󵄨󵄨Qkn 󵄨󵄨󵄨1|Qk |>1 ) = 0, n n n k=1

(ii) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 lim Jn2 := lim 𝔼n 󵄨󵄨󵄨 ∑ 𝔼n (Qkn 1|Qk |≤1 | ℱnk−1 )󵄨󵄨󵄨 = 0, n n n 󵄨󵄨 󵄨󵄨 1≤k≤n (iii) 󵄨󵄨 󵄨󵄨 2 󵄨 󵄨 lim Jn3 := lim 𝔼n 󵄨󵄨󵄨 ∑ 𝔼n ((Qkn ) 1|Qk |≤1 | ℱnk−1 ) − V 󵄨󵄨󵄨 = 0, n n n 󵄨󵄨 󵄨󵄨 1≤k≤n T

(iv)

where V = ∫0 c2 (t)dt > 0, and n

󵄨 󵄨3 lim Jn4 := lim ∑ 𝔼n (󵄨󵄨󵄨Qkn 󵄨󵄨󵄨 1|Qk |≤1 ) = 0. n

n

n

k=1

Then GT,n (x) → Φ( √x ) with the following rate of convergence: V

󵄨󵄨 x 󵄨󵄨󵄨󵄨 󵄨 sup󵄨󵄨󵄨GT,n (x) − Φ( )󵄨 √V 󵄨󵄨󵄨 x∈ℝ 󵄨󵄨 1

1

1

≤ C1 ((Jn1 + Jn2 ) 2 + (Jn3 ) 3 + (Jn4 ) 4 ),

where C1 = 2 +

2 13√2 12 + (√2π + 3/2 )V −1/2 + 2 V −1 . π π π

(2.26)

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 135

Remark 2.11. It is easy to see that condition (E1)(i) implies (1.73), condition (E1)(ii) implies (1.74) with b ≡ 0, and conditions (E1)(iii)–(iv) imply (1.75). Therefore weak convergence of measures ℙ(n) , ℙ

(Xn (t), t ∈ 𝕋) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ 𝕋),

n→∞

in the Skorokhod topology on the space D(𝕋) is immediate and evident. But the part of the proof that concerns the rate of convergence is quite technical and cumbersome; therefore we omit it here. For a detailed proof, see [108]. 2.1.4.2 Discrete multiplicative scheme of financial markets. Conditions of convergence with the rate of convergence of distribution functions Let T > 0 be given, 𝕋 = [0, T]. Consider the sequence of discrete-time financial markets consisting of two assets, a non-risky and a risky one (bond and stock), acting on 𝕋, with maturity date T. We suppose that the non-risky asset admits the representation ⌊ nt ⌋ T

rT Bn (t) = (1 + ) n

,

(2.27)

where r ≥ 0, n ≥ 1. Assume that the risky asset admits the representation Sn (t) =

∏ (1 + Rkn ),

1≤k≤[ nt ] T

(2.28)

where {Rkn > −1, 1 ≤ k ≤ n} are random variables on the probability space (Ω(n) , ℱ (n) , ℙ(n) ), n ≥ 1. Introduce the σ-fields ℱn0 = {0, Ω}, ℱnk = σ{Rin , 1 ≤ i ≤ k}. Without any doubt, Bn (t) → exp{rt}. Avoiding the problems with Bn , we are in a position to present the conditions of weak convergence of the multiplicative model (2.28) to the limit process of the form 1 S(t) = exp{rt + σW(t) − σ 2 t}, 2 together with the rate of convergence. Denote by ℚ and ℚ(n) , n ≥ 1, the measures that correspond to the processes S and Sn , n ≥ 1, respectively. Having in mind only mild conditions for weak convergence, without the rate of convergence, we can take Theorem 1.62, or even simply Theorem 1.41, as a starting point. However, in order to give the rate of convergence for the multiplicative scheme, we have to modify basic conditions. Denote 1 FT,n (x) = ℙ(n) (log Sn (T) − rT + σ 2 T ≤ x). 2

136 | 2 Rate of convergence of asset and option prices Theorem 2.12. Assume the following conditions are satisfied: (E2) (i) There exists such 0 > δ > −1 that Rkn ≥ δ, ℙ(n) -a. s., (ii) 󵄨 󵄨 lim Kn1 := lim ∑ 𝔼n (󵄨󵄨󵄨Rkn 󵄨󵄨󵄨1|Rk |> 1 ) = 0, n→∞ n 2

n→∞

1≤k≤n

(iii) 󵄨󵄨 󵄨󵄨 󵄨 󵄨 lim Kn2 := lim 𝔼n 󵄨󵄨󵄨 ∑ 𝔼n (Rkn 1|Rk |≤ 1 | ℱnk−1 ) − rT 󵄨󵄨󵄨 = 0, n→∞ n→∞ n 󵄨󵄨 󵄨󵄨 2 1≤k≤n (iv) for some σ > 0, 󵄨󵄨 󵄨󵄨 2 󵄨 󵄨 lim Kn3 := lim 𝔼n 󵄨󵄨󵄨 ∑ 𝔼n ((Rkn ) 1|Rk |≤ 1 | ℱnk−1 ) − σ 2 T 󵄨󵄨󵄨 = 0, n→∞ n→∞ n 󵄨󵄨 󵄨󵄨 2 1≤k≤n (v) 󵄨 󵄨3 lim Kn4 := lim ∑ 𝔼n (󵄨󵄨󵄨Rkn 󵄨󵄨󵄨 1|Rk |≤1 ) = 0. n n→∞

n→∞

1≤k≤n

Then ℚ(n) , ℚ

(Sn (t), t ∈ 𝕋) 󳨐󳨐󳨐󳨐󳨐⇒ (S(t), t ∈ 𝕋),

n → ∞,

in the Skorokhod topology on the space D(𝕋), and, moreover, 󵄨󵄨 󵄨󵄨 x 󵄨 󵄨 )󵄨󵄨󵄨 sup󵄨󵄨󵄨FT,n (x) − Φ( x 󵄨󵄨 σ √T 󵄨󵄨 ≤ Θn

1

2 1 1 󵄨 󵄨 13 := C1 (((2󵄨󵄨󵄨log(1 + δ)󵄨󵄨󵄨 + )Kn1 + Kn2 + Kn3 + Kn4 ) 3 2 2 3(1 + δ)

1 2|b|σ 2 T 2 2|b|T + ( (b2 T 2 + 2|b|rT 2 + ) + (1 + 2 n (1 + δ) n(1 + δ)2 4|b|T 2|b|T 2 1 2|b|T + )Kn1 + Kn + (1 + )K 3 n n 2 n(1 + δ)2 n 1

+(

1

3 8|bT|3 4 1 1 + )Kn4 ) + (8Kn4 + ) ), 4 2 (1 + δ) (1 + δ) n2

where b := − 21 σ 2 + r and C1 is defined in (2.26).

(2.29)

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model | 137

Proof. Denote Qkn = log(1 + Rkn ) − 1 − log 2. Then

bT n

and consider only such n ≥ 1 for which

|bT| n


1 = (log(1 + Rn ) − Rn >e −1 n bT + (− log(1 + Rkn ) + )1 k −1+ bTn . δe1+

bT n

−1

≤ 1Rk >1 ≤ 1|Rk |>1 . n

n

Moreover, we apply the evident inequality log(1 + x) ≤ x, x > −1 and the fact that for |bT| < 1 − log 2 we have |bT| < |Rkn | on the set {|Rkn | > 1}, obtaining n n (log(1 + Rkn ) − Furthermore, for

|bT| n

bT 󵄨 󵄨 󵄨 󵄨 )1 k 1+ bTn ≤ 2󵄨󵄨󵄨Rkn 󵄨󵄨󵄨1|Rk |>1 ≤ 2󵄨󵄨󵄨Rkn 󵄨󵄨󵄨1|Rk |> 1 . n −1 Rn >e n 2 n bT

< 1 − log 2 we have e−1+ n − 1 < − 21 ; therefore,

bT )1 k −1+ bTn δ 0 0 < (1 +

x ) n

−n

− e−x ≤

x2 . 2n

(2.38)

Second, integrating by parts we deduce that for any integrable non-negative random variable ξ K

𝔼(K − ξ )+ = ∫ ℙ(ξ ≤ x)dx. 0

Taking this into account, we can produce the following relations: −n 󵄨󵄨 󵄨 rT + + 󵄨󵄨 put 󵄨 󵄨 󵄨󵄨 put 󵄨󵄨πn (x) − π (x)󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼n (K − Sn (T)) (1 + ) − 𝔼(K − S(T)) exp{−rT}󵄨󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 K 󵄨󵄨 ≤ 󵄨󵄨󵄨∫ ℙ(n) (Sn (T) ≤ y)dy 󵄨󵄨 󵄨0 K

󵄨󵄨 −n 󵄨󵄨 log y − bT rT )dy󵄨󵄨󵄨 + K((1 + ) − exp{−rT}) − ∫ Φ( 󵄨󵄨 n σ √T 󵄨 0 󵄨󵄨 K(rT)2 log x − bT 󵄨󵄨󵄨󵄨 󵄨 ≤ + K sup󵄨󵄨󵄨ℙ(n) (Sn (T) ≤ x) − Φ( )󵄨󵄨, 󵄨󵄨 2n x 󵄨󵄨 σ √T

(2.39)

and (2.37) immediately follows from (2.39) and (2.36).

Remark 2.17. Evidently, under assumptions of Example 2.15, the rate of convergence of option prices can be bounded by n−1/8 up to a constant. 2.1.4.4 The rate of convergence of option prices for the pre-limit market created by independent Rnk with continuous distribution function Suppose that in each series {Rkn , 1 ≤ k ≤ n} are i. i. d. random variables with respect to the measure ℙ(n) . In this case, under some additional assumptions we can apply

142 | 2 Rate of convergence of asset and option prices asymptotic expansion of the distribution function of the sum of i. i. d. random variables that gives a more exact result than the martingale CLT, considered together with the rate of convergence in the previous subsections 2.1.4.2 and 2.1.4.3. Denote by vn (t) = 𝔼n exp{it log(1 + Rkn )} the characteristic function of the price jump. We assume that rnk = rn = in each series.

rT , n

1 ≤ k ≤ n,

Theorem 2.18. Assume that {Rkn , 1 ≤ k ≤ n} are i. i. d. random variables with respect to the measure ℙ(n) and the following conditions are satisfied: (i) there exist moments k m(1) n = 𝔼n log(1 + Rn ),

2

2

k (1) m(2) n = 𝔼n (log(1 + Rn )) − (mn ) ,

and 󵄨󵄨 k (1) 󵄨󵄨3 m(3) n = 𝔼n (󵄨󵄨log(1 + Rn ) − mn 󵄨󵄨 ), (ii) for any n > 1, lim supt→∞ |vn (t)| < 1, (2) 3/2 (iii) m(3) n = O((mn ) ) as n → ∞,

−1/2 √ (2) √ (iv) |bT − nm(1) ). n | + |σ T − nmn | = O(n

Then there exists C > 0 such that |πncall − π call | ≤ Cn−1/2 . Proof. Consider the sum Σn = (nm(2) n )

−1/2

( ∑ log(1 + Rkn ) − nm(1) n ) 1≤k≤n

and introduce the notation Qn (x) =

3 𝔼 (log(1 + Rkn ) − m(1) 1 − x22 n ) e (1 − x2 ) n . 3/2 √2π 6(m(2) n )

Let Fn (x) = ℙ(n) (log(1 + R1n ) − 𝔼n log(1 + R1n ) ≤ x). According to Theorem A.112, under condition (i) we have 󵄨󵄨 Q (x) 󵄨󵄨󵄨 󵄨󵄨 (n) 1 2 3 󵄨󵄨ℙ (Σn ≤ x) − Φ(x) − n 󵄨󵄨󵄨 = C(Kn + Kn + Kn ), 󵄨󵄨 √n 󵄨󵄨 where Kn1 = (m(2) n )

− 32 − 1 2

n (1 + |x|)

−3

∫ |z|≥√nm(2) n (1+|x|)

|y|3 dFn (y),

(2.40)

2.1 The rate of convergence of option prices when the limit is a Black–Scholes model |

−1 Kn2 = (m(2) n ) n (1 + |x|) −2

−4

143

|y|4 dFn (y),

∫ |y|≤√nm(2) n (1+|x|)

n

−4 󵄨 󵄨 1 Kn3 = (sup 󵄨󵄨󵄨fn (t)󵄨󵄨󵄨 + ) n6 (1 + |x|) , 2n |t|≥ςn

ςn =

m(2) n

12m(3) n

,

(1) fn (t) = 𝔼n exp{it(log(1 + Rkn ) − m(1) n } = vn (t) exp{−itmn }.

Under condition (iii), we have m(3) |Qn (x)| n ≤ = O(n−1 ). 3/2 n 6n(m(2) ) n

(2.41)

Let us study the terms Kn1 , Kn2 , and Kn3 . Note that m(1) n =

bT + O(n−3/2 ) = O(n−1 ), n 3/2

(2) m(3) n = O((mn )

m(2) n =

) = O(n−3/2 ),

σ2 T + O(n−2 ) = O(n−1 ), n

by conditions (iii)–(iv). Then Kn1 ≤ (m(2) n ) =

− 32 − 1 2

n (1 + |x|)

−3

− 32 − 1 2 (m(2) n ) n (1

∫ |y|3 dFn (y) ℝ

−1/2 + |x|) m(3) ). n = O(n

(2.42)

−3

Similarly, −2 −1 −3 3 √nm(2) Kn2 ≤ (m(2) n ∫ |y| dFn (y) n ) n (1 + |x|) ℝ

3

=

−2 −1 2 (m(2) n ) n (1

−3

+ |x|)

m(3) n

= O(n−1/2 ).

(2.43)

Let us consider Kn3 . By assumption (iii), we have 󵄨 󵄨 󵄨 󵄨 lim sup󵄨󵄨󵄨fn (t)󵄨󵄨󵄨 ≤ lim sup󵄨󵄨󵄨vn (t)󵄨󵄨󵄨 < 1 t→∞

for any n ≥ 1. Note that

m(3) n m(2) n

t→∞

1/2 = O((m(2) n ) ) → 0 as n → ∞. Hence, ςn =

n → ∞. Therefore, for sufficiently large n,

m(2) n 12m(3) n

→ ∞ as

󵄨 󵄨 sup 󵄨󵄨󵄨fn (t)󵄨󵄨󵄨 < 1.

|t|≥ςn

1 n ) decreases faster than n−p for any p > 0. Consequently, the factor (sup|t|≥ςn |fn (t)| + 2n So Kn3 = o(n−1/2 ), n → ∞. Combining this relation with (2.40)–(2.43), we arrive at

󵄨󵄨 (n) 󵄨 −1/2 󵄨󵄨ℙ (Σn ≤ x) − Φ(x)󵄨󵄨󵄨 = O(n )

144 | 2 Rate of convergence of asset and option prices uniformly in x. In other words, 󵄨󵄨 󵄨󵄨 log x − nm(1) 󵄨󵄨 󵄨󵄨 (n) −1/2 n ) 󵄨󵄨 = O(n ). 󵄨󵄨ℙ (Sn (T) ≤ x) − Φ( 1/2 󵄨󵄨 󵄨󵄨 (nm(2) ) n As in the proof of Theorem 2.16, taking (2.38) into account, we consider put options: −n 󵄨󵄨 󵄨󵄨󵄨 rT + 󵄨 󵄨󵄨 put put 󵄨 󵄨󵄨πn − π 󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼n (K − Sn (T))(1 + ) − 𝔼(K − S(T)) exp{−rT}󵄨󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 K −n 󵄨󵄨 rT = 󵄨󵄨󵄨∫ ℙ(n) (Sn (T) ≤ y)dy(1 + ) 󵄨󵄨 n 󵄨0 󵄨󵄨 󵄨󵄨 log y − bT )dy exp{−rT}󵄨󵄨󵄨 − Φ( 󵄨󵄨 σ √T 󵄨 2 󵄨 󵄨󵄨 󵄨󵄨 (n) log x − nm(1) K(rT) n 󵄨󵄨󵄨 + K sup󵄨󵄨󵄨ℙ (Sn (T) ≤ x) − Φ( ≤ ) 󵄨󵄨 1/2 2n x 󵄨󵄨 󵄨 (nm(2) ) n

(2.44)

K 󵄨󵄨 󵄨󵄨 K 󵄨󵄨 󵄨󵄨 log y − nm(1) log y − bT n 󵄨 )dy󵄨󵄨󵄨. )dy − ∫ Φ( + 󵄨󵄨∫ Φ( (2) 1/2 󵄨󵄨 󵄨󵄨 √ σ T (nmn ) 󵄨 󵄨0 0

It is sufficient to bound the last term in (2.44). Consider such n for which 1/2

(nm(2) n )

1 > σ √T 2

and denote C = supx∈[α,β] ex , where [α, β] is the interval with endpoints log K−bT . σ √T

log K−nm(1) n 1/2 (nm(2) n )

and

Under condition (iv) these endpoints are totally bounded, and therefore C is some constant. Then for n mentioned above we have K 󵄨󵄨 K 󵄨󵄨 (1) 󵄨󵄨 󵄨 󵄨󵄨∫ Φ( log y − bT )dy − ∫ Φ( log y − nmn )dy󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 (2) σ √T 󵄨󵄨 󵄨󵄨 (nmn )1/2 0 0 } exp{ logσK−bT √T

exp{

(1) log K−nmn (2) (nmn )1/2

}

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 Φ(log y)dy − Φ(log y)dy󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 ∫ ∫ 󵄨󵄨 󵄨󵄨 󵄨 󵄨 0 0 󵄨󵄨 󵄨󵄨 log K − nm(1) log K − bT 󵄨 󵄨 n } − exp{ }󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨exp{ (2) 1/2 󵄨󵄨 󵄨󵄨 σ √T (nmn ) (1) 󵄨󵄨󵄨 log K − bT log K − nmn 󵄨󵄨󵄨 󵄨󵄨 − ≤ C 󵄨󵄨󵄨 1/2 󵄨󵄨󵄨 󵄨󵄨 σ √T (nm(2) n ) 2C 1/2 󵄨 󵄨 󵄨 󵄨󵄨 ≤ 2 ((| log K| + |b|T)󵄨󵄨󵄨(nm(2) − σ √T 󵄨󵄨󵄨 + σ √T 󵄨󵄨󵄨nm(1) n ) n − bT 󵄨󵄨), σ T whence the proof follows with the same considerations as in Theorem 2.18.

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 145

c and Rkn has a distribution function with non-zero Example 2.19. Assume |Rkn | ≤ √n absolutely continuous component. Then conditions (i) and (ii) hold; see, e. g., [136]. Suppose additionally that 𝔼n Rkn = rn , 𝔼n (Rkn )2 = (σn )2 , and 𝔼n (Rkn )3 = (μn )3 satisfy the relations

󵄨 󵄨 |nrn − rT| + 󵄨󵄨󵄨n(σn )2 − σ 2 T 󵄨󵄨󵄨 = O(n−1/2 ) μ

and | σn | ≤ c. By very easy computations we can check that this is the case. Conditions n (iv) and (v) are satisfied.

2.2 The rate of convergence of option prices on the asset following the geometric Ornstein–Uhlenbeck process In this section we consider the asset that is modeled by the geometric Ornstein– Uhlenbeck process (Vasicek model); see Section 1.7. The sequence of pre-limit markets is modeled as the sequence of the discrete-time multiplicative stochastic processes. To construct these multiplicative processes, we take the Euler approximations of the Ornstein–Uhlenbeck process itself but replace the increments of the Wiener process with i. i. d. bounded vanishing random variables, not obligatory Bernoulli variables, as stated in Section 1.7. So, we slightly generalize the scheme considered in Section 1.7. Then, similarly to Section 1.7, it is shown that starting from the certain number, the pre-limit market is arbitrage-free. However, contrary to Section 1.7, now the pre-limit market can be incomplete. The limit market is arbitrage-free and complete. We connect the sequence of multiplicative discrete-time asset prices with the additive discrete-time stochastic processes constructed with the help of the modified Euler approximations so that they weakly converge in measure to the geometric Ornstein–Uhlenbeck process. The connection is so close that the multiplicative processes converge simultaneously with the additive ones. At the same time, additive processes can be reduced to the sums of non-identically distributed independent random variables. The rate of convergence of option prices is estimated via additive discrete-time stochastic processes and with the help of the classical results from [136] describing the rate of convergence to the normal law of the distribution function of the sum of non-identically distributed random variables. Section 2.2 is organized as follows. In Section 2.2.1 we recall what is the limit process. It is known from Lemma 1.87 that the limit market is arbitrage-free and complete. In Section 2.2.2 we describe pre-limit models of asset prices connected to the geometric Ornstein–Uhlenbeck process more generally than in Section 1.7 and establish that pre-limit markets are arbitrage-free and, generally speaking, incomplete, unlike the particular “binomial” case, where the jumps have a binomial distribution. To fix one probability measure in the conditions of incompleteness, we consider the minimal martingale measure. Also, we prove a theorem which states the weak convergence in

146 | 2 Rate of convergence of asset and option prices measure of asset prices, together with the rate of convergence that depends on the distribution of the jump. Furthermore, it is proved in Section 2.2.5 that in the case of the uniform distribution of the jump, and with respect to the minimal martingale measure, the rate of convergence of objective option prices for call and put options can be bounded from above by nC1/3 , while in the case of a Bernoulli distribution of the jump

it can be nC1/2 . The main tool is the theorem from Petrov’s book [136] about the rate of convergence of the sums of non-identically distributed independent random variables to the normal distribution. It is contained in the appendix; see Theorem A.114 and Lemma A.115. Section 2.2.6 refers to the fair prices and Bernoulli jumps. Here also the rate is nC1/2 . 2.2.1 Brief discussion of the limit Ornstein–Uhlenbeck asset price process

Let us recall and summarize the properties of the limit model; see also Section 1.7.1. Let T > 0, 𝕋 = [0, T], and let Ωℱ = (Ω, ℱ , 𝔽 = {ℱt , t ∈ 𝕋}, ℙ) be a complete filtered probability space satisfying the standard assumptions. Let W = {W(t), ℱt , t ∈ 𝕋} be an adapted Wiener process. Consider an adapted Ornstein–Uhlenbeck process with constant parameters on this stochastic basis. It is described as the unique strong solution to the following stochastic differential equation: dX(t) = (μ − X(t))dt + σdW(t),

X0 = x0 ∈ ℝ,

t ∈ 𝕋,

(2.45)

where μ ∈ ℝ and σ > 0. An explicit formula for X is t

X(t) = x0 e−t + μ(1 − e−t ) + σe−t ∫ es dW(s).

(2.46)

0

Assume that the non-discounted asset price S(t) follows the equality S(t) = exp{X(t) −

σ2 t}, 2

t ∈ 𝕋,

(2.47)

2

where the non-random term − σ2 t is added for technical simplicity. The respective discounted asset price Y(t) follows the equality Y(t) = exp{X(t) −

σ2 t − rt}, 2

t ∈ 𝕋.

It was proved in Lemma 1.65 that the market with bond B(t) = ert and stock S(t) is arbitrage-free and complete. Furthermore, the unique probability measure ℙ∗ ∼ ℙ

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 147

such that with respect to ℙ∗ , Y(t) is an ℱt -martingale has the Radon–Nikodym deriva∗ | , where tive of the form dℙ dℙ ℱT t

t

0

0

μ − r − X(s) dℙ∗ 󵄨󵄨󵄨󵄨 1 (μ − r − X(s))2 dW(s) − ∫ ds}, 󵄨󵄨 = exp{− ∫ dℙ 󵄨󵄨ℱt σ 2 σ2 2

̃ − σ t} with and with respect to ℙ∗ the process Y(t) has the form Y(t) = exp{x0 + σ W(t) 2 ̃ some Wiener process W. 2.2.2 Description and properties of the pre-limit discrete-time price processes Now we are in a position to construct a discrete scheme that weakly converges to the geometric Ornstein–Uhlenbeck process defined by (2.45)–(2.47). Similarly to Section 1.7.1, we consider the following discrete approximation scheme for the Ornstein– Uhlenbeck process that is based on Euler approximations of the solution of the stochastic differential equation (2.45) but the increments of the Wiener process are replaced with bounded vanishing i. i. d. random variables. Namely, assume that we have a sequence of probability spaces (Ω(n) , ℱ (n) , ℙ(n) ), n ≥ 1, and let {qnk , n ≥ 1, 0 ≤ k ≤ n} be the sequence of i. i. d. random variables defined on the corresponding probability space, satisfying the following condition: L1 (F) Each qnk is bounded; more precisely, |qnk | ≤ √n , 𝔼n qnk = 0, 𝔼n (qnk )3 = 0, and 𝔼n (qnk )2 =

L2 , n

where the constants L1 > 0 and L2 > 0 do not depend on n.

As an example, {qnk , n ≥ 1, 0 ≤ k ≤ n} can be uniformly distributed on the interval

[−√ 3Ln 2 , √ 3Ln 2 ], or they can have a Bernoulli distribution of the form qnk = ±√ Ln2 with

probability 21 . However, let us emphasize the distinction with Section 1.7.1, where the distribution of qnk was restricted only to the Bernoulli distribution. Let n > T. We introduce the recurrent scheme: xn0 ∈ ℝ,

Rkn := xnk − xnk−1 =

(μ − xnk−1 )T + σqnk , n

1 ≤ k ≤ n.

(2.48)

Let ℱn0 = {0, Ω(n) } and ℱnk = σ{Rin , 1 ≤ i ≤ k}. Denote 𝔽(n) = {ℱnk , 1 ≤ k ≤ n}, Xn (t) = xn0 +

∑

1≤k≤⌊ nt ⌋ T

Rkn = xn (⌊

tn ⌋)1t≥ T . n T

Now we construct the corresponding multiplicative scheme for the pre-limit price process as Sn (t) = exp{xn0 }

∏ (1 + Rkn ),

1≤k≤⌊ nt ⌋ T

t ∈ 𝕋.

(2.49)

148 | 2 Rate of convergence of asset and option prices As usual, ∑1≤k≤0 = 0, ∏1≤k≤0 = 1. The following simple result guarantees that Sn (t) > 0 for any t ∈ 𝕋. Lemma 2.20. Let condition (F) hold and let the sequence xn0 be bounded. Then the next statements hold: (a) There exists such number n0 ∈ ℕ and a constant C > 0 not depending on n that for C < 1. any n > n0 and for any 1 ≤ k ≤ n we have |Rkn | ≤ √n

(b) There exists a constant C > 0 not depending on n such that 𝔼n (xnk )4 ≤ C for n > T and 1 ≤ k ≤ n. Proof. (a) Let 0 < xn0 < c. It follows from (2.48) that xnk = xn0 (1 −

k

k

k−i

k T T T ) + μ(1 − (1 − ) ) + σ ∑ qni (1 − ) n n n i=1

,

(2.50)

whence xnk − μ = (xn0 − μ)(1 −

k

k−i

k T T ) + σ ∑ qni (1 − ) n n i=1

.

(2.51)

So, for any 1 ≤ k ≤ n, i

L k−1 T 󵄨󵄨 k 󵄨 󵄨󵄨xn − μ󵄨󵄨󵄨 ≤ (c + μ) + σ 1 ∑ (1 − ) √n i=0 n =c+μ+σ

k

L1 √n L √n T (1 − (1 − ) ) ≤ c + μ + σ 1 . T n T C √n

Therefore, for sufficiently large n, |xnk − μ| ≤ C√n and |Rkn | ≤ the proof follows. (b) Note that 𝔼(qni )4 ≤

L21 L2 . n2

(2.52)

for some C > 0, whence

Then it follows immediately from (2.50), the fact that

∑ki=1 qni (1− Tn )k−i creates a martingale, and the Burkholder–Gundy inequality that there exists C > 0 such that 4 𝔼n (xnk )

4 4

≤C+2 σ

k

𝔼n (∑ qni (1 i=1

4 4

≤ C + C2 σ k

k

4

k−i

T − ) n

2 𝔼n (∑(qni ) ) i=1

)

2

4

2

2

≤ C(1 + ∑ 𝔼n (qni ) + 𝔼n ∑(qni ) (qnk ) ) i=1

i=k̸

L2 L T2 ≤ C(1 + n ⋅ 1 2 2 + n2 ⋅ 2 ) ≤ C, n n and the lemma is proved.

(2.53)

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 149

Remark 2.21. We can clarify the bounds (2.52) as follows: k

L √n T 󵄨󵄨 k 󵄨 󵄨󵄨xn − μ󵄨󵄨󵄨 ≤ c + μ + σ 1 (1 − (1 − ) ) T n ≤c+μ+σ

n

L1 √n T (1 − (1 − ) ). T n

(2.54)

Note that for any ε > 0 there exists such n1 > 0 that for n ≥ n1 we have n

(1 −

T e−T T ) ≥ e−T e− n−T ≥ , n 1+ε

whence for such n L √n e−T 󵄨󵄨 k 󵄨 ). 󵄨󵄨xn − μ󵄨󵄨󵄨 ≤ c + μ + σ 1 (1 − T 1+ε

(2.55)

2.2.3 Incompleteness of the pre-limit market It was established in Section 1.7.2 that in the case when qnk have a Bernoulli distribution, the market with bond Bn (t) = ∏1≤k≤⌊ nt ⌋ (1 + rnk ) and stock Sn (t) from (2.49) is T arbitrage-free and complete, starting with some number n, under additional assump1 ) and |xn0 | ≤ C. Furthermore, it was established that the unique equivations rnk = o( √n lent martingale measure ℙ(n,∗) ∼ ℙ(n) has the Radon–Nikodym derivative n dℙ(n,∗) k = (1 + ρk−1 ∏ n qn ), dℙ(n) k=1

(2.56)

where the random variables ρk−1 n have the representation ρk−1 n =

nrnk − (μ − xnk−1 )T . σT

(2.57)

In the case of other distributions of qnk , the situation is more involved. We describe it with the help of notion of the minimal martingale measure; see Definitions A.30–A.31 for discrete time, Definition A.56 for continuous time, and Theorem A.32, again for discrete time. Note also that it was proved, e. g., in [59, 60], to which we refer the interested reader, that the minimal martingale measure is unique if it exists. 1 Theorem 2.22. Consider the model (2.48). Let condition (F) hold with rnk = o( √n ) uni-

formly in k and |xn0 | ≤ c. Let, moreover, following statements hold.

L21 (1 T

− e−T ) < 1 and

L41 (1 TL22

− e−T ) < 1. Then the

150 | 2 Rate of convergence of asset and option prices (i) Market (Bn (t), Sn (t)),

n ≥ 1,

t ∈ [0, T],

is asymptotically arbitrage-free, which means that there exists such n2 ∈ ℕ that (Bn (t), Sn (t)) is arbitrage-free for any n ≥ n2 . (ii) For such n ≥ n0 the market (Bn (t), Sn (t)) is, generally speaking, incomplete. More precisely, suppose that starting from n0 the distribution of qnk differs from the Bernoulli distribution. Then the pre-limit market is incomplete. (iii) The unique minimal martingale measure ℙ(n,∗) has a Radon–Nikodym derivative of m the form (2.56)–(2.57). Remark 2.23. Conditions

L21 (1−e−T ) T

L4

< 1 and TL12 (1−e−T ) < 1 are executed automatically 2

in the case of a Bernoulli distribution of the form qnk = ±√ Tn with probability 21 because in this case L1 = √T and L2 = T. In other cases it can be a limitation for the values of L1 , L2 , and T.

Proof. (i) If the equivalent martingale measure exists, then its Radon–Nikodym derivative has the form n dℙ(n,∗) = (1 + ΔMnk ), ∏ dℙ(n) k=1

(2.58)

where {Mnk , 1 ≤ k ≤ n} is some 𝔽(n) -martingale, ΔMnk > −1. One of the possibilities is k to put ΔMnk = ρk−1 n qn , and in this case we immediately get, similarly to Theorems 1.64 k and 1.66, equality (2.57). The only thing that we have to check is the inequality ρk−1 n qn > −1, and in this case (2.56) and (2.57) give us the martingale measure. In order to check 1 the above inequality, we use the assumption that rnk = o( √n ) uniformly in k, and we immediately obtain that for n exceeding n0 ∨ n1 , where n0 is taken from Lemma 2.20 and n1 is taken from Remark 2.21, nr k |μ − xnk−1 | L1 󵄨󵄨 k−1 k 󵄨󵄨 ) 󵄨󵄨ρn qn 󵄨󵄨 ≤ ( n + √n σT σ

L √nrnk L1 c + μ + σ 1T (1 − ≤ + σT σ √n

e−T ) 1+ε

L2 L1 e−T ), ≤ o(1) + 1 (1 − √n T 1+ε

where o(1) is a vanishing non-random sequence. Taking into account the fact that ε > L2

0 can be chosen sufficiently small, under condition T1 (1 − e−T ) < 1 we get the existence of the martingale measure for any market (Bn (t), Sn (t)) starting from some number (it equals n0 ∨ n1 in our case). (ii) In order to establish incompleteness of the market (Bn (t), Sn (t)), construct the measure having the Radon–Nikodym derivative of the form (2.58) but different from (2.56)–(2.57). In this connection, put ΔMnk = αnk−1 (qnk )3 , which is the martingale

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 151

increment due to the fact that 𝔼(qnk )3 = 0. Then, similarly to (2.57), we get the following equality for αnk−1 : αnk−1 =

nrnk − (μ − xnk−1 )T , σnm4n

where m4n = 𝔼(qnk )4 . If the distribution of qnk differs from the Bernoulli distribution, k then the equality αn0 (qnk )3 = ρk−1 n qn a. s., which is equivalent to the equality 2

(qnk ) =

𝔼n (qnk )4 nm4n 1qk =0̸ = n T (𝔼n (qnk )2 )2

a. s.,

is impossible. Now it is sufficient to establish that αnk−1 (qnk )3 > −1 starting from some number n0 . Note that 2 2

m4n ≥ (𝔼(qnk ) ) =

L22 . n2

1 Therefore, taking into account that rnk = o( √n ) uniformly in k, we get for any ε > 0 that for sufficiently large n L1 k 󵄨󵄨 k−1 k 3 󵄨󵄨 nrn + (c + μ + σ T (1 − max 󵄨󵄨αn (qn ) 󵄨󵄨 ≤ 1≤k≤n σnm4 √n

≤

nrnk

+ (c +

n L1 √n μ + σ T (1 σn1/2 L22

−

e−T )) 1+ε

L31 n3/2

e−T )) 3 1+ε L1

= o(1) +

L41 e−T (1 − ), 1+ε TL22

and this value is less than 1 for sufficiently small ε and consequently for sufficiently big n. Therefore, we indeed constructed the second martingale measure. (iii) In our case the price process has the form (2.49) with Rkn given by equality (2.48). It means that martingale N = Nn from Theorem A.32 has the form Nnk = ∑ki=1 βni qni with βni that is ℱni−1 -measurable. Now the proof immediately follows from Theorem A.32.

2.2.4 Weak convergence of asset price, with the rate of convergence Let ℚ(n) be the probability measure corresponding to the process Sn from (2.49) and let ℚ be the probability measure corresponding to the process S from (2.47). Let the following condition hold: xn0 → x0 , n → ∞. Then, according to Theorem 1.63, we have weak convergence of the probability measures ℚ(n) , ℚ

(Sn (t), t ∈ 𝕋) 󳨐󳨐󳨐󳨐󳨐⇒ (S(t), t ∈ 𝕋),

n → ∞.

152 | 2 Rate of convergence of asset and option prices In particular, it was established that 󵄨󵄨 󵄨󵄨 2 󵄨 󵄨 sup 󵄨󵄨󵄨 ∑ 𝔼n ((Rkn ) | ℱnk−1 ) − σ 2 t 󵄨󵄨󵄨 → 0 󵄨󵄨 t∈[0,T]󵄨󵄨 nt 1≤k≤⌊ ⌋ T

in probability. However, it is not sufficient for the estimation of the rate of convergence, and now we are in a position to establish that ∑1≤k≤n (Rkn )2 → σ 2 T in ℒ2 (ℙ), together with the rate of convergence. Lemma 2.24. Under condition (F) for n > T 2

2

𝔼n ( ∑ (Rkn ) − σ 2 T) ≤ 1≤k≤n

C T2 4 + Cn(𝔼(qn1 ) − 2 ). 2 n n

Proof. We start with the following evident transformation: 2

2

Σn := 𝔼n ( ∑ (Rkn ) − σ 2 T) 1≤k≤n

2

2

= 𝔼n ( ∑ (

(μ − xnk−1 )T + σqnk ) − σ 2 T) n

= 𝔼n ( ∑ (

(μ − xnk−1 )T (μ − xnk−1 )T k ) + 2σ ∑ qn n n 1≤k≤n

1≤k≤n

2

1≤k≤n

2

2

+ σ 2 ∑ (qnk ) − σ 2 T) 1≤k≤n

2

= 𝔼n ( ∑ ( 1≤k≤n

(μ − xnk−1 )T k (μ − xnk−1 )T ) + 2σ ∑ qn n n 1≤k≤n 2

+ σ 2 ∑ ((qnk ) − 1≤k≤n

≤ 3𝔼n ( ∑ ( 1≤k≤n

2

T )) n

2 2

2

(μ − xnk−1 )T (μ − xnk−1 )T k ) ) + 12σ 2 𝔼( ∑ qn ) n n 1≤k≤n 2

+ 3σ 4 𝔼n ( ∑ ((qnk ) − 1≤k≤n

2

T )) . n

Now, taking into account (2.53) and the elementary inequality 2

( ∑ ak ) ≤ n ∑ a2k , 1≤k≤n

1≤k≤n

we conclude that for some C > 0 2 2

𝔼n ( ∑ ( 1≤k≤n

4

(μ − xnk−1 )T (μ − xnk−1 )T ) ) ≤ n ∑ 𝔼n ( ) n n 1≤k≤n

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 153

≤ Cn2 n−4 ≤

C . n2

Recall that qnk are centered and independent and note that as evident consequence of (2.53), 𝔼n (μ − xnk−1 )2 ≤ C. Therefore, we obtain 2

2

(μ − xnk−1 )T k (μ − xnk−1 )T k qn ) = ∑ 𝔼n ( qn ) n n 1≤k≤n 1≤k≤n

𝔼n ( ∑

2

= ∑ 𝔼n ( 1≤k≤n

(μ − xnk−1 )T 2 ) 𝔼n (qnk ) ≤ Cnn−3 ≤ Cn−2 . n

At last, using again the independence of qnk and the equality 𝔼n (qnk )2 = 2

𝔼n ( ∑ ((qnk ) − 1≤k≤n

T , n

we obtain

2

T T2 4 )) = n(𝔼n (qn1 ) − 2 ), n n

and the proof follows. 2

Remark 2.25. As to the asymptotic value of 𝔼(qn1 )4 − Tn2 > 0, it depends on the distribution of qn1 . For example, it is zero in the case of a Bernoulli distribution with ℙ(n) (qn1 = ±√ Tn ) = terval

[−√ 3T , √ 3T ], n n

1 , 2

it equals

4T 2 5n2

and it equals

in the case of a uniform distribution on the in-

T2 4 4n2 +12n+5 n2

in the case when the distribution of qn1

has the density fn (x) = An x2n on the interval [−√ An =

(n + 1/2)nn+1/2

T n+1/2 ( 2n+3 )n+1/2 2n+1

Bn T B T , √ nn ], n

and

Bn =

with

2n + 3 . 2n + 1

In what follows we consider the uniform distribution as a base because of all these distributions only the uniform distribution satisfies the conditions of Theorem A.114.

2.2.5 The rate of convergence of objective option prices Denote by ℂn and ℂ standard call options and by ℙn and ℙ standard put options with strike price K ≥ 0 and maturity date T > 0 on the pre-limit and limit assets, respectively. Since the market is incomplete, we shall consider their prices with respect to call objective (physical) measures ℙ(n) and ℙ. Denote their objective prices as πn,obj (x0 ), put put call πobj (x0 ), πn,obj (x0 ), and πobj (x0 ), respectively. For technical simplicity, suppose that the bond price for the pre-limit model equals ⌊ tn ⌋ T

rT Bn (t) = (1 + ) n

(2.59)

154 | 2 Rate of convergence of asset and option prices and the limit bond price equals B(t) = ert . Furthermore, for the geometric Ornstein– Uhlenbeck process defined by (2.47), we have the following equalities: +

call πn,obj (xn0 ) = 𝔼n (exp xn0 ∏ (1 + Rkn ) − K) (1 + 1≤k≤n

rT ) , n

n ≥ 1,

rT ) , n

n ≥ 1,

−n

1 call πobj (x0 ) = 𝔼(exp{XT − σ 2 T} − K) e−rT , 2 +

+

put πn,obj (xn0 ) = 𝔼n (K − exp xn0 ∏ (1 + Rkn )) (1 + 1≤k≤n

−n

1 put πobj (x0 ) = 𝔼(K − exp{XT − σ 2 T}) e−rT . 2 +

Theorem 2.26. Let the following conditions hold: C (i) |xn0 − x0 | ≤ n1/30 with some constant C0 ≥ 0;

(ii) random variables qnk are independent and uniformly distributed on the interval , √ 3T ]. [−√ 3T n n

Then, starting with some n0 ∈ ℕ, there is an upper bound C1 󵄨󵄨 put put 0 󵄨 󵄨󵄨πobj (x0 ) − πn,obj (xn )󵄨󵄨󵄨 ≤ 1/3 n

(2.60)

call call for some C1 > 0. The same upper bound holds for |πobj (x0 ) − πn,obj (xn0 )|.

Proof. We consider only put options since they are bounded. The proof for call options follows from the put–call parity and conditions of the theorem. For technical simplicity and taking into account condition (i), suppose from now on that xn0 = x0 = 1. So, we need to bound from above the following value: −n + 󵄨󵄨 rT 󵄨 En := 󵄨󵄨󵄨𝔼n (K − exp xn0 ∏ (1 + Rkn )) (1 + ) 󵄨󵄨 n 1≤k≤n + 󵄨󵄨󵄨 1 − 𝔼(K − exp{X(T) − σ 2 T}) e−rT 󵄨󵄨󵄨. 󵄨󵄨 2

Applying inequality (2.38), we obtain after some elementary calculations that En ≤ (1 +

rT ) n

−n 󵄨

+ 󵄨󵄨 󵄨󵄨𝔼n (K − exp x 0 ∏ (1 + Rk )) n n 󵄨󵄨 󵄨 1≤k≤n +󵄨

1 − 𝔼(K − exp{X(T) − σ 2 T}) 2

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨

1 + 𝔼(K − exp{X(T) − σ 2 T}) 2

−n 󵄨󵄨 󵄨󵄨 󵄨󵄨(1 + rT ) − e−rT 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨 󵄨

+󵄨

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 155

+ 󵄨󵄨 󵄨 ≤ 󵄨󵄨󵄨𝔼n (K − exp xn0 ∏ (1 + Rkn )) 󵄨󵄨 1≤k≤n

1 − 𝔼(K − exp{X(T) − σ 2 T}) 2

󵄨󵄨 K(rT)2 󵄨󵄨 + . 󵄨󵄨 2n 󵄨

+󵄨

(2.61)

Integrating by parts we deduce as usual that for any integrable non-negative random variable ξ K

𝔼(K − ξ )+ = ∫ ℙ(ξ ≤ x)dx. 0

Therefore, +󵄨 + 󵄨󵄨 󵄨󵄨 1 2 󵄨󵄨 0 k 󵄨󵄨𝔼n (K − exp{xn } ∏ (1 + Rn )) − 𝔼(K − exp{X(T) − σ T}) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 1≤k≤n

󵄨󵄨 K 󵄨󵄨 1 = 󵄨󵄨󵄨∫(ℙ(exp{X(T) − σ 2 T} ≤ z) 󵄨󵄨 2 󵄨0 󵄨󵄨 󵄨󵄨 − ℙ(n) (exp{xn0 } ∏ (1 + Rkn ) ≤ z))dz 󵄨󵄨󵄨 󵄨󵄨 1≤k≤n 󵄨 K 󵄨󵄨󵄨 1 󵄨 = 󵄨󵄨󵄨∫(ℙ(X(T) − σ 2 T ≤ log z) 󵄨󵄨 2 󵄨0 󵄨󵄨 󵄨󵄨 − ℙ(n) (xn0 + log( ∏ (1 + Rkn )) ≤ log z))dz 󵄨󵄨󵄨 󵄨󵄨 1≤k≤n 󵄨 󵄨󵄨󵄨 K 1 󵄨 ≤ 󵄨󵄨󵄨∫(ℙ(X(T) − σ 2 T ≤ log z) 󵄨󵄨 2 󵄨0 − ℙ(n) (xn0 + ∑ Rkn − 1≤k≤n

󵄨󵄨 󵄨󵄨 1 2 ∑ (Rkn ) ≤ log z))dz 󵄨󵄨󵄨 󵄨󵄨 2 1≤k≤n 󵄨

󵄨󵄨 K 󵄨󵄨 + 󵄨󵄨󵄨∫(ℙ(n) (xn0 + ∑ log(1 + Rkn ) ≤ log z) 󵄨󵄨 1≤k≤n 󵄨0 − ℙ(n) (xn0 + ∑ Rkn − 1≤k≤n

󵄨󵄨 󵄨󵄨 1 2 ∑ (Rkn ) ≤ log z))dz 󵄨󵄨󵄨 󵄨󵄨 2 1≤k≤n 󵄨

󵄨󵄨 log K 󵄨󵄨 1 = 󵄨󵄨󵄨 ∫ ey (ℙ(X(T) − σ 2 T ≤ y) 󵄨󵄨 2 󵄨 −∞ − ℙ(n) (xn0 + ∑ Rkn − 1≤k≤n

󵄨󵄨 󵄨󵄨 1 2 ∑ (Rkn ) ≤ y))dy󵄨󵄨󵄨 󵄨󵄨 2 1≤k≤n 󵄨

156 | 2 Rate of convergence of asset and option prices 󵄨󵄨 log K 󵄨󵄨 + 󵄨󵄨󵄨 ∫ ey (ℙ(n) (xn0 + ∑ log(1 + Rkn ) ≤ y) 󵄨󵄨 1≤k≤n 󵄨 −∞ − ℙ(n) (xn0 + ∑ Rkn − =:

In1

+

In2 .

1≤k≤n

󵄨󵄨 1 2 󵄨󵄨 ∑ (Rkn ) ≤ y))dy󵄨󵄨󵄨 󵄨󵄨 2 1≤k≤n 󵄨

(2.62)

In order to bound In1 from above, let us rewrite both probabilities that it contains. Denote D(y) =

√2(y − x0 e−T − μ(1 − e−T ) + σ √1 − e−2T

σ2 T ) 2

.

Then, recalling that X(T) has a Gaussian distribution with mean x0 e−T + μ(1 − e−T ) and 2 −2T ) variance σ (1−e , we immediately obtain 2 1 ℙ(X(T) − σ 2 T ≤ y) = Φ(D(y)). 2 Furthermore, taking into account that |Φ(x) − Φ(y)| ≤ we get the following upper bound:

|x−y| √2π

and |D(y) − D(z)| ≤ C|y − z|,

󵄨󵄨 󵄨󵄨 σ2 T 1 2 󵄨󵄨 (n) 0 󵄨 k ≤ y)󵄨󵄨󵄨 ∑ (Rkn ) ≤ y) − ℙ(n) (xn0 + ∑ Rkn − 󵄨󵄨ℙ (xn + ∑ Rn − 󵄨󵄨 󵄨󵄨 2 2 1≤k≤n 1≤k≤n 1≤k≤n 󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 σ2 T 2 2 󵄨 󵄨 ≤ y) ≤ ℙ(n) (󵄨󵄨󵄨 ∑ (Rkn ) − σ 2 T 󵄨󵄨󵄨 > 1/3 ) + 󵄨󵄨󵄨ℙ(n) (xn0 + ∑ Rkn − 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 2 1≤k≤n 1≤k≤n σ2 T 1 󵄨󵄨󵄨 − ℙ(n) (xn0 + ∑ Rkn − ≤ y ± 1/3 )󵄨󵄨󵄨 󵄨󵄨 2 n 1≤k≤n 󵄨󵄨 󵄨󵄨 2 2 󵄨 󵄨 ≤ ℙ(n) (󵄨󵄨󵄨 ∑ (Rkn ) − σ 2 T 󵄨󵄨󵄨 > 1/3 ) 󵄨󵄨 󵄨󵄨 n 1≤k≤n 󵄨󵄨 󵄨󵄨 σ2 T 󵄨 󵄨 + 󵄨󵄨󵄨ℙ(n) (xn0 + ∑ Rkn − ≤ y) − Φ(D(y))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 1≤k≤n 󵄨󵄨 󵄨󵄨 1 1 σ2 T C 󵄨 󵄨 ≤ y ± 1/3 ) − Φ(D(y ± 1/3 ))󵄨󵄨󵄨 + 1/3 + 󵄨󵄨󵄨ℙ(n) (xn0 + ∑ Rkn − 󵄨󵄨 󵄨󵄨 n 2 n n 1≤k≤n =: Jn1 + Jn2 + Jn3 +

C . n1/3

It follows from (2.63) that 󵄨󵄨 󵄨󵄨 1 2 1 2 󵄨󵄨 󵄨 (n) 0 k ∑ (Rkn ) ≤ y)󵄨󵄨󵄨 󵄨󵄨ℙ(XT − σ T ≤ y) − ℙ (xn + ∑ Rn − 󵄨󵄨 󵄨󵄨 2 2 1≤k≤n 1≤k≤n 󵄨󵄨 󵄨󵄨󵄨 1 C 󵄨 ≤ 󵄨󵄨󵄨Φ(D(y)) − ℙ(n) (xn0 + ∑ Rkn − σ 2 T ≤ y)󵄨󵄨󵄨 + Jn1 + Jn2 + Jn3 + 1/3 󵄨󵄨 2 n 󵄨󵄨 1≤k≤n

(2.63)

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 157

C . n1/3

≤ Jn1 + 2Jn2 + Jn3 +

Now we obtain from Lemma 2.24 and Remark 2.25 that 2

2

Jn1 ≤ Cn2/3 𝔼n ( ∑ (Rkn ) − σ 2 T) 1≤k≤n

4

≤ C(n−4/3 + n5/3 (𝔼n (qn1 ) −

T2 )) ≤ Cn−1/3 . n2

Note that the latter upper bound gives the stated convergence rate; the others give better ones. Furthermore, since the terms Jni , i = 2, 3, are evaluated using the same arguments, we consider only Jn2 . Taking into account (2.50), we get the equalities xn0 + ∑ Rkn = xnn = xn0 (1 − 1≤k≤n

n

n

n−i

n T T T ) + μ(1 − (1 − ) ) + σ ∑ qni (1 − ) n n n i=1

.

Denote Xnk = σn1/2 qnk (1 − Tn )n−k , note that Xnk are bounded by σ √3T, and let 2

Bn = ∑ 𝔼n (Xnk ) = n 1≤k≤n

σ 2 (1 − (1 − Tn )2n ) 2−

T n

.

Then 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 k Jn2 = 󵄨󵄨󵄨ℙ(n) (B−1/2 X ≤ D (y)) − Φ(D(y)) ∑ 󵄨󵄨, n n n 󵄨󵄨 󵄨󵄨 1≤k≤n where Dn (y) =

√2 − Tn (y − μ(1 − (1 − Tn )n ) − xn0 (1 − Tn )n + σ √1 − (1 − Tn )2n

σ2 T ) 2

.

Applying inequality (2.38) and taking into account condition (i), it is easy to see that |D(y) − Dn (y)| ≤ C(1+|y|) . Therefore, taking into account that n−1/2 < n−1/3 , we get n1/2 󵄨󵄨 󵄨󵄨 C(1 + |y|) 󵄨 󵄨 Jn2 ≤ 󵄨󵄨󵄨ℙ(n) (B−1/2 . ∑ Xnk ≤ Dn (y)) − Φ(Dn (y))󵄨󵄨󵄨 + n 󵄨󵄨 󵄨󵄨 n1/3 1≤k≤n Now we are in a position to check conditions of Theorem A.114 and Lemma A.115 for Xnk . For condition (i) of Theorem A.114, lim inf n

Bn 1 − e−2T = σ2 >0 n 2

and lim sup n

1 n 󵄨󵄨 k 󵄨󵄨3 1 ∑ 𝔼󵄨X 󵄨 ≤ lim sup ⋅ n ⋅ C < ∞. n k=1 󵄨 n 󵄨 n n

158 | 2 Rate of convergence of asset and option prices Condition (ii) of Theorem A.114 evidently holds because Xnk are bounded. Instead of condition (iii) we check the inequality from Lemma A.115. This is in fact true because for n > T and t > 0, n−k 󵄨 󵄨󵄨 󵄨󵄨 T 󵄨 󵄨 󵄨 󵄨󵄨 itX k 󵄨 k 󵄨󵄨vk (t)󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼n e n 󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼n exp{i√nqn (1 − ) t}󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n 3T

√ n 󵄨󵄨 󵄨󵄨 n−k 󵄨󵄨 1 n 󵄨󵄨 T 󵄨 √ = 󵄨󵄨 √ ∫ exp{i nx(1 − ) t}dx 󵄨󵄨󵄨 󵄨󵄨 2 3T 󵄨󵄨 n 󵄨 󵄨 3T −√

n

√ 3T n

󵄨󵄨 󵄨󵄨 n−k 󵄨󵄨 n 󵄨󵄨 T = 󵄨󵄨󵄨√ ∫ cos(√nx(1 − ) t)dx 󵄨󵄨󵄨 󵄨󵄨 3T 󵄨󵄨 n 󵄨 󵄨 0 t √3T(1− T )n−k

n cos xdx| n | ∫0 1 ≤ ≤√ T n−k 3T √ √n(1 − n ) |t| |t| 3T(1 − Tn )n

and (1 − Tn )n is uniformly separated from zero. Therefore, the conditions of Theorem A.114 hold. Note that in our case 󵄨󵄨 Q (x) 󵄨󵄨 C 󵄨󵄨 1n 󵄨󵄨 , 󵄨󵄨 󵄨≤ 󵄨󵄨 √n 󵄨󵄨󵄨 √n and we get the estimate Jn2 ≤ we obtain

In1

C . As a result, substituting the above bounds into (2.62), n1/3 log K

C C ≤ 1/3 ∫ ey (1 + |y|)dy ≤ 1/3 . n n

(2.64)

−∞

In order to bound In2 from above, note that ∑ log(1 + Rkn ) = ∑ Rkn −

1≤k≤n

1≤k≤n

1 1 2 3 ∑ (Rk ) + αn ∑ (Rkn ) . 2 1≤k≤n n 3 1≤k≤n C3 √n

It follows from Lemma 2.20 that ∑1≤k≤n |Rkn |3 ≤ following bound for αn : |αn | ≤ for such n that

C √n

1

(1 −

max1≤k≤n |Rkn |)3

≤

and Taylor expansion gives the 1

(1 −

C 3 ) √n

≤8

≤ 1/2. Therefore, using (2.63), we obtain

󵄨󵄨 󵄨󵄨 1 2 󵄨󵄨 (n) 󵄨 k (n) k ∑ (Rkn ) ≤ y)󵄨󵄨󵄨 󵄨󵄨ℙ ( ∑ log(1 + Rn ) ≤ y) − ℙ ( ∑ Rn − 󵄨󵄨 󵄨󵄨 2 1≤k≤n 1≤k≤n 1≤k≤n 󵄨󵄨󵄨 1 2 ≤ 󵄨󵄨󵄨ℙ(n) ( ∑ Rkn − ∑ (Rkn ) ≤ y) 2 󵄨󵄨 1≤k≤n 1≤k≤n

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 159

− ℙ(n) ( ∑ Rkn − 1≤k≤n

C 󵄨󵄨󵄨󵄨 1 2 )󵄨 ∑ (Rkn ) ≤ y ± √n 󵄨󵄨󵄨 2 1≤k≤n

󵄨󵄨 󵄨󵄨 1 2 󵄨 󵄨 ≤ 󵄨󵄨󵄨ℙ(n) ( ∑ Rkn − ∑ (Rkn ) ≤ y) − Φ(D(y))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 2 1≤k≤n 1≤k≤n 󵄨󵄨 󵄨󵄨 1 C C 2 󵄨 󵄨 + 󵄨󵄨󵄨ℙ(n) ( ∑ Rkn − ) − Φ(D(y ± ))󵄨󵄨󵄨 ∑ (Rkn ) ≤ y ± 󵄨󵄨 󵄨󵄨 √n √n 2 1≤k≤n 1≤k≤n 󵄨󵄨 󵄨󵄨 C + |y| C 󵄨 󵄨 + 󵄨󵄨󵄨Φ(D(y)) − Φ(D(y ± ))󵄨󵄨󵄨 ≤ . √n 󵄨󵄨 √n 󵄨󵄨 C Similarly to (2.64), we can conclude that In2 ≤ √n . At last, from (2.61) and (2.62) together with (2.64) and the above estimate, we obtain (2.60), and the theorem is proved.

Remark 2.27. To analyze the proof of Theorem 2.26, we see that the rate of convergence of all terms except Jn1 is n−1/2 . But this term depends on the rate of convergence 2

of 𝔼n (qn1 )4 − Tn2 to zero. Therefore, we can formulate the following, more general, result.

Theorem 2.28. Let the following conditions hold: C (i) |xn0 − x0 | ≤ n1/20 with some constant C0 ≥ 0,

(ii) random variables qnk satisfy condition (F) and 4

𝔼n (qn1 ) −

C T2 ≤ β. n2 n

If β ≥ 5/2, then, starting with some n0 ∈ ℕ, there is an upper bound C1 󵄨󵄨 put put 0 󵄨 󵄨󵄨πobj (x0 ) − πn,obj (xn )󵄨󵄨󵄨 ≤ 1/2 n

(2.65)

call call for some C1 > 0. The same upper bound holds for |πobj (x0 ) − πn,obj (xn0 )|. In particular,

both bonds hold if qnk take values ±√ Tn with probability measure.

1 2

with respect to the objective

Proof. Indeed, applying Lemma 2.24, we can rewrite Jn1 as 󵄨󵄨 󵄨󵄨 2 2 󵄨 󵄨 ℙ(n) (󵄨󵄨󵄨 ∑ (Rkn ) − σ 2 T 󵄨󵄨󵄨 > α ) 󵄨󵄨 󵄨󵄨 n 1≤k≤n 2

2

≤ Cn2α 𝔼n ( ∑ (Rkn ) − σ 2 T) 1≤k≤n

4

≤ C(n2α−2 + n1+2α (𝔼n (qn1 ) −

T2 )) ≤ C(n2α−2 + n1+2α−β ). n2

Let α = 21 . Then 2α − 2 = −1 < − 21 and 1 + 2α − β ≤ − 21 for β ≥ follows.

5 , 2

whence the proof

160 | 2 Rate of convergence of asset and option prices 2.2.6 From objective measure to martingale measure. The rate of convergence of fair option prices In the preceding subsection, we obtained the rate of convergence of objective option prices, in particular, under the assumption that qnk assume values ±√ Tn with probabil-

ity 21 with respect to the objective measure, and the order of the rate of convergence was estimated in this case by O(n−1/2 ). If one wishes to get a rate of convergence of the same order with respect to the martingale measure, then it is necessary to ensure, for example, that the probabilities for the joint distribution ℙ(n) (⋂nk=1 {qnk = ±√ Tn }) are

such that the random variables qnk are jointly independent and assume values ±√ Tn

with probability 21 with respect to the martingale measure ℙ(n,∗) . Taking this idea into account, we first consider the model of price process without the assumption on the joint independence of the random variables {qnk , 1 ≤ k ≤ n} with respect to the objective probability measure. Let ℙ±k,n = ℙ(n) (qnk = ±√ Tn | ℱnk−1 ). If the independence with respect to the objective probability measure is not assumed, properties of the pre-limit model depend essentially on the behavior of ℙ±k,n . This is explained in the next result. Note that ℙ+k,n + ℙ−k,n = 1. Put h±k,n =

(μ − xnk−1 )T T ± σ√ n n

and ρk,n =

Rkn − h+k,n ℙ+k,n − h−k,n ℙ−k,n 4σ Tn ℙ+k,n ℙ−k,n

.

Theorem 2.29. Consider the market on which non-risky assets are described by (2.59) and risky assets are modeled by (2.48), without independence assumption, with rnk = 1 ) uniformly in k and |xn0 | ≤ c. o( √n (i) Let the following hold for n > T: (a) We have ℙ±k,n > 0 with probability 1 and 𝔼n |ρk,n (qnk − 𝔼n (qnk | ℱnk−1 ))| < ∞ for 1 ≤ k ≤ n. (b) There exists a constant C > 0 that does not depend on k and n for which C 󵄨󵄨 + 󵄨 󵄨󵄨2ℙk,n − 1󵄨󵄨󵄨 < 1/2 . n Then there exists a number of a series n0 > T for which this market is arbitrage-free and complete starting with n0 . (ii) Assume that, for some number n > T of a series, ℙ±k,n > 0 with probability one for 1 ≤ k ≤ n, and there exists k such that 󵄨 󵄨 𝔼n 󵄨󵄨󵄨ρk,n (qnk − 𝔼n (qnk | ℱnk−1 ))󵄨󵄨󵄨 = ∞.

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 161

Then there is no equivalent martingale measure and thus the market is not arbitragefree (the question on completeness is not discussed for this case). (iii) Let ℙ+k,n = 0 with positive probability or ℙ−k,n = 0 with positive probability. (c) If h+k,n = rnk

(2.66)

in the set A+k,n := {ω ∈ Ω : ℙ+k,n = 0} provided that ℙ+k,n = 0 with positive probability or h−k,n = rnk

(2.67)

in the set A−k,n := {ω ∈ Ω : ℙ−k,n = 0} provided that ℙ−k,n = 0 with positive probability, then C 󵄨󵄨 + 󵄨 󵄨󵄨2ℙk,n − 1󵄨󵄨󵄨 < 1/2 n in the set Ω \ A+k,n , while C 󵄨󵄨 − 󵄨 󵄨󵄨2ℙk,n − 1󵄨󵄨󵄨 < 1/2 n in the set Ω \ A−k,n . If, in addition, 𝔼n |ρk,n (qnk − 𝔼n (qnk | ℱnk−1 ))| < ∞ for 1 ≤ k ≤ n, then the market is arbitrage-free and incomplete. (d) The market is not arbitrage-free if ℙ+k,n = 0 with positive probability and equality (2.66) does not hold in the set A+k,n , or if ℙ−k,n = 0 with positive probability and equality (2.67) does not hold in the set A−k,n . Proof. According to Theorem A.26, all equivalent martingale measures ℙ(n,∗) have Radon–Nikodym derivatives of the form n dℙ(n,∗) = (1 + ΔMnk ), ∏ dℙ(n) k=1

(2.68)

where {Mnk , 1 ≤ k ≤ n} is some 𝔽(n) -martingale with respect to the objective measure and ΔMnk := Mnk − Mnk−1 > −1. In this case, the random variables ΔMnk = Mnk − Mnk−1 are measurable with respect to the σ-algebra ℱnk and thus there exists a Borel function f (x1 , x2 , . . . , xk ) such that ΔMnk = f (qn1 , qn2 , . . . , qnk ) k := f (qk−1 n , qn )

√ T )1k,n,+ + f (qk−1 √ T )1k,n,− , = f (qk−1 n , n ,− n n 1 2 k−1 k √T where qk−1 n = (qn , qn , . . . , qn ), 1k,n,± = 1(qn = ± n ).

162 | 2 Rate of convergence of asset and option prices ± √T ± Let gk,n = f (qk−1 n , ± n )ℙk,n . Then the condition that the process Mn is a martingale is rewritten as follows: + − gk,n + gk,n = 0.

(2.69)

Now formulate the condition that the discounted price process is a martingale with respect to the measure ℙ(n,∗) . It is formulated as follows: for all 1 ≤ k ≤ n, k

𝔼∗n (∏ i=1

1 + Rin 1 + rni

i k−1 󵄨󵄨 k−1 󵄨󵄨 ℱ ) = ∏ 1 + Rn , 󵄨󵄨 n 1 + rni i=1

where, as usual, we denote 𝔼∗n = 𝔼ℙ(n,∗) , 𝔼n = 𝔼ℙ(n) . Taking into account the relation 𝔼ℚ (ξ | G) =

ξ | G) 𝔼ℙ ( dℚ dℙ 𝔼ℙ ( dℚ | G) dℙ

,

we rewrite the above condition in the following form: 𝔼n (∏nj=1 (1 + ΔMnj ) ∏ki=1 j

1+Rin 1+rni

| ℱnk−1 )

𝔼n (∏nj=1 (1 + ΔMn ) | ℱnk−1 )

k−1

=∏ i=1

1 + Rin , 1 + rni

or 𝔼n ((1 + ΔMnk )(1 + Rkn ) | ℱnk−1 ) = 1 + rnk . The latter relation is equivalent to the equality 𝔼n (Rkn (1 + ΔMnk ) | ℱnk−1 ) = rnk .

(2.70)

Recalling the definition of all terms on the left-hand side of this equality and taking into account the recurrent scheme (2.48), we obtain + − h+k,n gk,n + h−k,n gk,n + h+k,n ℙ+k,n + h−k,n ℙ−k,n = rnk .

Combining this result with equality (2.69) we get a system of two linear equations with + − two unknowns, gk,n and gk,n . A solution of this system exists, is unique, and can be written as follows: + gk,n =

rnk − h+k,n ℙ+k,n − h−k,n ℙ−k,n 2σ √ Tn

,

− + gk,n = −gk,n .

(2.71)

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 163

Now we distinguish between the following three cases. (i) If ℙ+k,n > 0 and ℙ−k,n > 0 with probability one, then we get a unique formula for ΔMnk of the form √ T )1k,n,+ + f (qk−1 √ T )1k,n,− ΔMnk = f (qk−1 n , n ,− n n

(2.72)

rnk − h+k,n ℙ+k,n − h−k,n ℙ−k,n 1k,n,+ 1k,n,− = ( + − − ). ℙk,n ℙk,n 2σ √ T n

Recalling the notation ρk,n =

rnk − h+k,n ℙ+k,n − h−k,n ℙ−k,n 4σ Tn ℙ+k,n ℙ−k,n

,

we rewrite equality (2.72) as follows: ΔMnk = ρk,n (qnk − 𝔼n (qnk | ℱnk−1 )).

(2.73)

Note that the random variable ρk,n is ℱnk−1 -measurable. If condition (a) holds, then equality (2.73) defines a martingale, indeed. Next we check the condition ΔMnk > −1. The following relation follows immediately from (2.51): k

(xk−1 − μ) T 1 T T √ ( √ (xn0 − μ) + 1)(1 − ) − 1 ≤ n σ n n σ n

k

1 T T ≤ ( √ (xn0 − μ) − 1)(1 − ) + 1. σ n n

(2.74)

Note that inequality (2.74) holds without any assumption on the independence of qnk . Using the assumption xn0 ≤ C, we simplify the inequalities on the left- and right-hand sides of (2.74) as follows:

and

(xnk−1 − μ) T √ ≥ −1 + e−T + O(n−1/2 ) σ n

(2.75)

(xnk−1 − μ) T √ ≤ 1 − e−T + O(n−1/2 ). σ n

(2.76)

Then we use relations (2.75)–(2.76) to estimate the right-hand side of (2.72). For those elementary events ω where 1k,n,+ = 1 and hence 1k,n,− = 0, we obtain ΔMnk =

rnk − h+k,n ℙ+k,n − h−k,n ℙ−k,n 2σ √ Tn ℙ+k,n

1 − 2ℙ+k,n (xk−1 − μ) T rnk = n + √ + + . 2σℙk,n n 2σℙ+ √ T 2ℙ+k,n k,n n

(2.77)

164 | 2 Rate of convergence of asset and option prices By assumption (b), we have 2ℙ+k,n = 1 + O(n−1/2 ) and O(n−1/2 ) is estimated by with a constant C that does not depend on both k and n. Thus,

C n1/2

1 = 1 + O(n−1/2 ), 2ℙ+k,n where the latter term O(n−1/2 ) is also bounded by nC1/2 with the constant that does not depend on both k and n. Then inequality (2.75) implies (xnk−1 − μ) T √ ≥ −1 + e−T + O(n−1/2 ). 2σℙ+k,n n

(2.78)

The second and third terms on the right-hand side of (2.77) are bounded by O(n−1/2 ). For those elementary events ω where 1k,n,− = 1 and correspondingly 1k,n,+ = 0, the transformations and reasoning are the same if we use (2.76) instead of (2.75). Therefore, there exists a number n0 for which the market is arbitrage-free and complete for n > n0 . (ii) If condition (ii) holds, then the process Mnk , 1 ≤ k ≤ n, is not integrable and thus is not a martingale. On the other hand, the preceding reasoning makes it clear that there are no other martingales that generate martingale measures if ℙ±k,n > 0 with probability one. Therefore, in this series the market is not arbitrage-free and the question on its completeness is not discussed at all. (iii) (c) If equality (2.66) holds in the set A+k,n provided that ℙ+k,n = 0 with positive √T probability, then let f (qk−1 n , n ) be an arbitrary constant on this set and let

√T f (qk−1 n , − n ) be equal to zero on this set. Then equalities (2.71) hold on this

set and one can choose ΔMnk to be an arbitrary constant on this set. For the complement of A+k,n , we repeat the same reasoning as that used for the case of (i) and obtain an arbitrage-free and incomplete market. We follow a similar approach in the case where ℙ−k,n = 0 with positive probability. (d) If equality (2.66) does not hold in the set A+k,n , provided that ℙ+k,n = 0 with positive probability, or if equality (2.67) does not hold in the set A−k,n , provided that ℙ−k,n = 0 with positive probability, then equality (2.71) does not hold on these sets, that is, ΔMnk cannot be defined on these sets and thus the market is not arbitrage-free. The theorem is proved. Now we assume that condition (i) of Theorem 2.29 holds, that is, the market is arbitrage-free and complete. We are going to find sufficient conditions supplying that random variables {qnk , 1 ≤ k ≤ n} are independent and symmetric identically distributed with respect to the unique martingale measure ℙ(n,∗) .

2.2 Limit is Ornstein–Uhlenbeck, rate of convergence | 165

We introduce the notation for the set of all possible values of the families of random variables {qnk , 1 ≤ k ≤ n}: Ξ = {ξ = √ Tn (±1, . . . , ±1)}. Let ω(ξ ) be the elementary

events for which a family {qnk , 1 ≤ k ≤ n} take on the value ξ and denote the probability of every family with this property with respect to the objective measure by ℙ(n) (ξ ). Finally, denote by q(n) the family of random variables {qnk , 1 ≤ k ≤ n}. Lemma 2.30. If for every ω(ξ ) n

∏(1 + ΔMnk (ω(ξ )))ℙ(n) (ξ ) = 2−n , k=1

(2.79)

then the random variables {qnk , 1 ≤ k ≤ n} are independent symmetric and identically distributed with respect to the martingale measure ℙ(n,∗) . Proof. First, note that ΔMnk has a constant value on any ω(ξ ). Furthermore, to supply equalities ℙ(n,∗) (ξ ) = 2−n that are equivalent to the independence and symmetry of qnk with respect to ℙ(n,∗) , we write the following chain of equalities: 2−n = ℙ(n,∗) (ξ ) = 𝔼∗n 1ξ = 𝔼n n

n dℙ(n,∗) k 1ξ = 𝔼n ∏(1 + ΔMn )1ξ (n) dℙ k=1

= 𝔼n ∏(1 + ΔMnk (ω(ξ )))ℙ(n) (ξ ), k=1

whence the proof follows. Remark 2.31. Equalities (2.79) may simultaneously hold, that is, they do not contradict each other. This becomes obvious if one interchanges the measures. More precisely, define a measure Qn for which a family of independent symmetric identically distributed random variables {qnk , 1 ≤ k ≤ n} exists. Then determine increments ΔMnk of the martingale with respect to the measure Qn in the form of ΔMnk = ρkn qnk and such that 𝔼Qn ((1 + Rkn )(1 + ρkn qnk ) | ℱnk−1 ) = 1 + Rkn . Finally, put ℙ(n,∗) = Qn and choose ℙ(n) to be the measure whose Radon–Nikodym derivative is given by n n dQn = ∏(1 + ΔMnk ) = ∏(1 + ρkn qnk ). (n) dℙ k=1 k=1

The next result follows directly from Theorems 2.22 and 2.29 and Lemma 2.30. Theorem 2.32. Assume that: (i) there exists a constant C > 0 such that C 󵄨󵄨 0 󵄨 󵄨󵄨xn − x0 󵄨󵄨󵄨 ≤ 1/2 ; n

166 | 2 Rate of convergence of asset and option prices (ii) conditions (a) and (b) of Theorem 2.29(i) hold as well as all assumptions of Lemma 2.30. Then the non-arbitrage option prices satisfy the relations C1 󵄨󵄨 call call 0 󵄨 󵄨 put put 0 󵄨 󵄨󵄨π (x0 ) − πn (xn )󵄨󵄨󵄨 + 󵄨󵄨󵄨π (x0 ) − πn (xn )󵄨󵄨󵄨 ≤ 1/2 n starting with some n0 > T and for some C1 > 0.

2.3 Estimation of the rate of convergence of option prices by using the method of pseudomoments As we understood from Sections 2.1 and 2.2, if the distribution of the jumps of the prelimit asset price is not binomial, then one has to apply appropriate results on the rate of convergence of distribution functions of sums of independent random variables to the Gaussian distribution in order to obtain a good rate of convergence of option prices. A theoretical result of this kind is obtained in [121] by using the method of pseudomoments. In this section we apply this result and improve the upper bound for the rate of convergence up to O(n−1 ) without an assumption that the pre-limit increments are binomial (the latter assumption cannot be the case in many real situations). The section is organized as follows. In Section 2.3.1 we recall a representation and some properties of a financial market with discrete time and with a single risky asset, state the general results concerning the rate of convergence of distribution functions of the sum of i. i. d. r. v. to the standard normal distribution function in terms of pseudomoments (proofs are given in Appendix A.3.10.1), and apply this result to the rate of convergence of asset prices. Section 2.3.2 contains a theorem concerning the estimate of the rate of convergence of put and call option prices. An example of a distribution function of i. i. d. r. v. that allows to use the method of pseudomoments is discussed in Section A.3.10.2.

2.3.1 Rate of convergence in the CLT for i. i. d. random variables by the method of pseudomoments; rate of convergence of asset prices Let (Ω, ℱ , ℙ) be a probability space. Recall some notions from Section 1.4.3. So, we consider a sequence of financial markets with discrete time in the scheme of series defined on this space. We assume that there are only one non-risky asset and one risky asset in this market. More precisely, let T > 0 be given and let a parameter n ∈ ℕ. For every n ≥ 1, consider a partition π(n) = {0 = tn0 < tn1 < ⋅ ⋅ ⋅ < tnn = T} of the time interval [0, T]. The points of the partition are treated as trading moments in the

167

2.3 Method of pseudomoments |

financial markets. Now let {rnk , 1 ≤ k ≤ n} be a family of non-negative numbers treated as interest rates, so that the price of the non-risky asset at the moment tnk is given by k

Bkn = ∏(1 + rni ).

(2.80)

i=1

Furthermore, let, for every n ≥ 1, a family of random variables {Rkn , 1 ≤ k ≤ n} defined on (Ω, ℱ , ℙ) be given. We assume the following condition of boundedness: there exists a number 0 < c < 1 such that |Rkn | ≤ c for all n ≥ 1 and 1 ≤ k ≤ n. Consider a flow of σ-algebras ℱnk = σ{Rin , i = 1, . . . , k} generated by the above random variables. We assume that the price of the risky asset at the moment tnk is given by the equality k

Snk = Sn0 ∏(1 + Rin ).

(2.81)

i=1

Evidently, the discounted asset price process with discrete time has the form Xn (t) =

Sn0

⌋ ⌊ nt T

∏ k=1

1 + Rkn 1 + rnk

,

tnk ≤ t < tnk+1 ,

0 ≤ k ≤ n − 1,

where ⌊a⌋ is the biggest integer not exceeding a, ∏0k=1 = 1. For the sake of simplicity, we assume that Sn0 = 1. First, we assume that the conditions of Lemma 1.27 hold and the market is arbitrage-free. Second, we assume that conditions (A1)–(A5) hold and ensure the fulfillment of the functional limit theorem, namely, Theorem 1.41, so that ℙ(n,∗) , ℙ∗

(Xn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [0, T]), where X(t) = exp{σW(t) − 21 σ 2 t}. Then, in particular, Xn (T) weakly converges to exp{σW(T) − 21 σ 2 T}. We use this result in what follows to estimate the rate of convergence of prices of put and call options for the discrete-time model to the corresponding prices in the limit market with continuous time. To obtain an estimate of the rate of convergence, we use a result concerning i. i. d. r. v. proved by using the method of pseudomoments. This result can be stated as follows. Let {ξn , n ≥ 1} be a sequence of i. i. d. r. v. with 𝔼ξi = 0 and Var ξi = 1. Let F(x) and f (t) be their common distribution function and characteristic function, respectively. Let Φn (x), x ∈ ℝ, be the distribution function of the random variable Sn = n−1/2 ∑nk=1 ξk and let Φ(x), x ∈ ℝ, be the standard normal distribution function. Assume that the pseudomoments μk = ∫ xk dH(x), ℝ

k = 3, . . . , m,

m ∈ ℕ,

168 | 2 Rate of convergence of asset and option prices are finite for some m ≥ 3, where H(x) = F(x) − Φ(x). The pseudomoments truncated from above are given by ∫ |x|m+1 d|H|(x),

νn(1) (m) =

|x|≤√n

while those truncated from below are given by ∫ |x|m d|H|(x),

νn(2) (m) =

|x|>√n

where d|H|(x) is the differential of the total variation |H| of function H. Theorem 2.33. Assume that: (i) the characteristic function is integrable, that is, ∫ℝ |f (t)|dt = A < ∞; (ii) all pseudomoments of orders up to m are equal to zero, μk = 0,

k = 3, . . . , m,

for some m ≥ 3, and truncated pseudomoments are bounded, or more precisely, 1 3 νn (m) = max{νn(1) (m), νn(2) (m)} < e− 2 , 2 starting with a certain number n0 ∈ ℕ. Then 3

n

(2) (1) σA n−1 4e 2 e− 2 󵄨 󵄨 (2) νn (m) (1) νn (m) sup󵄨󵄨󵄨Φn (x) − Φ(x)󵄨󵄨󵄨 ≤ 2Cm + 2C + b1 + νn (m) =: Θn (m) m−1 m−2 m π π n x∈ℝ n 2 n 2

for all n ≥ n0 , where (1) Cm

=

12

m+1 2

Γ( m+1 ) 2

π(m + 1)!

b1 = exp{−

(2) (1) Cm = 2Cm−1 ,

,

π2 } < 1. + π)2

24A2 (2

Corollary 2.34. Let i. i. d. r. v. {ξn , n ≥ 1} with 𝔼ξi = 0 and Var ξi = 1 have a bounded probability density, p(x) ≤ A1 . Assume that condition (ii) of Theorem 2.33 holds. Then 3

n

(1) (2) 4e 2 e− 2 󵄨 󵄨 (1) νn (m) (2) νn (m) sup󵄨󵄨󵄨Φn (x) − Φ(x)󵄨󵄨󵄨 ≤ 2Cm + 2A1 bn−2 + νn (m) + 2Cm m−1 m−2 2 π n x∈ℝ n 2 n 2

for all n ≥ n0 , where b2 = exp{−

1

96A21 (2

+ π)2

} < 1.

(2.82)

2.3 Method of pseudomoments |

169

For the proof of Theorem 2.33 and Corollary 2.34, see Appendix A.3.10.1. An example of a distribution function for which all assumptions of Corollary 2.34 hold is given in Appendix A.3.10.2. We apply Corollary 2.34 to random variables with a bounded probability density whose support belongs to a certain interval. The following result holds for m = 3. Lemma 2.35. Let i. i. d. r. v. {ξn , n ≥ 1} have the following moments: 𝔼ξi = 𝔼(ξi )3 = 0, 𝔼(ξi )2 = 1, and 𝔼(ξi )4 = 3. Let the support of their common distribution belong to a certain interval [a, b] and let their distribution density be bounded by some constant A1 . Then there exists n0 depending on H(x) and σ such that n

2 e− 2 3 e−3/2 2√3e−3/2 󵄨 󵄨 + 2A1 bn−2 + sup󵄨󵄨󵄨Φn (x) − Φ(x)󵄨󵄨󵄨 ≤ Δn Φ := + 2 3/2 π n π n √πn x∈ℝ for all 3

|a| |b| 32e 2 ) ∨ 2 ∨ 2 ∨ n0 . n>( √2πσ 2 σ σ Proof. Note that C3(1)

122 Γ(2) 6 = , = π ⋅ 4! π

3

C2(1)

=

12 2 Γ( 32 ) π ⋅ 3!

=

2√3 √π

if m = 3, whence C32 =

4√3 . √π

Furthermore, since ∫ℝ x4 dΦ(x) = 3, there exists such n0 that νn(1) (3) =

∫

x4 d|H|(x) = 3 −

|x|σ√n

∫

|x|3 dΦ(x).

|x|>σ√n

If σ√n > |a| ∨ |b|, then ∫|x|>σ√n |x|3 dF(x) = 0. Now, if σ√n > 1, then ∫ |x|>σ√n

where x = σ 2 n.

3

∞

|x| dΦ(x) = 2 ∫ x σ√n

3e

2

− x2

√2π

∞

dx =

4 ∫ ye−y dy √2π 2 σ n

4 8 ≤ (x + 1)e−x ≤ xe−x , √2π √2π

(2.83)

170 | 2 Rate of convergence of asset and option prices Since ex >

x2 , 2

we conclude that 16 1 8 1 3 xe−x ≤ < e− 2 2 √2π √2π σ n 2

for 3

n>

32e 2 . √2πσ 2

Thus condition (ii) of Theorem 2.33 holds for 3

|a| |b| 32e 2 ) ∨ 2 ∨ 2 ∨ n0 , n>( √2πσ 2 σ σ and hence bound (2.82) follows. The latter bound can be transformed to the form of (2.83) by substituting the values of the coefficients obtained above. Remark 2.36. Note that the term 3 e−3/2 1 = O( ) π n n on the right-hand side of (2.83) dominates other terms in the asymptotic sense as n → ∞. For further applications, we recall the inequality |Φ(x) − Φ(y)| ≤ √12π |x − y|. Let random variables {ζnk , n ≥ 1, 1 ≤ k ≤ n} be such that: C (G) n−1/2 ∑nk=1 ζnk = n−1/2 ∑nk=1 ξnk + nn , where the random variables ξnk for every n satisfy assumptions of Lemma 2.35 and where |Cn | ≤ C is a bounded, possibly random, sequence.

Then 󵄨 󵄨 sup󵄨󵄨󵄨ℙ{n−1/2 Σnk=1 ζnk ≤ x} − Φ(x)󵄨󵄨󵄨 x∈ℝ

≤ Δ(1) n Φ :=

n

C 2√3e−3/2 2 e− 2 3 e−3/2 n−2 + + 2A b + . + 1 1 √2πn π n π n √πn3/2

Now we provide sufficient conditions imposed on the random variables {Rkn , 1 ≤ k ≤ n} under which the prices of assets satisfy condition (G) in a financial market with discrete time. For technical simplicity, we assume that the partitions πn are uniform, Rkn satisfy conditions of Lemma 1.25, and the interest rates are also uniform, rnk = rT . n Put (σn∗ )2 = Var∗n (Rkn ). First we prove an auxiliary result. Here the symbol C denotes a constant that can be different in different places and whose precise value does not matter for the proof.

2.3 Method of pseudomoments | 171

Theorem 2.37. Let there exist C > 0 such that random variables C 󵄨󵄨 k 󵄨󵄨 󵄨󵄨Rn 󵄨󵄨 ≤ √n and assume additionally that 5 (σ ∗ )2 3 + O(n−2 ), 𝔼∗n (Rkn ) = O(n− 2 ), n 3(σ ∗ )4 ∗ k 4 𝔼n (Rn ) = + O(n−3 ). n2

2

𝔼∗n (Rkn ) =

Then 󵄨󵄨 󵄨󵄨 C x 1 2 󵄨 󵄨 sup󵄨󵄨󵄨ℙ{log Xn (T) + n(σn∗ ) ≤ x} − Φ( )󵄨󵄨󵄨 ≤ . 2 x∈ℝ 󵄨󵄨 σ ∗ √T 󵄨󵄨 n Proof. As in the proof of Theorem 1.41, we assume that T = 1 and tnk = of generality. Put

k n

without loss

∗ 2 ζnk 1 r C r 2 (σ ) 1 C r2 3 2 3 = Rkn − (Rkn ) + (Rkn ) + α( )(Rkn ) − + 2 + β( ) 2 + n . √n √n 2 3 n 2n n n 2

The discounted price process is of a simpler form if the interest rate is uniform, namely n

Xn (1) = ∏ k=1

1 + Rkn 1 + Rkn

n

1 + Rkn r = (1 + ) n 1 + nr

−n n

=∏ k=1

∏(1 + Rkn ). k=1

With the above notation, the Taylor expansion for the random variables log Xn (1) + n(σn∗ )2 2

is given by

log Xn (1) +

n ζk n(σn∗ )2 =∑ n . √n 2 k=1

Moreover, n 󵄨󵄨 C 󵄨󵄨󵄨󵄨󵄨󵄨 k 󵄨󵄨3 C 󵄨 )󵄨󵄨R 󵄨 ≤ ∑ 󵄨󵄨󵄨α( 󵄨 √n 󵄨󵄨󵄨󵄨 n 󵄨 n k=1󵄨

and n 󵄨󵄨 󵄨 2 C 󵄨 C 󵄨󵄨 r ∑ 󵄨󵄨󵄨β( )󵄨󵄨󵄨 2 ≤ 2 . 󵄨󵄨 n 󵄨󵄨 n n k=1

Under our assumptions, Taylor expansion is given by log Xn (1) +

n n(σn∗ )2 ξk = ∑ n + Sn , √n 2 k=1

172 | 2 Rate of convergence of asset and option prices where Sn ≤ Cn . Here 𝔼∗n ξnk = 0 and 2

𝔼∗n (ξnk ) = n(Var∗n Rkn +

Cn (σ ∗ )2 Cn )= + , 2 2 n n

where |Cn | ≤ C. Therefore ζnk satisfy condition (G), and this completes the proof. 2.3.2 The rate of convergence of put and call option prices Let the options be considered in financial markets with discrete time that are defined by relations (2.80) and (2.81). We assume that rnk = rT in every series, where r > 0 is a n k certain number, and that {Rn , 1 ≤ k ≤ n} are independent and identically distributed random variables for which assumptions of Lemma 1.25 hold, that is, the market is arbitrage-free. Fix a martingale measure ℙ(n,∗) defined by relations (1.5) and (1.13). As shown in Lemma 1.29, the random variables {Rkn , 1 ≤ k ≤ n} are independent with respect to the probability measure ℙ(n,∗) and 𝔼∗n Rkn = rT . n Put Qkn =

1 + Rkn 1+

rT n

,

Ynk = √n(log Qkn +

(σn∗ )2 T ), 2

(2.84)

̃ n denote the distribution function of the sum and let Φ n

n−1/2 ∑ Ynk . k=1

The following result is an obvious corollary of Theorem 2.37. Lemma 2.38. Let i. i. d. r. v. {Rkn , 1 ≤ k ≤ n} satisfy assumptions of Theorem 2.37. Then 󵄨󵄨 󵄨󵄨 C x 󵄨̃ 󵄨 sup󵄨󵄨󵄨Φ )󵄨󵄨󵄨 ≤ . (x) − Φ( n x∈ℝ 󵄨󵄨 σ ∗ √T 󵄨󵄨 n

(2.85)

Now, consider European call and put options ℂ = (S(T) − K)+ and ℙ = (K − S(T))+ . Denote by πncall = 𝔼∗n (Xn (T) − K(1 +

rT ) ) n

−n +

the fair price of the discounted call option in the pre-limit model with respect to the martingale measure defined by equalities (1.5) and (1.13). Let π call denote the Black– Scholes price of a call option at the moment T whose strike price is K, whose interest rate is r, and whose variance is (σ ∗ )2 . The prices for the put options are denoted by πnput and π put , respectively.

2.3 Method of pseudomoments | 173

Theorem 2.39. Let i. i. d. r. v. {Rkn , 1 ≤ k ≤ n} satisfy assumptions of Theorem 2.37. Then for some fixed C > 0

C 󵄨󵄨 call call 󵄨 󵄨 put put 󵄨 󵄨󵄨πn − π 󵄨󵄨󵄨 + 󵄨󵄨󵄨πn − π 󵄨󵄨󵄨 ≤ . n Proof. Consider the put option and note that the parity condition for the pre-limit and limit options πncall = πnput + Sn0 − K(1 +

rT ) n

−n

and π call = π put + S0 − Ke−rT imply that the estimate of the rate of convergence is the same for the call option. Recall that we assume Sn0 = S0 = 1 and as usual in our proofs, we assume that T = 1. Now we take into account notation (2.84) and Theorem 2.37 and apply integration by parts, the Black–Scholes formula for T = 1, and the following inequality (see Lemma A.1 in the appendix): −n 󵄨 󵄨󵄨 󵄨󵄨 C r 󵄨󵄨 −r 󵄨󵄨e − (1 + ) 󵄨󵄨󵄨 ≤ . 󵄨󵄨 󵄨󵄨 n n

Then we get 󵄨󵄨 󵄨󵄨 put 󵄨 ∗ put 󵄨 󵄨󵄨πn − π 󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼n (K(1 + 󵄨󵄨 󵄨󵄨 󵄨 = 󵄨󵄨󵄨𝔼∗n (K(1 + 󵄨󵄨

r ) n

−n

r ) n

−n

+ 󵄨󵄨 󵄨 − Xn (1)) − π put 󵄨󵄨󵄨 󵄨󵄨

+ 󵄨󵄨 󵄨 − ∏ Qkn ) − π put 󵄨󵄨󵄨 󵄨󵄨 1≤k≤n

r −n

Ke−r 󵄨󵄨 K(1+ n ) 󵄨󵄨 log y + 21 (σ ∗ )2 󵄨󵄨 󵄨󵄨 (n,∗) k 󵄨 = 󵄨󵄨 ∫ ℙ )dy󵄨󵄨󵄨 { ∏ Qn ≤ y}dy − ∫ Φ( ∗ 󵄨󵄨 󵄨󵄨 σ 1≤k≤n 󵄨 0 󵄨 0 Ke−r

󵄨󵄨 (σ ∗ )2 1 2 󵄨 ≤ ∫ 󵄨󵄨󵄨ℙ(n,∗) { ∑ (log Qkn + n ) ≤ log y + n(σn∗ ) } 󵄨󵄨 2 2 1≤k≤n 0

− Φ( Ke−r

log y + 21 (σ ∗ )2 󵄨󵄨󵄨 C )󵄨󵄨󵄨dy + 󵄨󵄨 σ∗ n

󵄨󵄨 1 2 󵄨 = ∫ 󵄨󵄨󵄨ℙ(n,∗) {n−1/2 ∑ Ynk ≤ log y + n(σn∗ ) } 󵄨󵄨 2 1≤k≤n 0

− Φ(

log y + 21 (σ ∗ )2 󵄨󵄨󵄨 C )󵄨󵄨󵄨dy + 󵄨󵄨 σ∗ n

C . n The proof is complete. ≤

174 | 2 Rate of convergence of asset and option prices Remark 2.40. The key tool used in the proof of Theorem 2.39 is the upper bound (2.85). If this bound holds, then Theorem 2.39 follows independently of the method used to prove bound (2.85) (recall that we have used the method of pseudomoments). Even if the rate of convergence is higher than that in (2.85), the proof of Theorem 2.39 uses some additional estimates that obviously are of order 1/n, and thus one cannot improve the rate of convergence for prices of options in the model under consideration.

2.4 Market model with stochastic Ornstein–Uhlenbeck volatility: option pricing and discretization One of the promising directions of enhancement of the classical Black–Scholes model are construction and research of the diffusion models with volatility of risky assets being governed by a stochastic process. Empirical studies ([82, 142]) testify clearly in favor of the fact that the classical model with constant volatility is unable to capture important features of volatility observed in real financial markets. This drawback of the Black–Scholes model has been widely investigated and to some extent eliminated by the extension of the theory in three directions: models with time-dependent deterministic volatility, models with state-dependent volatility, and models with stochastic volatility. The first and second of the above categories may be viewed as intermediate ones between the classical model and the third category, although equipping the market with certain constraints (most essential is the limiting time period under consideration) that allow less complex models to produce results of acceptable precision. Starting with the pioneering works by Hull and White [80] and Heston [77], stochastic volatility models for asset prices have been a subject of intensive research activity, which is still vibrant from analytical, computational, and statistical points of view. Of course, option pricing is one of most relevant problems. Despite recent popularity of stochastic volatility modification of the Black– Scholes theory, the range of models under consideration is quite narrow. One of the first models of such type is presented in [80], where the authors assume the volatility of the price of risky assets to be governed by the square root of the geometric Brownian motion process. An expression for the price of European call options is derived under the following assumption: the volatility process is driven by a Brownian motion independent of the Brownian motion governing the price of risky assets. In [159] authors choose the Ornstein–Uhlenbeck process to drive volatility. The Ornstein–Uhlenbeck process is mean-reverting and there is strong evidence that the volatility in real financial markets has such feature ([61, 62]). Under this assumption, the authors of [159] describe the distribution of the price of risky assets and apply it to derive the estimate of the price of European call options. As an alternative to the above there is an option to choose the CIR process to govern the volatility process ([61, 62]). It should be noted that all works cited above despite containing some significant results rely upon simplified models of real-world volatility processes (e. g., ignoring the mean-reversion

2.4 Stochastic Ornstein–Uhlenbeck volatility | 175

property). We remark that there is a vast amount of further investigations which consider more sophisticated and thus more realistic models. An extensive overview of these results is given in [151]. Concerning the stochastic volatility modeling, note that there are approaches involving Gaussian ([129, 144]), non-Gaussian ([8, 9]), jump-diffusion, and Lèvy processes ([35, 103]), as well as time series ([29, 133, 150]). The references provided here are not in any way intended to be exhaustive or complete; we only illustrate the availability of various approaches. We would also like to mention the books [62, 87, 93] and references therein, as well as the paper [4]. A useful decomposition formula for option prices that is valid even when the Malliavin regularity conditions are not satisfied was obtained in [3]. Another desirable feature of the process, which is modeled to govern volatility, is non-negativity. One of the possible choices is to use the exponential function of the Ornstein–Uhlenbeck process to represent volatility (see [135, 148] and references therein). Questions of existence of equivalent (local) martingale measures are investigated in different frameworks and under different generalities in [62, 64, 88, 172]. Often authors after specifying the model state that a risk-neutral measure exists and continue investigation in the risk-neutral world without defining the measure. A significant part of the works (including the aforementioned) use Fourier transform to derive the analytical representation of the price of European call options. A great deal of information about development in the application of Fourier transform to option pricing problems can be found in [143]. With all of the above in mind, in this section we investigate the market defined by a diffusion model with stochastic volatility being a function governed by the Ornstein– Uhlenbeck process. For the general setting and quite mild assumptions, it is proved that markets satisfy two distinct no-arbitrage properties for different classes of trading strategies. In the particular case of uncorrelated Wiener processes we derive the analytical expression for the price of European call options. Since the resulting formula in an analytic expression for the option price is complicated and difficult to apply in most of the available software, we consider discrete-time approximation for the price of European call options when the stochastic volatility driven by the Ornstein–Uhlenbeck process is discretized. The discrete-time approximation is ready to be modeled even in the non-specific software. The problem of construction of discrete-time analogues for stochastic volatility models of financial markets is studied in a range of works, including [5, 27, 70, 80, 86, 165, 170]. Various techniques are implemented, e. g., multi-level Monte Carlo ([70]), conditional Monte Carlo ([27, 170]), exact simulation ([27, 165]), and Itô–Taylor approximations ([86]). In most of the works the authors construct discrete-time approximations both for processes which describe the evolution of the price of assets and for processes driving the volatility of asset prices. The model considered in this section allows to apply another approach: we only discretize the volatility process. The resulting discrete-time

176 | 2 Rate of convergence of asset and option prices volatility process is then averaged in a special way and substituted into the option pricing formula. The option price is determined conditionally on the path of the volatility process, thus the conditional Monte Carlo approach is used. The rate of convergence of the option price calculated using discrete-time volatility to the true option price for a given trajectory of volatility process is estimated. A similar approach is realized also in Section 2.5, when we, however, consider both approaches to discretization. Discretization of the model is naturally connected with the problem of discretetime approximations to the solutions of stochastic differential equations. These matters are widely investigated and systematized in [92, 137, 169]. The simplest discretetime approximation is the stochastic generalization of Euler approximation for deterministic differential equations proposed in [114], which is also referred to as Euler– Maruyama scheme. Another effective method suitable for implementation is the Milstein scheme ([116]). As the model under consideration is diffusion with additive noise, the above schemes coincide (see below). It is worth noting that Euler and Milstein schemes both belong to the class of Itô–Taylor approximations and have orders of convergence 0.5 and 1, respectively. For some diffusions the approximation schemes might be enhanced to provide higher-order convergence, but this usually results in greatly increased computation times. Although exact simulations provide more precision compared to Euler approximation, we use the latter. This is motivated by the fact that Euler approximation is cheaper in terms of computation time and by our desire to assess the rate of convergence of conditional option prices when the volatility is discretized using the Euler scheme. This section is structured as follows: in Section 2.4.1 the general model is defined while the main definitions and subsidiary results concerning the description of the market are contained in Section 2.4.2. Most of the information presented in Section 2.4.2 can be found in more detail in [48] and [152]. Section 2.4.3 investigates matters of existence of equivalent (local) martingale measures and arbitrage properties of our model with stochastic volatility. In Section 2.4.4 the special case of the general model is considered where the Wiener drivers of the main process and volatility process are independent and the problem of pricing European call options is raised. Section 2.4.5 covers the derivation of an analytical expression for the option price. Sections 2.4.6 and 2.4.7 contain the description of the discretization scheme for the volatility and the comparison of the prices of European call options considered for discrete-time and continuous volatility processes, in order to derive the estimate of the strong convergence order. Section 2.4.8 provides numerical results of the simulation. In Section 2.4.9 we demonstrate the precision of discrete-time approximation for the case of deterministic volatility.

2.4 Stochastic Ornstein–Uhlenbeck volatility | 177

2.4.1 Diffusion model with stochastic volatility governed by the Ornstein–Uhlenbeck process Let {Ω, ℱ , 𝔽 = {ℱt = ℱt(B,W) , t ≥ 0}, ℙ} be a complete probability space with filtration generated by correlated Wiener processes {B(t), W(t), 0 ≤ t ≤ T}. We consider the model of the market where one risky asset is traded, its price evolves according to the geometric Brownian motion {S(t), 0 ≤ t ≤ T}, and its volatility is driven by stochastic processes. More precisely, the market is described by the following pair of stochastic differential equations: dS(t) = μS(t)dt + σ(Y(t))S(t)dB(t),

dY(t) = −αY(t)dt + kdW(t).

(2.86) (2.87)

We denote by x and y deterministic initial values of the processes specified by equations (2.86) and (2.87), respectively. We impose the following assumptions: (H1) Wiener processes B and W are correlated with correlation coefficient ρ ∈ [−1, 1], that is, dB(t)dW(t) = ρdt. (H2) The volatility function σ: ℝ → ℝ+ is measurable and bounded away from zero by a constant c, σ(x) ≥ c > 0,

x ∈ ℝ,

T

and satisfies the conditions ∫0 σ 2 (Y(t))dt < ∞ a. s. (H3) The coefficients α, μ, and k are positive. For example, the conditions mentioned in assumption (H2) are fulfilled for the measurable function σ(x) for which the inequality c ≤ σ 2 (x) ≤ C holds for 0 < x < T and some constants 0 < c < C. Moreover, given the square integrability of σ(Y(s)), the solution of the differential equation (2.86) is presented by the formula t

t

0

0

1 S(t) = x exp (μt − ∫ σ 2 (Y(s))ds + ∫ σ(Y(s))dB(s)), 2

(2.88)

which yields that S is a continuous process. Hence, the product σ(Y(t))S(t) is squareT integrable in the sense that ∫0 σ 2 (Y(t))S2 (t) dt < ∞ a. s. The unique solution Y of the Langevin equation (2.87) is the Ornstein–Uhlenbeck process. It was introduced in a more general form in Section 1.7. Its properties make it a suitable tool for modeling volatility in financial markets. One of the most important features is the meanreverting property, which holds for α > 0. More precisely, if the current value of the process is less than the (long-term) mean, which equals μ in the general equation (1.77), the drift will be positive; if the current value of the process is greater than

178 | 2 Rate of convergence of asset and option prices the (long-term) mean, the drift will be negative. In other words, the mean acts as an equilibrium level for the process. Recall that the Ornstein–Uhlenbeck process is Gaussian and admits the explicit representation t

Y(t) = ye

−αt

+ k ∫ e−α(t−s) dW(s). 0

Its characteristics are 𝔼Y(t) = ye−αt ,

Cov(Y(t), Y(s)) =

k 2 −α|t−s| (e − e−α(t+s) ). 2α

So, in our case the long-term mean is zero since μ = 0 (compare with equation (1.77)), and, in accordance with this fact, 𝔼Y(t) = ye−αt → 0 as t → ∞. Concerning the process W, we can represent it in the form W(t) = ρB(t) + √1 − ρ2 Z(t), where Z is a Wiener process independent of B. In what follows we will use such representation. Note that ℱ (B,W) = ℱ (B,Z) , where filtration {ℱt(B,Z) , 0 ≤ t ≤ T} is generated by independent Wiener processes B and Z.

2.4.2 Definitions and auxiliary results Consider the market in continuous time 𝕋 = [0, T], with one risky asset and one riskfree asset traded in it. Evolutions of prices of both assets are given by the semimartingale process S = {S(t), t ∈ 𝕋} and the deterministic process B(t) = ert , respectively, where r is a constant risk-free rate of return. We introduce the discounted price process X(t) = e−rt S(t), t ∈ 𝕋. Obviously, X(0) = S(0) = x > 0. Agents acting in the market may buy or sell risky assets and make their decisions concerning the structure of their portfolios based upon the information available at the moment of decision. This principle can be formalized with the following definition (for the concept of predictability see Definition A.55). Definition 2.41. A trading strategy is a predictable process π = {π(t), t ∈ 𝕋}. The value π(t) of this process represents the amount of asset S in a portfolio at time t. Let a semimartingale X admit the decomposition X = X(0) + A + M, where A is a bounded variation process and M is a local martingale. Denote by [M]t , t ∈ 𝕋, the quadratic variation of M (see Section A.3.1.2). Definition 2.42. Let π be a predictable process. We say that: t (i) π ∈ Lvar (A) if for each ω ∈ Ω, ∫0 π(s)d|A|(s) < ∞, t ∈ 𝕋;

2.4 Stochastic Ornstein–Uhlenbeck volatility | 179

(ii) π ∈ Lqloc (M), q ≥ 1, if there exists a sequence of stopping times τn approaching ∞ as n → ∞ such that τn

q/2

2

𝔼[∫ π (s)d[M]s ]

< ∞;

0

(iii) π ∈ Lq (X) if there exists a representation X = X(0) + A + M such that π ∈ Lvar (A) ∩ Lqloc (M). Definition 2.43. A trading strategy is called admissible (relative to the price process X) if π ∈ L1 (X). In what follows we assume that the price process X is continuous. Definition 2.44. An admissible strategy is said to be self-financing (relative to the price process X), or equivalently π ∈ SF(X), if its value X π (t) = π(t)X(t) has representation t ⋅ X π (t) = X π (0) + ∫0 π(s)dX(s), where ∫0 π(s)dX(s) is the integral of the predictable process with respect to a continuous semimartingale, defined in Section A.3.1.2. Furthermore, we define two special classes of trading strategies along with corresponding classes of ℱT -measurable payoff functions ψ = ψ(ω) that can be majorized by returns of strategies belonging to each class. Definition 2.45. For each a ≥ 0 define Πa (X) = {π ∈ SF(X) : X π (t) ≥ −a, t ∈ 𝕋} and T

Ψ+ (X) = {ψ ∈ ℒ∞ (Ω, ℱT , ℙ) : ψ ≤ ∫ π(s)dX(s) for some π ∈ Π+ (X)}, 0

where Π+ (X) = ⋃a≥0 Πa (X). Definition 2.46. Let g(x) = g0 + g1 x, g0 ≥ 0, g1 ≥ 0. Define Πg (X) = {π ∈ SF(X) : X π (t) ≥ −g(X(t)), t ∈ 𝕋} and T

Ψg (X) = {ψ ∈ Lg (Ω, ℱT , ℙ) : ψ ≤ ∫ π(s)dX(s) for some π ∈ Πg (X)}, 0

where Lg (Ω, ℱT , ℙ) is the set of ℱT -measurable random variables ψ such that |ψ| ≤ g(X(T)).

180 | 2 Rate of convergence of asset and option prices In the spirit of [152] p. 648, introduce the norm 󵄨 󵄨 󵄨 󵄨 ‖ψ‖∞ = ess sup󵄨󵄨󵄨ψ(ω)󵄨󵄨󵄨 = inf{c ≥ 0 : ℙ(󵄨󵄨󵄨ψ(ω)󵄨󵄨󵄨 > c) = 0}. This norm makes ℒ∞ (Ω, ℱ , ℙ) a complete (and therefore, Banach) space. Also, introduce the norm 󵄩󵄩 ψ 󵄩󵄩 󵄩 󵄩󵄩 ‖ψ‖g = 󵄩󵄩󵄩 󵄩. 󵄩󵄩 g(X(T)) 󵄩󵄩󵄩 We shall denote the closures of the sets Ψ+ (X) and Ψg (X) with respect to norms ‖ ⋅ ‖∞ and ‖ ⋅ ‖g by Ψ+ (X) and Ψg (X), respectively. Now taking into account the notation presented in [152], we proceed to main definitions of absence of arbitrage for the general semimartingale markets in continuous time. Definition 2.47. We say that the property NA+ holds (or equivalently that a market is NA+ ) if Ψ+ (X) ∩ ℒ+∞ (Ω, ℱT , ℙ) = {0}, where ℒ+∞ (Ω, ℱT , ℙ) is the subset of non-negative random variables in ℒ∞ (Ω, ℱT , ℙ). Definition 2.48. We say that the property NAg holds (or equivalently that a market is NAg ) if Ψg (X) ∩ ℒ+∞ (Ω, ℱT , ℙ) = {0}. The property NA+ is consistently used by F. Delbaen and W. Schachermayer [46–48], who call it the No Free Lunch with Vanishing Risk (NFLVR) property. This name can be explained as follows. In the discussion of the absence of arbitrage in its NA+ -version we take for “test” functions non-negative functions ψ ∈ ℒ+∞ (Ω, ℱT , ℙ) ∩ Ψ+ (X) T

that are smaller or equal to the “returns” ∫0 π(s)dX(s) for π ∈ Π+ (X). The sense of NAg can be explained in a similar way. There are two theorems which establish necessary and sufficient conditions for the absence of arbitrage in the market in terms of equivalent (local) martingale measures. An important condition which will be addressed below is the local boundedness of the price process. Definition 2.49. A probability measure ℚ, which is equivalent to the objective measure ℙ, is called an equivalent (local) martingale measure if the discounted price process is a (local) martingale under the measure ℚ.

2.4 Stochastic Ornstein–Uhlenbeck volatility | 181

Definition 2.50. A stochastic process X is called locally bounded if there exists a sequence {τn , n ≥ 1} of stopping times, increasing a. s. to +∞, such that the stopped processes X τn (t) = X(t ∧ τn ) are uniformly bounded for each n ≥ 1. Theorem 2.51 ([152]). Let semimartingale X be locally bounded. Then the market is NA+ if and only if there exists an equivalent local martingale measure (ELMM). Theorem 2.52 ([152]). Let semimartingale X be locally bounded. Then the market is NAg if and only if there exists an equivalent martingale measure (EMM). The following theorem is a corollary of Proposition 6.1 from [64], which is cited in Section A.3.4; see Proposition A.72. This result defines the construction of the ELMM in the model (2.86)–(2.87). Theorem 2.53. A probability measure ℚ, which is equivalent to the objective measure ℙ on ℱT , is an ELMM for the process X defined by the model (2.86)–(2.87) on ℱT if and T only if there exists a predictable process ν = {ν(t), t ∈ 𝕋}, such that ∫0 ν2 (s)ds < ∞ ℙ-a. s. and the following holds: the local martingale {L(t), t ∈ 𝕋}, L(t) = dℚ/dℙ|ℱt , with t

t

L(t) = exp(∫(r − μ)/σ(Y(s))dB(s) + ∫ ν(s)dZ(s) t

0

0

1 − ∫((r − μ)2 /σ 2 (Y(s)) + ν(s)2 )ds), 2

(2.89)

0

satisfies 𝔼L(T) = 1. Denote by ℒℳX (ℙ) and ℳX (ℙ) the sets of ELMM and EMM in the market modeled by (2.86)–(2.87). It is obvious that ℳX (ℙ) ⊂ ℒℳX (ℙ).

2.4.3 Absence of arbitrage in the market with stochastic volatility In this section we investigate the absence of arbitrage in the model (2.86)–(2.87). Theorem 2.54. A market defined by the model (2.86)–(2.87) with assumptions (H1)– (H3) has the following properties: (i) It satisfies the NA+ property. (ii) It satisfies the NAg property provided that for some ELMM ℚ the following inequality holds: T

𝔼ℚ ∫ σ 2 (Y(s))X 2 (s)ds < ∞. 0

(2.90)

182 | 2 Rate of convergence of asset and option prices Proof. (i) As X is locally bounded due to its continuity, Theorem 2.51 yields that in order to prove the first part of the present theorem it suffices to show that ℒℳX (ℙ) ≠ 0. Consider the process L(t) defined by (2.89) with ν = {ν(t), t ∈ 𝕋} = 0. Let L(T) = on the σ-field ℱT . In view of Theorem 2.53 it suffices to dℚ/dℙ|ℱT , a restriction of dℚ dℙ show that under such choice of ν the equality 𝔼L(T) = 1

(2.91)

holds. In turn, it suffices to verify the following Novikov condition: T

1 𝔼 exp( ∫((r − μ)2 /σ 2 (Y(s)) + ν2 (s))ds) < ∞. 2

(2.92)

0

It follows from the boundedness away from zero of the function σ (assumption (H2)) and our choice of ν that inequality (2.92) is in place. Hence, ℚ ⊂ ℒℳX (ℙ), which proves part (i) of the theorem. (ii) Now let us show that the measure ℚ from (2.92) with ν = 0 is an EMM. Denote α(s) := (r − μ)/σ(Y(s)). Knowing that the measure ℚ is equivalent to the measure ℙ and is defined by Radon–Nikodym derivative L(T) = dℚ | , we may apply the Girdℙ ℱT t

sanov theorem (see Theorem A.69) to derive that the processes Bℚ (t) := B(t)−∫0 α(s)ds, Z ℚ (t) := Z(t), 0 ≤ t ≤ T, are Wiener processes with respect to ℚ. The asset price process under the measure ℚ is a solution of the stochastic differential equation dS(t) = rS(t)dt + σ(Y(t))S(t)dBℚ (t),

which yields the following representation for the discounted price process X(t) := e−rt S(t): t

X(t) = x + ∫ σ(Y(s))X(s)dBℚ (s).

(2.93)

0

Hence, provided that assumption (H2) as mentioned above yields square-integrability of σ(Y(s))S(s) on [0, T], {X(t), t ∈ [0, T]} is a martingale. Therefore, ℚ ⊂ ℳS (ℙ), and by Theorem 2.52 we deduce that the market (X, Y) is NAg . In order to prove NAg and NA+ properties of the market we have proved the existence of one EMM. However we actually have a family of equivalent martingale measures in the market. The discounted price process is in the form of (2.93), which is always a martingale given the square-integrability of σ(Y(s))X(s) and any admissible choice of process ν.

2.4 Stochastic Ornstein–Uhlenbeck volatility | 183

Lemma 2.55. Let the market be defined by (2.86)–(2.87) with assumptions (H1)–(H3) and with additional condition (2.90). Measure ℚ is such that the following inequality holds: T

1 𝔼 exp( ∫ ν2 (s)ds) < ∞. 2 ℚ

0

Then ℚ ∈ ℳX (ℙ). As we have more than one equivalent martingale measure in the market it is straightforward that the market is incomplete. Each EMM in the market is defined by the process ν(s) = ν(s, Y(s), S(s)) associated with it. In financial literature the process ν(s) is called market price of volatility risk. Under EMM ℚν , the pair of processes (S(t), Y(t)) have the following representation: dS(t) = rS(t)dt + σ(Y(t))S(t)dBℚ (t), dY(t) = (−αY(t) − k(ρ

μ−r + ν(t)√1 − ρ2 ))dt + kdW ℚ (t), σ(Y(t))

(2.94)

where the processes t

Bℚ (t) = B(t) + ∫ 0

μ−r ds, σ(Y(s))

W ℚ (t) = ρBℚ (t) + √1 − ρ2 Z ℚ (t),

t ∫0

and Z ℚ (t) = Z(t) + ν(s)ds, are Wiener processes with respect to ℚ according to the Girsanov theorem (see Theorem A.69), where Bℚ and Z ℚ are independent. In the risk-neutral model (2.94) the volatility process is not the Ornstein–Uhlenbeck process anymore. So, generally speaking, there is no analytic solution to the corresponding differential equation. Therefore, below we consider a particular case of the general model which is defined by a set of assumptions concerning the form and behavior of certain parameters of the model. 2.4.4 The case of independent Wiener processes Let us define a modified set of assumptions: (I1) Wiener processes B and W are independent, that is, ρ = 0; (I2) = (H2) and (I3) = (H3). Assumption (I1) simplifies the risk-neutral model to the following form: dS(t) = rS(t)dt + σ(Y(t))S(t)dBℚ (t),

dY(t) = (−αY(t) − kν(t))dt + kdZ ℚ (t),

(2.95)

184 | 2 Rate of convergence of asset and option prices t

t

μ−r

where Bℚ (t) = B(t) + ∫0 σ(Y(s)) ds and Z ℚ (t) = Z(t) + ∫0 ν(s)ds are independent Wiener processes with respect to ℚ. Our purpose is to price a European call option in the model (2.95). We limit further investigation to the valuation with respect to the minimal martingale measure; see Definition A.56. Theorem 2.56. EMM ℚ in the market defined by the model (2.95) is a minimal martingale measure if and only if the process ν corresponding to ℚ is identically zero. Proof. Suppose ν(t) = 0, t ∈ 𝕋. Let B be an 𝔽-adapted Wiener process with respect to measure ℙ. If N is a square-integrable ℙ-martingale, then we can apply its Galtchouk– Kunita–Watanabe decomposition, (A.11), to derive t

N(t) = N(0) + ∫ l(u)dB(u) + Z(t), 0

where B and Z are strongly orthogonal, so that ⟨B, Z⟩t = 0, Let N be strongly orthogonal to

t ∫0

a. s.,

t ∈ 𝕋.

σ(u)dB(u). Then t

⋅

0 = ⟨N, ∫ σ(u)dB(u)⟩ = ∫ l(u)σ(u)du, t

0

whence l(t) = 0, t ∈ 𝕋 a. s. Then for L(t) =

dℚ | dℙ ℱt

0

and γ(t) := (r − μ)/σ(t) we have

d(N(t)L(t)) = N(t)dL(t) + L(t)dN(t) + d⟨N, L⟩t

= N(t)dL(t) + L(t)dN(t) + γ(t)l(t)dt = N(t)dL(t) + L(t)dN(t).

The process N(t)L(t) is a local ℙ-martingale; hence, N(t) is a local ℚ-martingale. By definition ℚ is a minimal martingale measure. The reverse statement of the theorem comes straightforward from the uniqueness of the minimal martingale measure. As a result, we consider the scheme dS(t) = rS(t)dt + σ(Y(t))S(t)dBℚ (t),

dY(t) = −αY(t)dt + kdZ ℚ (t), t

(2.96)

t

μ−r

where Bℚ (t) = B(t) + ∫0 σ(Y(s)) ds and Z ℚ (t) = Z(t) + ∫0 ν(s)ds are independent Wiener processes with respect to ℚ. The solution of the differential equation defining the evolution of the asset price has the following representation: t

t

1 S(t) = x exp{rt + ∫ σ(Y(s))dB (s) − ∫ σ 2 (Y(s))ds}, 2 ℚ

0

0

0 ≤ t ≤ T.

(2.97)

2.4 Stochastic Ornstein–Uhlenbeck volatility | 185

For a fixed trajectory of Y the argument of the exponential function in the righthand side of (2.97) is a Gaussian process, and S(t), t ∈ 𝕋, has a log-normal distribution with 1 log S(t) ∼ 𝒩 (log x + (r − σ̄ 2 (t))t, σ̄ 2 (t)t), 2 t

where σ̂ 2 (t) = σ̂ 2 (t)(Y) := 1t ∫0 σ 2 (Y(s))ds, t ∈ 𝕋. This fact is crucial for the derivation of the expression for the value of European call options. The value of a European call option at time 0 with respect to the minimal martingale measure ℚ is defined by the general formula π call (x) = e−rT E ℚ (Sℚ (T) − K) . +

We apply the telescopic property of mathematical expectation and conditional mathematical expectation to transform the previous expression as follows: π call (x) = e−rT 𝔼ℚ {𝔼ℚ {(Sℚ (T) − K) | Y(s), 0 ≤ s ≤ T}}.

(2.98)

+

The inner expectation is conditional on the path of Y(s), 0 ≤ s ≤ T, and therefore, it actually is the Black–Scholes price for a model with deterministic time-dependent volatility. According to Lemma 2.1 in [123], the inner expectation in (2.98) has the following representation: πinner = 𝔼ℚ {(Sℚ (T) − K) | Y(s), 0 ≤ s ≤ T} +

= elog x+rT Φ(

log(x/K) + (r + 21 σ̄ 2 (0))T ̄ √T σ(0)

) − KΦ(

log(x/K) + (r − 21 σ̄ 2 (0))T ̄ √T σ(0)

=: elog x+rT Φ(d1 ) − KΦ(d2 ),

) (2.99)

T

̄ where σ(t)(Y) := √ T1 ∫t σ 2 (Y(s))ds ≥ 0, Φ is the standard normal cumulative distribu-

tion function,

d1 = d2 =

log(x/K) + (r + 21 σ̄ 2 (0))T ̄ √T σ(0)

,

̄ √T σ(0)

.

log(x/K) + (r − 21 σ̄ 2 (0))T

and

Note that σ̄ 2 (0)(Y) = σ̂ 2 (T)(Y). The former notation may be viewed as the volatility averaged from the current moment to maturity while the latter is the volatility averaged from the initial moment to the current one. Note that the inner conditional expectation is an increasing function of σ̄ 2 (0) (see Lemma 1.49 in this book and Lemma 3.1 in [123]), which is the type of behavior one may expect to be exhibited by the Black–Scholes price of European call options.

186 | 2 Rate of convergence of asset and option prices Taking into account the form of the inner integral, in order to derive the analytic expression for the price of an option V0 , it is necessary to deal with expectations of Φ(⋅) in (2.99) with respect to the trajectory of Y. Instead of trying to evaluate the integral analytically it is possible to use the Monte Carlo method. 2.4.5 Derivation of an analytic expression for the option price From equalities (2.98)–(2.99) one can see that in order to derive the formula for the option price it is necessary to present the exact formula for the expectation of Φ(⋅) in (2.99) with respect to the trajectory of Y. In this connection, we apply the inverse Fourier transform after rearrangement of the right-hand side of (2.99). We introduce deterministic functions σj = σj (s), j = 1, 4, of the form σ1,2 (s) =

s s2 − 2(log (x/K) + rT) , ∓√ √T T

(2.100)

σ3,4 (s) =

−s √ s2 + 2(log (x/K) + rT) . ∓ √T T

(2.101)

We define the domains of each of the above functions to guarantee the non-negativity of the expressions under square root, that is, s2 ≥ 2(log (x/K) + rT) for σ1 and σ2 and s2 ≥ −2(log (x/K) + rT) for σ3 and σ4 . Lemma 2.57. Suppose the market is defined by the model (2.95) with assumptions (I1)– (I3), ℚ is a minimal martingale measure, and π call (x) is the price of a European call option at time 0. Then the following representations hold: 1) For log (x/K) + rT ≥ 0 and k = √2(log (x/K) + rT), π

call

∞

1 ̄ (x) = xe (Φ(k) + < σ1 (s)) ∫ (ℚ(σ(0) √2π rT

k

2

̄ + ℚ(σ(0) > σ2 (s)))e−s /2 ds) ∞

2 1 ̄ − K(Φ(0) + ( ∫ ℚ(σ(0) < σ4 (s))e−s /2 ds √2π

0

0

2

̄ − ∫ ℚ(σ(0) > σ4 (s))e−s /2 ds)); −∞

(2.102)

2.4 Stochastic Ornstein–Uhlenbeck volatility | 187

2) for log (x/K) + rT < 0 and l = √−2(log (x/K) + rT), ∞

2 1 1 ̄ π call (x) = xerT ( + ( ∫ ℚ(σ(0) > σ2 (s))e−s /2 ds 2 √2π

0

0

2

̄ − ∫ ℚ(σ(0) < σ2 (s))e−s /2 ds))

(2.103)

−∞ −l

− K(Φ(−l) −

1 ̄ < σ3 (s)) ∫ (ℚ(σ(0) √2π −∞

2

̄ + ℚ(σ(0) > σ4 (s)))e−s /2 ds). Proof. From (2.98) and (2.99) we have π call (x) = xerT 𝔼ℚ (Φ(d1 )) − K𝔼ℚ (Φ(d2 )), where d1 and d2 are defined as follows: d1 =

log(x/K) + (r + 21 σ̄ 2 (0))T d1

Φ(d1 ) =

̄ √T σ(0)

,

̄ √T, d2 = d1 − σ(0)

(2.104)

2 1 ∫ e−s /2 ds √2π

−∞

d1

0

1 1 1 −s2 /2 −s2 /2 = + ds. 1{d1 >0} ∫ e ds − 1{d1 d1 )e−s /2 ds. √2π

−∞

Probabilities from the integrands may be represented as follows: 1 ̄ √T + log (x/K) + rT > 0), ℚ(s < d1 ) = ℚ( σ̄ 2 (0)T − sσ(0) 2 1 ̄ √T + log (x/K) + rT < 0). ℚ(s > d1 ) = ℚ( σ̄ 2 (0)T − sσ(0) 2

(2.105)

188 | 2 Rate of convergence of asset and option prices Similarly, for Φ(d2 ), we have d2

Φ(d2 ) =

2 1 ∫ e−s /2 ds √2π

−∞

d2

0

1 1 1 −s2 /2 −s2 /2 ds − ds. 1{d2 >0} ∫ e 1{d2 d2 )e−s /2 ds. √2π

−∞

Probabilities from the integrands may be represented as follows: 1 ̄ √T − log (S/K) − rT < 0), ℚ(s < d2 ) = ℚ( σ̄ 2 (0)T + sσ(0) 2 1 ̄ √T − log (S/K) − rT > 0). ℚ(s > d2 ) = ℚ( σ̄ 2 (0)T + sσ(0) 2 Solutions of the quadratic equations, which correspond to the above quadratic inequalities, do not necessarily exist; therefore, we consider different cases: 1) The discriminant D12 := s2 T − 2T(log (x/K) + rT) is a quadratic form with respect to s. There are two possibilities: 1.1) Let log (x/K) + rT > 0. Then for k = √2(log (x/K) + rT) we have D12 < 0 if s ∈ (−k; k) and D12 > 0 if s ∈ (−∞; −k) ∪ (k; ∞). Therefore k

∞

0

k

2 1 1 ̄ 𝔼 (Φ(d1 )) = + (∫ e−s /2 ds + ∫ (ℚ(σ(0) < σ1 (s)) 2 √2π

ℚ

2

̄ + ℚ(σ(0) > σ2 (s)))e−s /2 ds) −k

2 1 ̄ ̄ − < σ2 (s)) − ℚ(σ(0) < σ1 (s)))e−s /2 ds. ∫ (ℚ(σ(0) √2π

−∞

1.2) Let log (x/K) + rT ≤ 0. Then for any s ∈ ℝ we have D12 > 0; so ∞

1 1 ̄ 𝔼 (Φ(d1 )) = + < σ1 (s)) ∫ (ℚ(σ(0) 2 √2π ℚ

0

2.4 Stochastic Ornstein–Uhlenbeck volatility | 189

2

̄ + ℚ(σ(0) > σ2 (s)))e−s /2 ds 0

2 1 ̄ ̄ < σ2 (s)) − ℚ(σ(0) < σ1 (s)))e−s /2 ds. − ∫ (ℚ(σ(0) √2π

−∞

2

2) The discriminant D34 := s T + 2T(log (x/K) + rT) is a quadratic form with respect to s. There are two possibilities: 2.1) Let log (x/K) + rT < 0. Then for l = √−2(log (x/K) + rT), we have D34 < 0 for s ∈ (−l; l) and D34 > 0 for s ∈ (−∞; −l) ∪ (l; ∞). Therefore ∞

1 1 ̄ < σ4 (s)) 𝔼 (Φ(d2 )) = + ∫ (ℚ(σ(0) 2 √2π ℚ

l

2

̄ − ℚ(σ(0) < σ3 (s)))e−s /2 ds −

0

2 1 (∫ e−s /2 ds √2π

−l

−l

2

̄ ̄ < σ3 (s)) + ℚ(σ(0) > σ4 (s)))e−s /2 ds). + ∫ (ℚ(σ(0) −∞

2.2) Let log (x/K) + rT ≥ 0. Then for any s ∈ ℝ we have D34 > 0, so ∞

𝔼ℚ (Φ(d2 )) =

1 1 ̄ + < σ4 (s)) ∫ (ℚ(σ(0) 2 √2π 0

2

̄ − ℚ(σ(0) < σ3 (s)))e−s /2 ds 0

2 1 ̄ ̄ < σ3 (s)) + ℚ(σ(0) > σ4 (s)))e−s /2 ds. − ∫ (ℚ(σ(0) √2π

−∞

Combining the above cases, we get the following expressions for the option price: 1) In the case where log (x/K) + rT ≥ 0 we conclude that π

call

∞

1 ̄ (x) = xe (Φ(k) + ( ∫ (ℚ(σ(0) < σ1 (s)) √2π rT

k

̄ + ℚ(σ(0) > σ2 (s)))e −k

−s2 /2

ds 2

̄ ̄ − ∫ (ℚ(σ(0) < σ2 (s)) − ℚ(σ(0) < σ1 (s)))e−s /2 ds)) −∞ ∞

2 1 1 ̄ ̄ ( ∫ (ℚ(σ(0) < σ4 (s)) − ℚ(σ(0) < σ3 (s)))e−s /2 ds − K( + 2 √2π

0

0

2

̄ ̄ − ∫ (ℚ(σ(0) < σ3 (s)) + ℚ(σ(0) > σ4 (s)))e−s /2 ds)). −∞

(2.106)

190 | 2 Rate of convergence of asset and option prices 2) In the case where log (x/K) + rT < 0 we conclude that π

call

∞

1 1 ̄ ( ∫ (ℚ(σ(0) < σ1 (s)) (x) = xe ( + 2 √2π rT

0

2

̄ + ℚ(σ(0) > σ2 (s)))e−s /2 ds 0

2

̄ ̄ − ∫ (ℚ(σ(0) < σ2 (s)) − ℚ(σ(0) < σ1 (s)))e−s /2 ds)) (2.107)

−∞ ∞

1 ̄ − K(Φ(−l) + < σ4 (s)) ( ∫ (ℚ(σ(0) √2π l

̄ − ℚ(σ(0) < σ3 (s)))e

−s2 /2

−l

ds 2

̄ ̄ < σ3 (s)) + ℚ(σ(0) > σ4 (s)))e−s /2 ds)). − ∫ (ℚ(σ(0) −∞

̄ Recalling that σ(0) ≥ 0 and noting that some of the probabilities presented above are identically zero we simplify (2.106) and (2.107) to the forms (2.102) and (2.103), respectively. Let Sj ⊂ ℝ be the domains of positivity of functions σj (s), 1 ≤ j ≤ 4. It is easy to check that the functions appearing in integrals (2.102)–(2.103) are positive on the integration domains. Assume that the probability density function of σ̄ 2 (0) is piecewise continuous on ℝ. Then due to the Fourier inversion theorem for almost all s ∈ Sj the probabilities in the integrands in (2.102)–(2.103) have the following representation: ̄ ℚ(σ(0) < σj (s)) = ℚ(σ̄ 2 (0) < σj2 (s)) σj2 (s)

∞

−∞

−∞

1 ε2 u2 = lim )ϕ(u)du)dy, ∫ ( ∫ exp(iyu − ε→0 2π 2

(2.108)

2

where ϕ(u) = 𝔼ℚ (eiuσ̄ (0) ) is the characteristic function of σ̄ 2 (0), i2 = −1. We are now in a position to state the main result of this subsection. Theorem 2.58. Suppose the market is defined by the model (2.95) with assumptions (I1)–(I3), ℚ is the minimal martingale measure, and π call (S) is the price at time 0 of a European call option. Let the probability density function of σ̄ 2 (0) be piecewise continuous on ℝ. Then the following representations hold:

2.4 Stochastic Ornstein–Uhlenbeck volatility | 191

1)

For log (x/K) + rT ≥ 0 and k = √2(log (x/K) + rT) we have π

call

2 ∞ σ1 (s) ∞

1 (x) = lim (xe (Φ(k) + ( ∫ ( ∫ ∫ exp(iyu ε→0 (2π)3/2 rT

k

∞

2 2

−

−∞ −∞

∞

2 εu ε 2 u2 )ϕ(u)dudy + ∫ ∫ exp(iyu − )ϕ(u)dudy)e−s /2 ds)) 2 2

σ22 (s) −∞ 2

∞ σ4 (s) ∞

1 1 − K( + ( ∫ ∫ ∫ exp(iyu 2 (2π)3/2 0 −∞ −∞

2 2

−

2 εu )ϕ(u)dudye−s /2 ds 2

0

∞

∞

+ ∫ ∫ ∫ exp(iyu − −∞ σ 2 (s) −∞

2 ε 2 u2 )ϕ(u)dudye−s /2 ds))). 2

4

2) For log (x/K) + rT < 0 and l = √−2(log (x/K) + rT) we have ∞ ∞

∞

1 1 π call (x) = lim (xerT ( + ( ∫ ∫ ∫ exp(iyu ε→0 2 (2π)3/2 0 σ22 (s) −∞

−

2 ε2 u2 )ϕ(u)dudye−s /2 ds 2 2 0 σ2 (s) ∞

− ∫ ∫ ∫ exp(iyu − −∞ −∞ −∞

2 ε 2 u2 )ϕ(u)dudye−s /2 ds)) 2

−l

σ32 (s) ∞

−∞

−∞ −∞

1 ε 2 u2 − K(Φ(−l) − )ϕ(u)dudy ( exp(iyu − ∫ ∫ ∫ 2 (2π)3/2 ∞

∞

+ ∫ ∫ exp(iyu − σ42 (s) −∞

2 2

2 εu )ϕ(u)dudy)e−s /2 ds)), 2

2

where ϕ(u) = 𝔼ℚ (eiuσ̄ (0) ) is the characteristic function of the random variable σ̄ 2 (0); σi = σi (s), i = 1, 4, are of the form (2.100), (2.101). ̄ Remark 2.59. If σ(0) ∈ ℒ2 (ℝ), then the limit in Theorem 2.58 may be moved inside the integrals. Thus, ϵ may be equated to zero and the expression for the option price is simplified. Remark 2.60. Under the assumption that σ is bounded we can rewrite the analytical expression in terms of moments of σ̄ 2 (0).

192 | 2 Rate of convergence of asset and option prices Indeed, in this case σ̄ 2 (0) is bounded as well, so the characteristic function ϕ(u) admits the Taylor series expansion around zero: ∞ j j

ϕ(u) = 1 + ∑ j=1

iu m, j! j

(2.109)

where mn is the nth moment of random variable σ̄ 2 (0). Moments of random variable σ̄ 2 (0) can be represented applying the fact that finite-dimensional distributions of Ornstein–Uhlenbeck processes are Gaussian vectors. Bearing in mind that the covariance matrix of the process Y is non-degenerate and consists of the elements of the form j

(Σn,l )n,l=1 =

k2 exp (−α(tn + tl )) 2α × (exp(2α min(tn , tl )) − 1),

(2.110)

we get the following representation for the moments of random variable σ̄ 2 (0): T

T

0

0 ℝj

σ 2 (y1 ) . . . σ 2 (yj ) 1 mj = j ∫ . . . ∫ ∫ T (2π)j/2 |Σ|1/2 1

× e− 2 (y−μ)

Σ (y−μ)

⊤ −1

(2.111)

dydt1 . . . dtj ,

where y = (y1 , . . . , yj ), dy = dy1 × ⋅ ⋅ ⋅ × dyj , μ = (Y(0)e−αy1 , . . . , Y(0)e−αyj ). We have demonstrated that there is an analytic solution to the problem of pricing of European call options in the model. However, the resulting formula is complicated and cumbersome. Therefore, our further investigation will be aimed at comparison of numerical results produced by it with approximate calculations and at possible simplifications. 2.4.6 Discrete approximation of volatility processes The continuous-time model (2.96) admits a variety of discrete-time approximations. Here we apply an Euler–Maruyama scheme also referred to as Euler scheme. The Euler–Maruyama approximation of the solution of the Langevin equation (the second equation from (2.96)) is the Markov chain Y (m) defined as follows: – We create the partition of the interval [0, T] into m equal subintervals of width Δt = T/m. – The initial value of the scheme is fixed: Y (m) (0) = y. (m) – We will use Yl+1 as a shorthand for Y (m) ((l + 1)T/m), 0 ≤ l ≤ m − 1, which is recursively defined by (m) Yl+1 = (1 − αΔt)Yl(m) + kΔZlℚ ,

where ΔZlℚ = Z ℚ ((l + 1)T/m) − Z ℚ (lT/m).

(2.112)

2.4 Stochastic Ornstein–Uhlenbeck volatility | 193

The continuous-time process Y (m) (t) is a step-type process defined by Y (m) (t) = Y (m) (⌊tm/T⌋T/m),

t ∈ 𝕋,

where, as always, ⌊x⌋ denotes a biggest integer not exceeding x. 2.4.7 The price of European call options The price of a European call option π call (x) at the initial moment of time in model (2.96) is provided by formulas (2.98)–(2.99). Our aim is to estimate the error arising as a result of approximation of the exact formulas (2.98)–(2.99) by application of Euler approximation to the process which drives volatility. Thus, we need to assess the expectation of R given by 󵄨 󵄨 R := 󵄨󵄨󵄨πinner − π̂inner (m)󵄨󵄨󵄨,

(2.113)

where πinner is given by formula (2.99), m is the number of discretization points dividing the time interval [0, T] into equal intervals, and π̂inner (m) denotes the price of the option in a discrete setting calculated using a formula similar to (2.99): π̂inner (m) = elog x+rT Φ(d1(m) ) − KΦ(d2(m) ),

(2.114)

where d1(m) = d2(m)

=

2 )T log(x/K) + (r + 21 σ̄ m

σ̄ m √T

,

σ̄ m √T

.

2 log(x/K) + (r − 21 σ̄ m )T

(2.115)

The above uses the following notation: 1/2

1 m T σ̄ m = ( ∑ σ 2 (Yl(m) ) ) T l=1 m

1/2

1 m = ( ∑ σ 2 (Yl(m) )) , m l=1

(2.116)

where Yl(m) is defined in (2.112). It is unlikely that we are not able to find an exact or even approximate value for R. However, what really makes the investigation of the above bundle of models interesting is the rate of convergence of the discrete setting to the continuous one. In order to assess the rate of convergence the expression for the upper bound of R in terms of m needs to be derived. Comparing (2.99) and (2.114) it can be seen that the approximation error arises solely due to the difference between σ̄ and σ̄ m . So the first step would be to assess the upper bound of expectation of the absolute value of this difference with respect to m. After that R might be expressed in terms of Rσ := 𝔼|σ̄ − σ̄ m |.

194 | 2 Rate of convergence of asset and option prices Lemma 2.61. Let σ 2 (x) satisfy the Hölder condition: 󵄨 󵄨󵄨 2 γ 2 󵄨󵄨σ (x) − σ (y)󵄨󵄨󵄨 ≤ |x − y| ,

(2.117)

where 0 < γ ≤ 1 and L is some positive constant. Then 𝔼Rσ ≤ Cm−0.5γ , where C is some positive constant. Proof. As σ̄ m and σ̄ are both square root functions it would be more convenient to work 2 with σ̄ m and σ̄ 2 . To this end we will use Hölder’s inequality: 1/2 1/2 󵄨 T 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 1 m 2 (m) 󵄨 2 󵄨 ̄ ̄ 𝔼|σm − σ| = 𝔼󵄨󵄨( ∫ σ (Y(s))ds) − ( ∑ σ (Yi )) 󵄨󵄨󵄨 󵄨󵄨 T 󵄨󵄨 m i=1 󵄨 󵄨 0

󵄨󵄨 1/2 󵄨󵄨 T 󵄨󵄨 󵄨󵄨 1 1 m ≤ 𝔼(󵄨󵄨󵄨 ∫ σ 2 (Y(s))ds − ∑ σ 2 (Yi(m) )󵄨󵄨󵄨) 󵄨󵄨 󵄨󵄨 T m i=1 󵄨 󵄨 0

󵄨󵄨 T 󵄨 1/2 󵄨󵄨 1 1 m 2 (m) 󵄨󵄨󵄨󵄨 2 󵄨 ≤ (𝔼󵄨󵄨 ∫ σ (Y(s))ds − ∑ σ (Yi )󵄨󵄨) . 󵄨󵄨 T 󵄨󵄨 m i=1 󵄨 0 󵄨

Now we represent the integral as a sum of integrals over shorter intervals. As the second summand does not depend on s we may move it inside the integral sign multiplying it by the inverse to the interval length: 󵄨󵄨m−1 󵄨󵄨 1 𝔼|σ̄ m − σ|̄ ≤ (𝔼󵄨󵄨󵄨 ∑ ( 󵄨󵄨 T 󵄨 i=0

(i+1)T/m

󵄨󵄨m−1 󵄨󵄨 1 = (𝔼󵄨󵄨󵄨 ∑ ( 󵄨󵄨 T 󵄨 i=0

(i+1)T/m

∫

σ 2 (Y(s))ds −

󵄨 1/2 1 2 (m) 󵄨󵄨󵄨󵄨 σ (Yi+1 ))󵄨󵄨) 󵄨󵄨 m 󵄨

σ 2 (Y(s))ds −

1 m mT

iT/m

∫ iT/m

(i+1)T/m

∫ iT/m

󵄨󵄨 1/2 󵄨󵄨 (m) σ 2 (Yi+1 )ds)󵄨󵄨󵄨) 󵄨󵄨 󵄨

󵄨󵄨 m−1 (i+1)T/m 󵄨󵄨 1/2 󵄨󵄨 󵄨󵄨 1 2 2 (m) 󵄨 = (𝔼󵄨󵄨 ∑ ∫ (σ (Y(s)) − σ (Yi+1 ))ds󵄨󵄨󵄨) . 󵄨󵄨 󵄨󵄨 T 󵄨 󵄨 i=0 iT/m Now we can apply the Hölder property of σ 2 (x): 󵄨󵄨 m−1 (i+1)T/m 󵄨󵄨 1/2 󵄨󵄨 1 󵄨󵄨 2 2 (m) (𝔼󵄨󵄨󵄨 ∑ ∫ (σ (Y(s)) − σ (Yi+1 ))ds󵄨󵄨󵄨) 󵄨󵄨 T 󵄨󵄨 󵄨 i=0 iT/m 󵄨 m−1 L ≤ ( 𝔼( ∑ T i=0

=(

L m−1 ∑ T i=0

∫ iT/m

(i+1)T/m

where L is some positive constant.

1/2

(i+1)T/m

∫ iT/m

󵄨󵄨 (m) 󵄨γ 󵄨󵄨Y(s) − Yi+1 󵄨󵄨󵄨 ds)) 1/2

󵄨 (m) 󵄨󵄨γ 𝔼󵄨󵄨󵄨Y(s) − Yi+1 󵄨󵄨 ds) ,

2.4 Stochastic Ornstein–Uhlenbeck volatility | 195

Recall that Yi(m) is a shorthand for Y (m) (iT/m) = Y (m) (s),

s ∈ [iT/m, (i + 1)T/m),

(m) and Proposition A.129 from the appendix yields that 𝔼|Y(s) − Yi+1 | ≤ C1 m−1 , where C1 (m) γ is some positive constant. We use Hölder’s inequality to derive that 𝔼|Y(s) − Yi+1 | ≤ γ −γ C1 m , and we arrive at 1/2

L T γ 𝔼|σ̄ m − σ|̄ ≤ ( m C1 m−γ ) T m

= Cm−γ/2 ,

γ

for C := √LC1 , which proves the lemma. The above lemma enables us to prove the main result of this subsection. γ

Theorem 2.62. Let σ 2 (x) satisfy Hölder condition (2.117). Then 𝔼R ≤ Dm− 2 , where D is some positive constant. Proof. Function Φ(x) has a continuous bounded derivative on ℝ; hence we can use its Lipschitz property and obtain 󵄨 󵄨 𝔼R = 𝔼󵄨󵄨󵄨πinner − π̂inner (m)󵄨󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 𝔼(SerT 󵄨󵄨󵄨Φ(d1 ) − Φ(d1(m) )󵄨󵄨󵄨 + K 󵄨󵄨󵄨Φ(d2 ) − Φ(d2(m) )󵄨󵄨󵄨) 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ 𝔼(sup󵄨󵄨󵄨f (x)󵄨󵄨󵄨(SerT 󵄨󵄨󵄨d1 − d1(m) 󵄨󵄨󵄨 + K 󵄨󵄨󵄨d2 − d2(m) 󵄨󵄨󵄨)), x x2

where f (x) = √12π e− 2 is the probability density function of the standard normal distribution. In the above representation 󵄨󵄨 󵄨󵄨 1 1 log(x/K) + rT 1 √ 󵄨󵄨 󵄨 󵄨 (m) 󵄨 + ) T(σ̄ − σ̄ m )󵄨󵄨󵄨 (2.118) 󵄨󵄨d1 − d1 󵄨󵄨󵄨 = 󵄨󵄨󵄨( − √T 2 󵄨󵄨 󵄨󵄨 σ̄ σ̄ m 󵄨󵄨 1 log(x/K) + rT √T 󵄨󵄨 󵄨󵄨 󵄨 + (2.119) ≤ |σ̄ − σ̄ m |󵄨󵄨󵄨 󵄨 󵄨󵄨 σ̄ σ̄ m √T 2 󵄨󵄨󵄨 󵄨󵄨 log(x/K) + rT √T 󵄨󵄨 󵄨󵄨 󵄨 + ≤ |σ̄ − σ̄ m |󵄨󵄨󵄨 (2.120) 󵄨, 󵄨󵄨 2 󵄨󵄨󵄨 c2 √T

where c is a positive constant and the last inequality is due to the assumption that σ(x) is bounded away from zero for any x ∈ ℝ (see assumption (H2) above). Hence, using Lemma 2.61 we get 󵄨 󵄨 𝔼󵄨󵄨󵄨d1 − d1(m) 󵄨󵄨󵄨 ≤ C1 𝔼|σ̄ − σ̄ m | ≤ D1 m−γ/2 ,

where C1 := | log(x/K)+rT + √ c T

√T | 2

Similarly, 𝔼|d2 − d2(m) |

and D1 are positive constants.

≤ D2 m−γ/2 , where D2 is a positive constant, and we arrive at

󵄨 󵄨 𝔼R = 𝔼󵄨󵄨󵄨πinner − π̂inner (m)󵄨󵄨󵄨 ≤

γ γ γ 1 (D1 SerT m− 2 + D2 Km− 2 ) = Dm− 2 , √2π

for a positive constant D. The theorem is proved.

(2.121)

196 | 2 Rate of convergence of asset and option prices 2.4.8 Numerical examples Theorem 2.58 provides an analytic representation for the price of European call options for the stochastic volatility model under consideration. However, using it to calculate the price of an option is rather difficult and time consuming. Below we present the results of calculation of the price of European call options using simulation techniques. The calculation process is performed in MATLAB 7.9.0 and is structured as follows: 1. The choice of discrete ranges of values of input parameters. 2. The choice of the function σ(Y(s)). 3. For each combination of input parameters we generate 1000 trajectories of the Ornstein–Uhlenbeck process by splitting the time interval into subintervals of length Δt = 0.001 and modeling values of the Ornstein–Uhlenbeck process in these points (that is, generating normally distributed variables with known mean and standard deviation using relationship (2.112)). For each trajectory, (2.114) is 2 applied to calculate σ̄ m and a price of an option. Results for all trajectories are then averaged and discounted to provide the sample average of the price denoted call by 𝔼̂ π̂ m . Average volatility over all trajectories and time intervals is denoted by 2 ̂ 𝔼σ̄ m . To begin with, let us recall the notation of input parameters along with ranges of values assigned to them in the process of simulation: T – time to maturity, T = 0.25; 0.5; 1; k – volatility of the Ornstein–Uhlenbeck process, k = 0.1; 0.5; 1; α – mean-reversion rate, α = 1; 100; r – interest rate, r = 0; 0.01; 0.02; K – strike price, K = 0.8; 1; 1.2; x – initial price of stock, x = 1; y – initial value of the Ornstein–Uhlenbeck process, y = 0.1. In order to produce numerical results we choose the following options for function σ(x): 1. σ 2 (x) = a|x| + b, where a = {0, 1}, b = {0, 0.2, 1}; 2. σ 2 (x) = ex + c, c = 0.02. These functions correspond to Tables 2.1 and 2.2, respectively. The results of simulations are split into groups by mean-reversion rate α and function σ(x). Meaningless and non-interesting results provided by some distinct combinations of inputs are ignored. A mean-reversion of 1 corresponds to slow reverting models and fast meanreverting models are characterized by α = 100. Matters of speed of mean-reversion are addressed, for example, in [62].

2.4 Stochastic Ornstein–Uhlenbeck volatility | 197 Table 2.1: σ 2 (x) = a|x| + b. T

k

r

K

a

b

̄2 𝔼̂ σm

call 𝔼̂ π̂m

̄2 𝔼̂ σm

call 𝔼̂ π̂m

0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1

0.1 0.1 0.1 0.5 0.5 0.5 1 1 1 0.1 0.1 0.1 0.5 0.5 0.5 1 1 1 0.1 0.1 0.1 0.5 0.5 0.5 1 1 1

0 0 0 0 0 0 0 0 0 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 1 1 1 1 1 1 1 1 1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1 1 1 1 1 1 1 1 1

0.088 0.082 0.073 0.147 0.185 0.216 0.264 0.338 0.412 0.289 0.281 0.273 0.346 0.375 0.414 0.459 0.532 0.617 1.089 1.079 1.073 1.148 1.178 1.216 1.262 1.341 1.414

α=1 0.204 0.213 0.227 0.211 0.235 0.280 0.224 0.264 0.334 0.108 0.151 0.210 0.117 0.172 0.254 0.134 0.203 0.305 0.141 0.228 0.347 0.147 0.240 0.371 0.157 0.260 0.402

0.009 0.007 0.007 0.031 0.030 0.029 0.059 0.058 0.058 0.209 0.207 0.207 0.231 0.230 0.229 0.259 0.258 0.258 1.009 1.007 1.007 1.031 1.030 1.029 1.059 1.058 1.058

α = 100 0.200 0.200 0.200 0.200 0.201 0.207 0.201 0.207 0.221 0.092 0.130 0.184 0.097 0.137 0.193 0.102 0.145 0.204 0.134 0.218 0.335 0.136 0.221 0.339 0.138 0.225 0.344

2 One might observe that under faster mean-reversion the average volatility 𝔼̂ σ̄ m and consequently the price of the option are lower, which is exactly what is expected from the model. Tables 2.3 and 2.4 illustrate how the price of the option changes with decreasing time steps in the discrete model. In view of Section 2.4.7 it is also of certain interest to compare calculations obtained over one trajectory but under different discretization steps. We constructed 2000 trajectories with a time step size of 10−6 : 1000 for the case α = 1 and 1000 for the case α = 100. These trajectories are considered to be “true” continuous-time trajecto2 ries of Ornstein–Uhlenbeck process Y(t). The corresponding values of σ̄ m are consid2 ̄ ered to be “true” continuous-time values of σ . The calculations were then performed for wider discretization intervals using the points of constructed trajectories. Thus, 2 2 the samples of discretization errors for σ̄ m were derived. Probably, the estimate of σ̄ m

198 | 2 Rate of convergence of asset and option prices Table 2.2: σ 2 (x) = ex + c. T

k

r

K

̄2 𝔼̂ σm

call 𝔼̂ π̂m

̄2 𝔼̂ σm

call 𝔼̂ π̂m

0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1

0.1 0.1 0.1 0.5 0.5 0.5 1 1 1 0.1 0.1 0.1 0.5 0.5 0.5 1 1 1 0.1 0.1 0.1 0.5 0.5 0.5 1 1 1

0 0 0 0 0 0 0 0 0 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 1 1 1 1 1 1 1 1 1 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2

1.113 1.103 1.088 1.135 1.131 1.119 1.184 1.212 1.238 1.112 1.103 1.086 1.121 1.129 1.128 1.178 1.206 1.216 1.110 1.103 1.087 1.133 1.128 1.115 1.162 1.201 1.255

α=1 0.303 0.372 0.465 0.305 0.374 0.468 0.307 0.380 0.478 0.209 0.291 0.401 0.209 0.294 0.405 0.213 0.299 0.412 0.143 0.231 0.349 0.145 0.233 0.352 0.147 0.239 0.367

1.024 1.022 1.021 1.025 1.023 1.022 1.027 1.025 1.024 1.024 1.022 1.021 1.025 1.023 1.022 1.026 1.025 1.023 1.024 1.022 1.021 1.025 1.023 1.021 1.027 1.025 1.023

α = 100 0.297 0.363 0.456 0.297 0.363 0.456 0.297 0.363 0.456 0.201 0.281 0.390 0.201 0.281 0.390 0.201 0.281 0.390 0.135 0.220 0.338 0.135 0.220 0.338 0.135 0.220 0.338

Table 2.3: Rate of convergence. σ 2 (x) = |x| + 0.2, K = 1, r = 0.02, k = 0.1, T = 1. Δt

α

̄2 𝔼̂ σm

d1̄ (m)

d2̄ (m)

call 𝔼̂ π̂m

10−2 10−3 10−4 10−5 10−6 10−2 10−3 10−4 10−5 10−6

1 1 1 1 1 100 100 100 100 100

0.272367 0.271043 0.272534 0.271837 0.271421 0.208910 0.206599 0.206439 0.206413 0.206443

0.299043 0.298506 0.299123 0.298822 0.298667 0.272291 0.271267 0.271196 0.271184 0.271198

−0.222056 −0.221338 −0.222179 −0.221753 −0.221560 −0.184776 −0.183264 −0.183159 −0.183142 −0.183162

0.213552 0.213073 0.213631 0.213351 0.213220 0.189047 0.188073 0.188005 0.187994 0.188007

2.4 Stochastic Ornstein–Uhlenbeck volatility | 199 Table 2.4: Rate of convergence. σ 2 (x) = ex + 0.2, K = 1, r = 0.02, k = 0.1, T = 1. Δt

α

̄2 𝔼̂ σm

d1̄ (m)

d2̄ (m)

call 𝔼̂ π̂m

10−2 10−3 10−4 10−5 10−6 10−2 10−3 10−4 10−5 10−6

1 1 1 1 1 100 100 100 100 100

1.265414 1.269504 1.266274 1.266030 1.265635 1.201083 1.201047 1.201026 1.201036 1.201023

0.556279 0.579243 0.584925 0.566934 0.576169 0.566092 0.566500 0.566203 0.566693 0.566052

−0.519082 −0.543620 −0.549670 −0.530485 −0.540343 −0.529585 −0.530021 −0.529703 −0.530228 −0.529542

0.431865 0.432472 0.431990 0.431948 0.431892 0.422128 0.422123 0.422120 0.422121 0.422119

is more valuable in such context as one would not usually calculate the price of an option over one trajectory. However the estimate of volatility is usually derived from past data, which is in essence one distinct realization of the space of all possible scenarios. Tables 2.5 and 2.6 provide characteristics of the samples of discretization errors. Errors are measured as a percentage of the “true” value. Table 2.5: Characteristics of samples of errors. σ 2 (x) = |x| + 0.2, K = 1, r = 0.02, k = 0.1, T = 1. 10−2

10−3

10−4

10−5

Average St. error Median St. deviation Excess Skewness Min Max Count

0.08710 % 0.0000427 0.0009575 0.0013517 −0.217306 0.0492335 −0.29706 % 0.52352 % 1000

0.00834 % 0.0000042 0.0000834 0.0001334 −0.191189 −0.002248 −0.03669 % 0.04766 % 1000

0.00081 % 0.0000004 0.000008 0.0000137 −0.143295 0.023124 −0.00303 % 0.00502 % 1000

α=1 0.00008 % 0 0.0000007 0.0000013 −0.021156 0.0577173 −0.00036 % 0.00044 % 1000

Average St. error Median St. deviation Excess Skewness Min Max Count

0.07790 % 0.000043 0.0008379 0.0013602 −0.234452 −0.024765 −0.30504 % 0.46265 % 1000

0.00742 % 0.0000044 0.0000728 0.0001379 −0.302723 0.0922374 −0.03231 % 0.04974 % 1000

0.00083 % 0.0000004 0.0000083 0.0000136 −0.352995 0.0055451 −0.00323 % 0.00454 % 1000

α = 100 0.00007 % 0 0.0000007 0.0000014 −0.054568 0.0229423 −0.00037 % 0.00050 % 1000

200 | 2 Rate of convergence of asset and option prices Table 2.6: Characteristics of sample of errors. σ 2 (x) = ex + 0.2, K = 1, r = 0.02, k = 0.1, T = 1. 10−2

10−3

10−4

10−5

Average St. error Median St. deviation Excess Skewness Min Max Count

0.02496 % 0.0000113 0.0002559 0.0003584 0.1947561 −0.1691937 −0.09961 % 0.12871 % 1000

0.00268 % 0.0000011 0.0000266 0.0000354 0.1687356 −0.0097185 −0.00861 % 0.01464 % 1000

0.00026 % 0.0000001 0.0000027 0.0000035 −0.0576859 −0.1643507 −0.00088 % 0.00126 % 1000

α=1 0.00002 % 0.00000001 0.0000002 0.0000003 0.0700827 −0.0522007 −0.00011 % 0.00013 % 1000

Average St. error Median St. deviation Excess Skewness Min Max Count

0.02692 % 0.0000118 0.0002712 0.0003735 0.17242 −0.0299531 −0.09174 % 0.16291 % 1000

0.00268 % 0.0000012 0.0000265 0.0000377 0.070383 −0.0195205 −0.01068 % 0.01411 % 1000

0.00025 % 0.0000001 0.0000027 0.0000036 0.3383414 −0.1914745 −0.00112 % 0.00139 % 1000

α = 100 0.00002 % 0 0.0000002 0.0000003 0.0853763 −0.0371876 −0.00011 % 0.00014 % 1000

It can be seen from the above tables that approximation results do not differ significantly for various time steps. Even the widest investigated discretization interval provides acceptable precision for most applications.

2.4.9 Approximation precision check for the case of deterministic volatility In this subsection we compare the option prices obtained for the Euler scheme (2.112) with the true prices of European call options for different sets of parameters for the case of deterministic time-dependent volatility. The models with deterministic time-dependent volatility are natural extensions of the Black–Scholes model. The expression for the price of the option is the same as in the classical model except for the fact that instead of constant volatility it operates with average (or root mean square) volatility over the time interval to maturity (see, e. g., [107, 171]). Thus, the formula remains similar to (2.99) and (2.114). It has been shown that deterministic volatility does not reflect the real world’s stochastic dynamics correctly ([28, 149]) and such models have begun falling out of favor in the mid 1980s. The shift to stochastic volatility models was boosted by the rapid development of computational tools.

2.4 Stochastic Ornstein–Uhlenbeck volatility | 201

Nevertheless, deterministic volatility is suitable for the purpose of our investigation as we can calculate the exact price of the option for the continuous-time model. In order to analyze the deterministic time-dependent volatility case it looks natural to let the Brownian noise term in the definition of process Y vanish. Thus, we get dY(t) = −αY(t)dt,

(2.122)

which is a familiar linear differential equation solved by Y(t) = ye−αt .

(2.123)

For the same transformation functions σ and sets of parameters as in the subsection 2.4.8 we calculate the prices of the European call option in the continuous case call using (2.99) (denoted 𝔼πdet ) and compare them with the prices of the same option calculated using (2.114)–(2.116) with (m) Yl+1 = (1 − αΔt)Yl(m) .

(2.124)

We use the time step of 0.01 and only 10 simulations per combination of inputs. As before all calculations are performed in MATLAB 7.9.0. Table 2.7 presents the results of calculations. Comparison of two approaches reveals that the Euler–Maruyama scheme provides good approximation for the exact option price. The fast mean-reversion results coincide when rounded to the sixth digit. Table 2.7: Approximate option prices versus true option prices for deterministic volatility. T 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1 0.25 0.5 1

α 1 1 1 1 1 1 1 1 1 100 100 100 100 100 100 100 100 100

r 0 0 0 0.01 0.01 0.01 0.02 0.02 0.02 0 0 0 0.01 0.01 0.01 0.02 0.02 0.02

K 0.8 0.8 0.8 1 1 1 1.2 1.2 1.2 0.8 0.8 0.8 1 1 1 1.2 1.2 1.2

a 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

b 0 0 0 0.2 0.2 0.2 1 1 1 0 0 0 0.2 0.2 0.2 1 1 1

call 𝔼̂ π̂m

call 𝔼πdet

σ 2 (x) = a|x| + b 0.203891 0.203888 0.211556 0.211549 0.223003 0.222994 0.107942 0.107935 0.150207 0.150199 0.206464 0.206457 0.141317 0.141313 0.227633 0.227629 0.345261 0.345257 0.200000 0.200000 0.200000 0.200000 0.200000 0.200000 0.091044 0.091044 0.128449 0.128449 0.181507 0.181507 0.133108 0.133108 0.217100 0.217100 0.333759 0.333759

call 𝔼̂ π̂m

call 𝔼πdet

σ 2 (x) = ex + 0.2 0.316223 0.316220 0.390150 0.390147 0.490305 0.490302 0.224736 0.224733 0.312794 0.312791 0.429067 0.429064 0.159958 0.159954 0.253710 0.253706 0.379955 0.379952 0.309950 0.309950 0.382107 0.382106 0.481610 0.481610 0.217149 0.217149 0.303457 0.303457 0.419198 0.419198 0.152065 0.152065 0.243748 0.243748 0.369312 0.369312

202 | 2 Rate of convergence of asset and option prices Remark 2.63. In this section we consider the price of the option at the initial moment of time. However, all the above considerations are applicable for any valuation date t between the initial moment of time and maturity. Some obvious changes need to be T

1 ̄ := √ T−t made, e. g., the function σ(t) ∫t σ 2 (Y(s))ds ≥ 0 needs to be introduced instead

of σ̄ and T needs to be substituted by T − t in (2.98)–(2.99) and (2.114)–(2.116).

2.5 Option pricing with fractional stochastic volatility and discontinuous payoff function of polynomial growth On the one hand, as is clear from Section 2.4, stochastic volatility models are widely used because of their flexibility. On the other hand, it was mentioned in Section 1.9 that the presence of memory in financial markets has already been convincingly recorded. Therefore, a natural idea is to insert the memory of a financial model in its stochastic volatility, and thus achieve two goals: firstly, to take into account the memory of the market, and secondly, to avoid arbitrage in the model. The most simple and suitable for this purpose is an fBm with Hurst index H ∈ (0, 1), where H ∈ (0, 1/2) corresponds to so-called short memory, while H ∈ (1/2, 1) corresponds to so-called long memory. For the definition and some properties of fBm, see Section A.3.3. The financial market models where the asset price includes stochastic volatility with long memory in the volatility process are subject of extensive research activity; see, e. g., [20], where a wide class of fractionally integrated GARCH and EGARCH models for characterizing financial market volatility was studied, [34] for affine fractional stochastic volatility models, and [33], where the Heston model with fractional Ornstein–Uhlenbeck stochastic volatility was studied. As was mentioned in [34], long memory included into the volatility model allows to explain some option pricing puzzles such as steep volatility smiles in long-term options and co-movements between implied and realized volatility. Although the long memory effect corresponds to the case H > 1/2, empirical evidence suggests values H < 1/2 (“rough volatility”); see for example [10] or the discussion in [66], where an approximation formula is proposed. When dealing with “rough volatility,” one may find the decomposition approach developed in [13] useful. Concerning the approach that permits to get the semiclosedform solution for the call option price in the Heston model with jumps in the asset price and semimartingale stochastic volatility with long memory, see [138]. Note again that as a rule, the option pricing in stochastic volatility models needs some approximation procedures including Monte Carlo methods. The present section contains a comprehensive and diverse approach to the exact and approximate option pricing of the asset price model that is described by the linear model with stochastic volatility driven by the fractional Ornstein–Uhlenbeck process with Hurst index H ∈ (0, 1). We assume that the Wiener processes, one of which is

2.5 Fractional stochastic volatility | 203

driving the asset price and another one is the underlying Wiener process for the fBm driving stochastic volatility, are correlated with a constant correlation coefficient. We consider three possible levels of the representation and approximation of the option price, with the corresponding rate of convergence of the discretized option price to the original one. We can rigorously treat the class of discontinuous payoff functions of polynomial growth. As an example, our model allows to analyze linear combinations of a digital and call option. Moreover, we provide rigorous estimates for the rates of convergence of option prices for polynomial discontinuous payoffs f and Hölder volatility coefficients, a crucial feature considering settings for which exact pricing is not possible. The first level of approximation corresponds to the case when the price is presented as a functional of both driving stochastic processes, the Wiener process and the fBm. We discretize and simulate the trajectories of both the Wiener process and the fBm (the double discretization) and estimate the rate of convergence for the discretized model. In these settings we apply the Malliavin calculus technique, following [4], to transform the option price to the form that does not contain discontinuous functions. The second level corresponds to the case when we discretize and simulate only the trajectories of the fBm involved in the Ornstein–Uhlenbeck stochastic volatility process (the single discretization), basically conditioning on the stochastic volatility process, then calculating the corresponding option price as a functional of the fBm trajectory, and finally estimating the rate of convergence of the discretized price. This approach allows to simulate only the trajectories of the fBm. Corresponding simulations are presented and compared to those obtained by the first level. We conjecture that the single discretization gives better simulation results. In general conditioning is widely used in option pricing in various situations (see, e. g., [15] and references therein), so it is not surprising that it helps here as well. The third level potentially permits to avoid simulations, because it is possible to provide an analytical expression for the option price as an integral including the density of the functional which depends on stochastic volatility. However, the density we obtain within the Malliavin calculus framework is complicated from a computational point of view. So, our subject now is a financial market characterized by a finite maturity time T and composed by a risk-free bond, or bank account, β = {β(t), t ∈ [0, T]}, whose dynamic reads as β(t) = ert , where r ∈ ℝ+ represents the risk-free interest rate, and a risky asset S = {S(t), t ∈ [0, T]} whose stochastic price dynamic is defined over the probability space {Ω, ℱ , 𝔽 = {ℱt , t ∈ [0, T]}, ℙ} by the following system of stochastic differential equations: dS(t) = bS(t)dt + σ(Y(t))S(t)dW(t),

(2.125)

dY(t) = −αY(t)dt + dB (t),

(2.126)

H

t ∈ [0, T].

Here W = {W(t), t ∈ [0, T]} is a standard Wiener process, b ∈ ℝ, α > 0 are constants, and Y = {Y(t), t ∈ [0, T]} characterizes the stochastic volatility term of our model,

204 | 2 Rate of convergence of asset and option prices being the argument of the function σ. The process Y is Ornstein–Uhlenbeck, driven by an fBm BH = {BH (t), t ∈ [0, T]} with Hurst parameter H ∈ (0, 1), and is assumed to be correlated with W. We would also like to emphasize that a market model described by the system of equations (2.125) and (2.126) is incomplete because of two sources of uncertainty, whether or not it is arbitrage-free. Therefore, in what follows we focus our attention on the objective measure instead of using an equivalent martingale one. Concerning a minimal martingale measure, contrary to Section 2.4, where we assumed that the Wiener drivers are independent, now they are correlated, and therefore the minimal martingale measure has a more involved form (see Section 2.5.1) and does not help much in calculations. A discussion of the arbitrage-free property of the market under consideration (with additional restrictions of the coefficients), a presentation of the class of martingale measures, and a formula for the minimal martingale measure is contained in Section 2.5.1. For more detail on the arbitrage-free property of the markets with stochastic volatility, see, e. g., [98]. Concerning the payoff function, we consider a measurable one defined by f : ℝ+ → ℝ+ and depending on the value S(T) of the stock at maturity time T. Our main goal is to calculate and approximate 𝔼f (S(T)) using the aforementioned levels, also providing rigorous estimates for the corresponding rate of convergence for the first and second levels. The section is organized as follows: in Section 2.5.1 we give additional assumptions on the components of the model and formulate auxiliary results; Section 2.5.2 contains the necessary elements of the Malliavin calculus supplying the further option pricing; Section 2.5.3 contains the main results on the rate of convergence of the discretized option pricing approach; Section 2.5.4 contains the main results concerning the rate of convergence of the discretized option pricing problem when one uses conditioning on the trajectory of the fBm; Section 2.5.5 is devoted to the analytical derivation of the option price in terms of the density of the volatility functional; and finally, Section 2.5.6 provides the numerical simulations associated to the approaches described in Sections 2.5.3 and 2.5.4. 2.5.1 Payoff function: additional assumptions, auxiliary properties. Discussion of asset price model, absence of arbitrage, martingale measures, incompleteness 2.5.1.1 Assumptions on the payoff function and volatility coefficient In all considerations we assume that the payoff function f : ℝ+ → ℝ+ satisfies the following conditions: (J1) (i) f is a measurable function of polynomial growth, f (x) ≤ Cf (1 + x p ), for some constants Cf > 0 and p > 0;

x ≥ 0,

2.5 Fractional stochastic volatility |

205

(ii) function f is locally Riemann integrable, possibly having discontinuities of the first kind. Moreover we assume that the function σ : ℝ → ℝ satisfies the following conditions: (J2) there exists Cσ > 0 such that: (i) σ is bounded away from 0, σ(x) ≥ σmin > 0, (ii) σ has moderate polynomial growth, i. e., there exists q ∈ (0, 1) such that σ(x) ≤ Cσ (1 + |x|q ),

x ∈ ℝ,

(iii) σ is uniformly Hölder continuous, so that there exists r ∈ (0, 1] such that 󵄨󵄨 󵄨 r 󵄨󵄨σ(x) − σ(y)󵄨󵄨󵄨 ≤ Cσ |x − y| ,

x, y ∈ ℝ,

(iv) σ ∈ C(ℝ) is differentiable a. e. with respect to the Lebesgue measure on ℝ, and its derivative is of polynomial growth: there exists q′ > 0 such that 󵄨󵄨 ′ 󵄨󵄨 q′ 󵄨󵄨σ (x)󵄨󵄨 ≤ Cσ (1 + |x| ) a. e. with respect to the Lebesgue measure on ℝ. Remark 2.64. 1) Concerning the relations between properties (ii) and (iii), note that we allow r = 1 in (iii) whereas (ii) follows from (iii) only in the case r < 1. 2) Concerning the relations between properties (iii) and (iv), neither of these properties implies the other one unless r = 1. Indeed, on the one hand, a typical trajectory of a Wiener process is Hölder up to order 21 but nowhere differentiable. On the other hand, even continuous differentiability does not imply the uniform Hölder property. 3) Concerning assumption (i), we need it for theoretical calculations because in the process of smoothing the payoff function we divide by σ several times. 2.5.1.2 Properties of the fractional Ornstein–Uhlenbeck process fBm admits a compact interval representation via some Wiener process B; see Appendix A.3.3. Specifically, t

BH (t) = ∫ k(t, s)dB(s), 0

H−1/2

t k(t, s) = cH (( ) s

(t − s)H−1/2 t

H− 32

− (H − 1/2)s−H+1/2 ∫ u cH = (

2HΓ( 32 − H)

s

Γ(H + 21 )Γ(2 − 2H)

1/2

) .

(2.127) H− 21

(u − s)

du)1s 1/2 the kernel k(t, s) is simplified to t

1 1 3 1 k(t, s) = (H − )cH s 2 −H ∫ uH− 2 (u − s)H− 2 du1s 0. The next result is almost evident; however, we formulate it and give a short proof for the reader’s convenience. Lemma 2.65. (i) Equation (2.126) has a unique solution of the form t

Y(t) = Y(0)e

+ ∫ e−α(t−s) dBH (s)

−αt

0

:= Y(0)e

−αt

H

t

+ B (t) − α ∫ e−α(t−s) BH (s)ds.

(2.129)

0

Moreover, for any a > 0 and any ϱ < 2, 󵄨 󵄨ϱ 𝔼 exp{a sup 󵄨󵄨󵄨Y(t)󵄨󵄨󵄨 } < ∞.

(2.130)

t∈[0,T]

(ii) Equation (2.125) has a unique solution of the form t

S(t) = S(0) exp{bt + ∫ σ(Y(s))dW(s) − 0

t

1 ∫ σ 2 (Y(s))ds}. 2

(2.131)

0

(iii) For any m ∈ ℤ we have 𝔼(S(T))m < ∞, and consequently for any m > 0 the following relation holds: 𝔼(f (S(T)))m < ∞. Moreover, for any m ∈ ℤ we have t T supt∈[0,T] 𝔼 exp{m ∫0 σ(Y(s))dB(s)} < ∞ and 𝔼 exp{m ∫0 σ 2 (Y(s))ds} < ∞. Proof. (i) The representation (2.129) for the fractional Ornstein–Uhlenbeck process Y is well known; see, e. g., [31]. For H > 1/2 it is straightforward, for H < 1/2 we should

2.5 Fractional stochastic volatility |

207

integrate by parts. It is a continuous Gaussian process with 2

sup 𝔼(Y(t)) < ∞.

(2.132)

t∈[0,T]

The finiteness of any exponential moments of the form (2.130) follows directly from (2.132) and the corresponding result from [57, Example 1.3.4] or [105, Theorem 4.1]. Consider (ii). To establish the representation (2.131) for S, we need only to prove t t that the integrals ∫0 σ(Y(s))dW(s) and ∫0 σ(Y(s))S(s)dW(s) are well defined, while the form of the representation is straightforward, because W is a square-integrable martingale with respect to the flow 𝔽B,V generated by the couple of independent prot cesses B and V. Concerning ∫0 σ(Y(s))dW(s), it follows from (2.130) and condition (J2)(i) that t

t

󵄨 󵄨2q ∫ 𝔼σ (Y(s))ds ≤ C ∫ 𝔼(1 + 󵄨󵄨󵄨Y(s)󵄨󵄨󵄨 )ds < ∞, 2

0

0

t

and consequently ∫0 σ(Y(s))dW(s) is well defined. Moreover, the following moments of any order are finite: supt∈[0,T] 𝔼σ 2n (Y(t)) < ∞. Again, for any ϱ < 2 exponential inequality (2.130) follows from [57]. Therefore, taking into account that q < 1, we get that for any k > 0 t

t

󵄨 󵄨2q 𝔼 exp{k ∫ σ (Y(s))ds} ≤ 𝔼 exp{Cσ k ∫(1 + 󵄨󵄨󵄨Y(s)󵄨󵄨󵄨 )ds} 2

(2.133)

0

0

󵄨2q 󵄨 ≤ C𝔼 exp{Cσ Tk sup 󵄨󵄨󵄨Y(s)󵄨󵄨󵄨 } < ∞. s∈[0,T]

It follows immediately from (2.133) and from Novikov’s condition (see Theorem A.70) that for any n ∈ ℤ t

2

t

𝔼 exp{2n ∫ σ(Y(s))dW(s) − 2n ∫ σ 2 (Y(s))ds} = 1, 0

0

and consequently t

sup 𝔼Sn (t) ≤ C sup 𝔼 exp{n ∫ σ(Y(s))dW(s) −

t∈[0,T]

t∈[0,T]

0

t

t

n ∫ σ 2 (Y(s))ds} 2 0

2

t

2

1/2

≤ C sup ((𝔼 exp{2n ∫ σ(Y(s))dW(s) − 2n ∫ σ (Y(s))ds}) t∈[0,T]

0

0

(2.134)

208 | 2 Rate of convergence of asset and option prices t

2

1/2

2

× (𝔼 exp{(2n − n) ∫ σ (Y(s))ds}) ) 0

1/2

t

= C sup (𝔼 exp{(2n2 − n) ∫ σ 2 (Y(s))ds}) t∈[0,T]

< ∞.

0

Furthermore, applying both final and intermediate bounds from (2.134), we obtain T

2

2

4

4

1 2

(2.135)

∫ 𝔼(σ (Y(s))S (s))ds ≤ T( sup 𝔼S (t) sup 𝔼σ (Y(t))) < ∞, t∈[0,T]

0

t∈[0,T]

and finally the proof of (ii) follows from (2.134) and (2.135). To establish (iii), it is sufficient to prove that for any m ∈ ℤ t

sup 𝔼 exp{m ∫ σ(Y(s))dB(s)} < ∞.

t∈[0,T]

0

But we can proceed as before: it follows from (2.133) that for any n ∈ ℤ and any t ∈ [0, T] t

t

0

0

1 𝔼 exp{m ∫ σ(Y(s))dB(s) − m2 ∫ σ 2 (Y(s))ds} = 1, 2 and consequently t

sup 𝔼 exp{m ∫ σ(Y(s))dB(s)}

t∈[0,T]

0

t

t

1/2

≤ sup (𝔼 exp{2m ∫ σ(Y(s))dB(s) − 2m2 ∫ σ 2 (Y(s))ds}) t∈[0,T]

t

0

0

1/2

× (𝔼 exp{2m2 ∫ σ 2 (Y(s))ds}) 0

< ∞. Remark 2.66. Analyzing the bounds obtained in (2.134), we can immediately conclude that for any m ∈ ℤ t

t

0

0

1 sup 𝔼 exp{m(∫ σ(Y(s))dW(s) − ∫ σ 2 (Y(s))ds)} < ∞ 2 t∈[0,T]

2.5 Fractional stochastic volatility | 209

and t

t

0

0

1 sup 𝔼 exp{m(∫ σ(Y(s))dB(s) − ∫ σ 2 (Y(s))ds)} < ∞. 2 t∈[0,T] Remark 2.67. We can generalize assertion (ii) of Lemma 2.65 to the following one: for any function ψ = ψ(x) : ℝ → ℝ of polynomial growth 󵄨 󵄨 sup 𝔼(󵄨󵄨󵄨ψ(S(t))󵄨󵄨󵄨) < ∞.

t∈[0,T]

Also, it follows from (i) that for any function ψ = ψ(x) : ℝ → ℝ of polynomial growth supt∈[0,T] 𝔼(|ψ(Y(t))|) < ∞, and in particular, 󵄨 󵄨 sup 𝔼(󵄨󵄨󵄨σ(Y(t))󵄨󵄨󵄨) < ∞.

t∈[0,T]

2.5.1.3 Arbitrage-free property, class of martingale measures, minimal martingale measure, incompleteness We formulate the arbitrage properties of the market (2.125)–(2.126) in the following statement. Theorem 2.68. Let the volatility coefficient σ satisfy assumption (J2). Then the market described by (2.125)–(2.126) has the following properties: (i) It is arbitrage-free and incomplete. (ii) Any probability measure ℚ with Radon–Nikodym derivative T

T

T

0

0

0

1 2 dℚ = exp{∫ ν1 (s)dB(s) + ∫ ν2 (s)dV(s) − ∑ ∫ νi2 (s)ds} dℙ 2 i=1

(2.136)

with non-anticipative bounded coefficients νi satisfying the equation ρν1 (s) + μν2 (s) =

r−b σ(Y(s))

(2.137)

is a martingale measure. r−b r−b (iii) Taking ν1 (s) = ρ σ(Y(s)) and ν2 (s) = μ σ(Y(s)) in (2.136), we get the minimal martingale measure. Proof. Let a probability measure ℚ be defined via (2.136) with functions νi satisfying (2.137). Then the discounted process D(t) := e−rt S(t) ⋅

dℚ 󵄨󵄨󵄨󵄨 󵄨 dℙ 󵄨󵄨󵄨ℱt

210 | 2 Rate of convergence of asset and option prices gets the form t

t

D(t) = 1 + ∫(ρσ(Y(s)) + ν1 (s))dB(s) + ∫(μσ(Y(s)) + ν2 (s))dV(s), 0

0

and therefore it is a martingale with respect to the measure ℙ. Therefore, e−rt S(t) is a martingale with respect to the measure ℚ, and we established both (i) and (ii). To check (iii), note that in our case, the minimal martingale measure, according to [147], is defined via the relation t

dℚ 󵄨󵄨󵄨󵄨 󵄨 = 1 − ∫ Z(s)α(s)dM(s), dℙ 󵄨󵄨󵄨ℱt 0

t

where the price process has the form S(t) = S(0)+M(t)+∫0 α(s)d⟨M⟩s . In our case M(t) = t

t

t

∫0 S(s)σ(Y(s))dW(s), ∫0 α(s)d⟨M⟩s = (b − r) ∫0 S(s)ds, and therefore α(s) = t

t

b−r , S(s)σ 2 (Y(s))

r−b whence 1 − ∫0 Z(s)α(s)dM(s) = 1 + ∫0 Z(s) σ(Y(s)) dW(s). This means that the minimal martingale measure Q (that is unique) has a Radon–Nikodym derivative t

t

0

0

1 (r − b)2 dℚ 󵄨󵄨󵄨󵄨 r−b dW(s) − ∫ 2 ds}, 󵄨󵄨 = exp{∫ dℙ 󵄨󵄨ℱt σ(Y(s)) 2 σ (Y(s)) r−b r−b and this equality is satisfied if we choose ν1 (s) = ρ σ(Y(s)) and ν2 (s) = μ σ(Y(s)) . The theorem is proved.

2.5.2 Malliavin calculus with application to option pricing For the elements of Malliavin calculus see Section A.3.5. To apply Malliavin calculus to the asset price S under consideration, note that we have a two-dimensional case with two independent Wiener processes (V, B). With evident modifications, denote by (DV , DB ) the stochastic derivative with respect to the two-dimensional Wiener process (V, B). Denote also t

t

0

0

1 X(t) = log S(t) = log S(0) + bt − ∫ σ 2 (Y(s))ds + ∫ σ(Y(s))dW(s) 2 t

= log S(0) + bt − t

1 ∫ σ 2 (Y(s))ds 2 0

t

+ μ ∫ σ(Y(s))dV(s) + ρ ∫ σ(Y(s))dB(s). 0

0

2.5 Fractional stochastic volatility | 211

Lemma 2.69. (i) The stochastic derivatives of the fBm BH equal DVu BH (t) = 0,

DBu BH (t) = k(t, u)1u 0 such that 󵄨󵄨 F(S(T)) F(Sn (T)) 󵄨󵄨2 󵄨 󵄨󵄨 −2rH 𝔼󵄨󵄨󵄨 − . 󵄨 ≤ CF ⋅ n 󵄨󵄨 S(T) Sn (T) 󵄨󵄨󵄨 Proof. We can write 󵄨󵄨 F(S(T)) F(Sn (T)) 󵄨󵄨2 󵄨󵄨 󵄨 𝔼󵄨󵄨󵄨 − 󵄨 󵄨󵄨 S(T) Sn (T) 󵄨󵄨󵄨 󵄨󵄨 F(S(T)) F(S(T)) 󵄨󵄨2 󵄨󵄨 F(S(T)) F(Sn (T)) 󵄨󵄨2 󵄨 󵄨󵄨 󵄨 󵄨󵄨 ≤ 2𝔼󵄨󵄨󵄨 − n − 󵄨󵄨 + 2𝔼󵄨󵄨󵄨 n 󵄨 := 2I1 + 2I2 . 󵄨󵄨 S(T) 󵄨󵄨 S (T) S (T) 󵄨󵄨 Sn (T) 󵄨󵄨󵄨

(2.156)

220 | 2 Rate of convergence of asset and option prices Now we estimate the right-hand side of (2.156) term by term. For I1 we have I1 = 𝔼(F(S(T))((S(T))

−1

4

2

− (Sn (T)) )) −1

≤ (𝔼(F(S(T))) 𝔼((S(T))

−1

−1 4 1/2

− (Sn (T)) ) ) .

(2.157)

On the one hand, since f and F both have polynomial growth, we see that 𝔼(F(S(T)))4 < ∞ according to Remark 2.67. On the other hand, −1

𝔼((S(T))

−1 4

− (Sn (T)) )

= (S(0)) e

−4 −4bT

T

T

0

0

1 𝔼(exp{ ∫ σ 2 (Y n (s))ds − ∫ σ(Y n (s))dW(s)} 2

T

T

4

1 − exp{ ∫ σ 2 (Y(s))ds − ∫ σ(Y(s))dW(s)}) . 2 0

(2.158)

0

Using the inequalities 󵄨󵄨 x y󵄨 x y 󵄨󵄨e − e 󵄨󵄨󵄨 ≤ (e + e )|x − y|,

x, y ∈ ℝ

(see Lemma A.2(i) in the appendix), and (x + y)2n ≤ C(n)(x2n + y2n ),

x, y ∈ ℝ,

n ∈ ℕ,

along with results outlined in Remarks 2.67 and 2.74, the Burkholder–Davis–Gundy and Hölder inequalities, conditions (J2)(ii) and (iii), and relation (2.154) with ν = 16q, from (2.158) we obtain −1

𝔼((S(T))

−1 4

− (Sn (T)) ) T

T

1 ≤ C𝔼(exp{ ∫ σ 2 (Y n (s))ds − ∫ σ(Y n (s))dW(s)} 2 T

0

0

T

4

1 − exp{ ∫ σ 2 (Y(s))ds − ∫ σ(Y(s))dW(s)}) 2 0

0

T

T

≤ C𝔼((exp{2 ∫ σ 2 (Y n (s))ds − 4 ∫ σ(Y n (s))dW(s)} T

0

T

0

T

1 + exp{2 ∫ σ (Y(s))ds − 4 ∫ σ(Y(s))dW(s)})( ∫ σ 2 (Y n (s))ds 2 T

0

2

T

0

T

0

4

1 − ∫ σ(Y (s))dW(s) − ∫ σ 2 (Y(s))ds + ∫ σ(Y(s))dW(s)) ) 2 0

n

0

0

2.5 Fractional stochastic volatility | 221 T

T

≤ C(𝔼(exp{4 ∫ σ (Y (s))ds − 8 ∫ σ(Y n (s))dW(s)} T

2

n

0

0

T

1/2

+ exp{4 ∫ σ 2 (Y(s))ds − 8 ∫ σ(Y(s))dW(s)})) 0

0

T

T

1 × [𝔼( ∫ σ 2 (Y n (s))ds − ∫ σ(Y n (s))dW(s) 2 0

T

0

T

8 1/2

1 − ∫ σ 2 (Y(s))ds + ∫ σ(Y(s))dW(s)) ] 2 0

0

T

T

1 ≤ C[𝔼( ∫ σ 2 (Y n (s))ds − ∫ σ(Y n (s))dW(s) 2 0

T

0

T

8 1/2

1 − ∫ σ 2 (Y(s))ds + ∫ σ(Y(s))dW(s)) ] 2 0

T

0

2

n

T

8

2

≤ C[𝔼(∫ σ (Y (s))ds − ∫ σ (Y(s))ds) 0

0

T

8 1/2

T

+ 𝔼(∫ σ(Y n (s))dW(s) − ∫ σ(Y(s))dW(s)) ] 0

0

T

8

≤ C[T 7 𝔼(∫(σ 2 (Y n (s)) − σ 2 (Y(s))) ds) T

0

4 1/2

2

n

+ C𝔼(∫(σ(Y (s)) − σ(Y(s))) ds) ] 0

T

8

= C[T 7 (∫ 𝔼{(σ(Y n (s)) − σ(Y(s)))(σ(Y n (s)) + σ(Y(s)))} ds) 0 3

T

n

8

1/2

+ CT 𝔼(∫(σ(Y (s)) − σ(Y(s))) ds)] T

0

󵄨 󵄨16r 󵄨 󵄨16q 󵄨 󵄨16q 1/2 ≤ C[∫(𝔼󵄨󵄨󵄨Y(s) − Y n (s)󵄨󵄨󵄨 𝔼(C + 󵄨󵄨󵄨Y(s)󵄨󵄨󵄨 + 󵄨󵄨󵄨Y n (s)󵄨󵄨󵄨 )) ds 0

222 | 2 Rate of convergence of asset and option prices 1/2

T

󵄨8r 󵄨 + 𝔼(∫󵄨󵄨󵄨Y(s) − Y n (s)󵄨󵄨󵄨 ds)] T

0

1/2

󵄨16r 1 󵄨 󵄨8r 󵄨 ≤ C[∫((𝔼󵄨󵄨󵄨Y(s) − Y n (s)󵄨󵄨󵄨 ) 2 + 𝔼󵄨󵄨󵄨Y(s) − Y n (s)󵄨󵄨󵄨 )ds] .

(2.159)

0

Applying (2.155) consequently with θ = 8 and θ = 16 we get that the last expression in (2.159) does not exceed C( n1 )4rH , thus from (2.157) we obtain I1 ≤ Cn−2rH .

(2.160)

Now we continue with I2 from (2.156): 4 1/2

−4 1/2

I2 ≤ [𝔼(F(S(T)) − F(Sn (T))) ] [𝔼(Sn (T)) ] . The second multiplier is bounded according to Remark 2.74; therefore it follows from condition (J1)(i) that n

S(T)∨Sn (T)

4 1/2

I2 ≤ C[𝔼(F(S(T)) − F(S (T))) ]

= C[𝔼(

∫

4 1/2

f (x)dx) ]

S(T)∧Sn (T)

p p 4 1/2 󵄨 󵄨4 ≤ C(Cf )2 [𝔼(󵄨󵄨󵄨S(T) − Sn (T)󵄨󵄨󵄨 (1 + (S(T)) + (Sn (T)) ) )] p 8 1/4 󵄨 󵄨8 ≤ C[𝔼󵄨󵄨󵄨S(T) − Sn (T)󵄨󵄨󵄨 𝔼(1 + Sp (T) + (Sn (T)) ) ] .

According to Lemma 2.65 and Remark 2.74, p

p 8

sup 𝔼(1 + (S(T)) + (Sn (T)) ) < ∞, n∈ℕ

whence we obtain 󵄨 󵄨8 1/4 I2 ≤ C(𝔼󵄨󵄨󵄨S(T) − Sn (T)󵄨󵄨󵄨 ) . To evaluate the right-hand side of this inequality, we can proceed as in the proof of (2.159) and the subsequent inequalities, because neither the opposite sign of the exponents nor the eighth power instead of the fourth leads to serious discrepancies in the estimations. Therefore we get T

󵄨32r

󵄨 I2 ≤ C[∫((𝔼󵄨󵄨󵄨Y(s) − Y n (s)󵄨󵄨󵄨 0

≤ C(n−16rH )

1/8

1 2

󵄨16r

󵄨 ) + 𝔼󵄨󵄨󵄨Y(s) − Y n (s)󵄨󵄨󵄨

= Cn−2rH .

Bounds (2.160) and (2.161) complete the proof.

1 8

)ds] (2.161)

2.5 Fractional stochastic volatility | 223

Using previous lemmas, we are now in a position to state the main result of this subsection, namely to provide the rate of convergence of the discretized option price to the exact one represented by Ef (S(T)), under the double discretization. Theorem 2.77. Let conditions (J1) and (J2) hold. There exists a constant C > 0 not depending on n such that 󵄨󵄨 F(Sn (T)) Z n (T) 󵄨󵄨󵄨󵄨 󵄨󵄨 (1 + ))󵄨󵄨 ≤ Cn−rH . 󵄨󵄨μ𝔼f (S(T)) − 𝔼( n 󵄨󵄨 󵄨󵄨 S (T) T Proof. By Lemma 2.72 we can write 󵄨󵄨 F(Sn (T)) Z n (T) 󵄨󵄨󵄨󵄨 󵄨󵄨 (1 + ))󵄨󵄨 󵄨󵄨μ𝔼f (X(T)) − 𝔼( n 󵄨󵄨 󵄨󵄨 S (T) T n 󵄨󵄨 F(S(T)) Z(T) F(S (T)) Z n (T) 󵄨󵄨󵄨󵄨 󵄨 (1 + )) − ( n (1 + ))󵄨󵄨 = 𝔼󵄨󵄨󵄨( 󵄨󵄨 󵄨󵄨 S(T) T S (T) T 󵄨 󵄨 1 󵄨󵄨 F(S(T)) 󵄨󵄨 (Z(T) − Z n (T))󵄨󵄨󵄨 ≤ 𝔼󵄨󵄨󵄨 󵄨󵄨 T 󵄨󵄨 S(T) n 󵄨󵄨󵄨 Z (T) F(S(T)) F(Sn (T)) 󵄨󵄨󵄨󵄨 )( − )󵄨 + 𝔼󵄨󵄨󵄨(1 + T S(T) Sn (T) 󵄨󵄨󵄨 󵄨󵄨 2

1/2

F(S(T)) 1 2 ) 𝔼(Z(T) − Z n (T)) ) ≤ (𝔼( T S(T) 2

+ (𝔼(

2 1/2

F(S(T)) F(Sn (T)) Z n (T) − ) 𝔼(1 + )) . S(T) Sn (T) T

According to Lemma 2.65, Remark 2.67, and the Cauchy–Schwarz inequality, < ∞. Obviously,

𝔼( F(S(T)) )2 S(T)

2

sup 𝔼(Z n (T)) < n≥1

T

2 σmin

.

Now the proof follows from Lemmas 2.75 and 2.76.

2.5.4 The rate of convergence of approximate option prices in the case when only fractional Brownian motion is discretized Now we implement the second approach (second level) to approximate the option price. The implementation is based on the fact that the logarithm of the asset price is conditionally Gaussian given the trajectory of the fBm. This allows to exclude Wiener process W from the consideration and to calculate the option price explicitly in terms of the trajectory of fBm BH . Therefore, we can discretize and simulate only the trajectories of BH (the single discretization). Theorem 2.79 gives the explicit option pricing formula as the functional of the trajectory of fBm BH , and Theorem 2.80 gives the rate

224 | 2 Rate of convergence of asset and option prices of convergence. Comparing to Theorem 2.77, we see that the rate of convergence admits a bound of the same order, influenced by the behavior of volatility. Let us introduce the following notations: define a covariance matrix μ2 σY2 μT

CX,Z = (

μT ), σZ2

and let σY2

T

T

2

= ∫ σ (Y(s))ds,

1 mY = X0 + bT − σY2 + ρ ∫ σ(Y(s))dB(s), 2 0

0

T

σZ2 = ∫ σ −2 (Y(s))ds, 0

Δ = |CX,Z | = μ2 (σY2 σZ2 − T 2 ).

Evidently, Δ ≥ 0. We assume additionally that the following assumption is satisfied: (J4) Δ = μ2 (σY2 σZ2 − T 2 ) > 0 with probability 1, in particular, μ > 0. Note that the random vector T

T

1 (X(T), Z(T)) = (X(0) + bT − ∫ σ 2 (Y(s))ds + ρ ∫ σ(Y(s))dB(s) 2 T

0

T

0

+ μ ∫ σ(Y(s))dV(s), ∫ σ −1 (Y(s))dV(s)) 0

(2.162)

0

is Gaussian conditionally to the σ-field ℱTH . The conditional mean vector equals (mY , 0), and the conditional covariance matrix is CX,Z . The following lemma presents the common conditional density of (X(T), Z(T)). Note that under assumption (J4) the distribution of (X(T), Z(T)) is non-degenerate in ℝ2 . Lemma 2.78. Let assumption (J4) hold. Then the common conditional density pX,Z (x, z) of (X(T), Z(T)), conditionally to the given trajectory {Y(t), t ∈ [0, T]}, equals pX,Z (x, z) =

1 2πΔ

1 2

exp{−

1 2 (σ (x − mY )2 + μ2 σY2 z 2 − 2Tμ(x − mY )z)}. 2Δ Z

(2.163)

Proof. The proof immediately follows from the general formula for the density of a two-dimensional Gaussian vector: f (x1 , x2 ) =

(1 − ϱ2 )−1/2 2πσ1 σ2

(2.164) (x1 − m1 )2 (x2 − m2 )2 2ϱ(x1 − m1 )(x2 − m2 ) 1 × exp(− [ + − ]), σ1 σ2 2(1 − ϱ2 ) σ12 σ22

2.5 Fractional stochastic volatility | 225

with the mean vector and covariance matrix σ2 ( 1 ϱσ1 σ2

m ( 1) , m2

ϱσ1 σ2 ), σ22

respectively. In our case the covariance matrix equals μ2 σY2 μT

CX,Z = (

μT ), σZ2 T

the mean vector (mY , 0) = (log S(0) + bT − 21 σY2 + ρ ∫0 σ(Y(s))dB(s), 0), and ϱ = Now (2.163) follows immediately from (2.164).

T . σY σZ

The next result states that the option price can be presented as a functional of σY2 T

and ∫0 σ(Y(s))dB(s) only.

Theorem 2.79. Under conditions (J1)–(J4) the following equality holds: 1

μ𝔼g(X(T)) = (2π)− 2 ∫ G(x)𝔼( ℝ

(x − mY ) (x − mY )2 exp{− })dx 2μ2 σY2 μ2 σY3

− 21

= (2π) 𝔼((σY )−1 ∫ G((x + mY )σY ) ℝ

2

x − 2μx 2 e dx). μ2

(2.165)

Proof. Applying Lemma 2.72, equality (2.145), and Lemma 2.78, we obtain T

μT𝔼g(X(T)) = 𝔼(G(X(T)) ∫ 0

1 dV(u)) σ(Y(u)) T

= 𝔼(𝔼(G(X(T)) ∫ 0

󵄨󵄨 1 dV(u) 󵄨󵄨󵄨 ℱTH )) 󵄨 σ(Y(u))

(2.166)

󵄨󵄨 = 𝔼(𝔼(∫ G(x)zpX,Z (x, z)dxdz 󵄨󵄨󵄨 ℱTH )) 󵄨 ℝ2

= 𝔼 ∫ G(x)zpX,Z (x, z)dxdz = 𝔼 ∫ G(x)(∫ zpX,Z (x, z)dz)dx. ℝ2

ℝ

ℝ

The inner integral can be significantly simplified. Indeed, denote x̃ = x −mY . Then ∫ zpX,Z (x, z)dz = ℝ

=

1 2πΔ 1 2πΔ

1 2

1 2

∫ z exp{− ℝ

∫ z exp{− ℝ

1 2 2 ̃ (σ x̃ + μ2 σY2 z 2 − 2μT xz)}dz 2Δ Z 2

1 T x̃ T 2 x̃ 2 ((μσY z − ) − 2 + σZ2 x̃ 2 )}dz 2Δ σY σY

226 | 2 Rate of convergence of asset and option prices

= =

1 2πΔ 1 2πΔ

1 2

1 2

exp{− exp{−

2

2 2 2 1 x̃ 2 σY σZ − T T x̃ } ∫ z exp{− (μσY z − ) }dz 2 2Δ 2Δ σY σY ℝ

2

2

μσY x̃ T x̃ } ∫ z exp{−( ) }dz. z− 1 1 2μ2 σY2 √2Δ 2 √2Δ 2 σY ℝ

Since 2

∫ xe−(ax−b) dx = ℝ

b √π, a2

we obtain ∫ zpX,Z (x, z)dz = ℝ

T x̃ x̃ 2 exp{− }. 2μ2 σY2 μ2 σY3 √2π

(2.167)

Combining (2.166) and (2.167), we get the proof. In order to state the main result of the present subsection, let us define the following quantities: 2 σY,n

T

= ∫ σ 2 (Y n (s))ds, 0

n 1 2 mY,n = X(0) + bT − σY,n + ρ ∑ σ(Y(tkn ))ΔBk 2 k=0 T

1 2 = X(0) + bT − σY,n + ρ ∫ σ(Y n (s))dB(s). 2 0

Theorem 2.80. Under conditions (J1)–(J4) we have 󵄨󵄨 󵄨󵄨 (x − mY,n ) (x − mY,n )2 󵄨󵄨 󵄨󵄨 −1 −rH exp{− })dx 󵄨󵄨μ𝔼g(X(T)) − (2π) 2 ∫ G(x)𝔼( 2 3 󵄨󵄨 ≤ Cn . 2 󵄨󵄨 󵄨󵄨 2μ2 σY,n μ σY,n

(2.168)

ℝ

Proof. To simplify notation, without loss of generality, let us assume that X(0) + bT = 0. Then, using (2.165), we obtain 󵄨󵄨 󵄨󵄨 (x − mY,n ) (x − mY,n )2 󵄨󵄨 󵄨 −1 exp{− })dx 󵄨󵄨󵄨 󵄨󵄨μ𝔼g(X(T)) − (2π) 2 ∫ G(x)𝔼( 2 3 2 2 󵄨󵄨 󵄨󵄨 2μ σY,n μ σY,n ℝ

󵄨󵄨 (x − m ) (x − mY )2 󵄨 = (2π) 󵄨󵄨󵄨∫ G(x)𝔼( 2 3 Y exp{− } 󵄨󵄨 2μ2 σY2 μ σY − 21

ℝ

−

(x − mY,n ) 3 μ2 σY,n

exp{−

(x − mY,n )2 2 2μ2 σY,n

󵄨󵄨 󵄨 })dx 󵄨󵄨󵄨 󵄨󵄨

2.5 Fractional stochastic volatility | 227

1

≤ (2π)− 2 ∫ G(x)[𝔼( ℝ

(x − mY,n )2 1 󵄨󵄨󵄨󵄨 x − mY x − mY,n 󵄨󵄨󵄨󵄨 exp{− − }) 󵄨 󵄨 󵄨 󵄨 3 2 󵄨󵄨 μ2 󵄨󵄨 σY3 2μ2 σY,n σY,n

󵄨󵄨 (x − m ) (x − mY,n )2 󵄨󵄨󵄨 (x − mY )2 󵄨 } − exp{− })󵄨󵄨󵄨]dx + 𝔼󵄨󵄨󵄨 2 3 Y (exp{− 2 󵄨󵄨 2μ2 σY2 2μ2 σY,n 󵄨󵄨 μ σY 1

:= (2π)− 2 (∫ G(x)(J1n (x) + J2n (x))dx). ℝ

To bound J1n (x) from above, denote n Eexp (x) = (𝔼 exp{−

Eexp (x) = (𝔼 exp{−

(x − mY,n )2 2 μ2 σY,n

1/2

}) , 1/2

(x − mY )2 }) . μ2 σY2

By the Cauchy–Schwarz inequality, J1n (x) ≤

2 1/2 󵄨󵄨 x − m x − mY,n 󵄨󵄨󵄨 1 󵄨󵄨 Y 󵄨󵄨 ) E n (x). 𝔼( − 󵄨 exp 󵄨󵄨 3 μ2 󵄨󵄨󵄨 σY3 σY,n 󵄨

(2.169)

Furthermore, x − mY x − mY,n 1 −3 2 1 2 2 −3 ) + (x + σY,n )(σY−3 − σY,n ) − = σY (σY − σY,n 3 2 2 σY3 σY,n T

−3 + ρ(∫(σ(Y(s)) − σ(Y n (s)))dB(s))(σY−3 − σY,n ).

(2.170)

0

2 σY,n

2 Since |a31 −a32 | ≤ |a21 −a22 |(a1 +a2 ), a1 , a2 > 0, and also the lower bounds σY2 ≥ Tσmin , 2 ≥ Tσmin hold, we can conclude that 2 |σY2 − σY,n | −1 󵄨󵄨 −3 󵄨 −2 −3 󵄨 −2 󵄨 −1 −1 (σY,n + σY−1 ) 󵄨󵄨σY − σY,n 󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨σY − σY,n 󵄨󵄨󵄨(σY,n + σY ) = 2 2 σY σY,n

≤

2 2|σY2 − σY,n | 3

2 T 2 σ3 σY,n min

󵄨 2 󵄨󵄨 ≤ C 󵄨󵄨󵄨σY2 − σY,n 󵄨󵄨.

(2.171)

2 By the penultimate bound from (2.171) and since σY,n is bounded from below uniformly in n, we get

󵄨 (|x| + 1 σ 2 )|σ 2 − σ 2 | 󵄨󵄨 1 2 Y Y,n 󵄨󵄨 −3 −3 󵄨󵄨󵄨 2 Y,n σ )(σ − σ ) (x + 󵄨󵄨 3 Y Y,n 󵄨󵄨󵄨 ≤ 󵄨󵄨 3 2 2 Y,n 󵄨 σY,n T 2 σmin

|x| 1 󵄨 󵄨 2 2 󵄨 2 󵄨󵄨 ≤ C( 2 + )󵄨󵄨󵄨σY2 − σY,n 󵄨󵄨 ≤ C(1 + |x|)󵄨󵄨󵄨σY − σY,n 󵄨󵄨󵄨. σY,n 2

(2.172)

228 | 2 Rate of convergence of asset and option prices Summarizing (2.170)–(2.172) we get T 󵄨󵄨 x − m x − mY,n 󵄨󵄨󵄨 󵄨 󵄨󵄨 2 󵄨󵄨 Y 󵄨 ≤ C(1 + |x| + (σ(Y(s)) − σ(Y n (s)))dB(s))󵄨󵄨󵄨σY2 − σY,n − ∫ 󵄨󵄨. 󵄨󵄨 󵄨󵄨 3 󵄨󵄨 σY,n 󵄨󵄨 σY3

(2.173)

0

Furthermore, applying the Burkholder–Davis–Gundy inequality and Remark 2.67 together with Lemma 2.73, we obtain T

𝔼((∫(σ(Y(s)) − σ(Y 0

n

(s)))dB(s))(σY2

−

2 2 σY,n ))

4

T

4

1/2

2 ≤ (𝔼(∫(σ(Y(s)) − σ(Y n (s)))dB(s)) 𝔼(σY2 − σY,n ) ) 0

T

n

2

2

≤ C(𝔼(∫(σ(Y(s)) − σ(Y (s))) ds) 0

𝔼(σY2

−

1/2 4 2 σY,n ) )

4 1/2

2 ≤ C(𝔼(σY2 − σY,n ) ) .

(2.174)

Finally, we obtain from (2.169)–(2.174) 4 1/4 n Eexp (x).

2 J1n (x) ≤ C(1 + |x|)(𝔼(σY,n − σY2 ) )

(2.175)

Similarly to (2.153) and (2.159), by applying condition (J2) and Lemma 2.73(iii) and (iv), together with the standard Hölder inequality, we get 4

T

4

2 𝔼(σY,n − σY2 ) = 𝔼(∫(σ 2 (Y n (s)) − σ 2 (Y(s)))ds) 0

T

4

≤ C𝔼 ∫(σ 2 (Y n (s)) − σ 2 (Y(s))) ds 0

T

8r

8 1/2

≤ Cσ C ∫[𝔼(Y n (s) − Y(s)) 𝔼(σ(Y n (s)) + σ(Y(s))) ] ds T

0 8r 1/2

≤ C ∫[𝔼(Y n (s) − Y(s)) ] ds ≤ Cn−4rH .

(2.176)

0

Combining inequality (2.176) with (2.175) we get n J1n (x) ≤ Cn−rH (1 + |x|)Eexp (x),

and consequently n (x)dx. ∫ G(x)J1n (x)dx ≤ Cn−rH ∫ G(x)(1 + |x|)Eexp ℝ

ℝ

(2.177)

2.5 Fractional stochastic volatility | 229

Let us show that the integral in the right-hand side of (2.177) is bounded in n ≥ 1. In this 2 connection, denote ϰ = Tμ2 σmin . Applying the standard Hölder inequality together with polynomial growth of G(x) and relations (2.150) from Remark 2.74, we obtain ∫ G(x)(1 +

n |x|)Eexp (x)dx

2 −(2p+1)|x|

2

≤ (∫ G (x)(1 + |x|) e

ℝ

ℝ

× (∫ e

(2p+1)|x|

𝔼 exp{−

dx)

1 2

(x − mY,n )2 2 μ2 σY,n

ℝ

≤ C(𝔼 ∫ e(2p+1)|x+mY,n | exp{− ℝ

1/2

1/2

x2 }dx) ϰ

+ C(𝔼 ∫ e−(2p+1)(x+mY,n ) exp{− ℝ

1/2

x2 }dx) ϰ

= C(𝔼e(2p+1)mY,n ∫ e(2p+1)x exp{− ℝ

}dx)

x2 }dx) ϰ

≤ C(𝔼 ∫ e(2p+1)(x+mY,n ) exp{− ℝ

1/2

+ C(𝔼e−(2p+1)mY,n ∫ e−(2p+1)x exp{− ℝ

≤ C.

1/2

x2 }dx) ϰ

1/2

x2 }dx) ϰ

(2.178)

Construction of an upper bound for J2n (x) is similar. Indeed, 󵄨󵄨 2 2 󵄨 2 2 󵄨󵄨exp{−u } − exp{−v }󵄨󵄨󵄨 ≤ (exp{−u } + exp{−v })|u − v|(|u| + |v|). In our case u=

x − mY , √2μσY

v=

x − mYn √2μσY

(2.179)

.

n

Note that 󵄨󵄨 T 󵄨󵄨 2 󵄨󵄨 󵄨󵄨 2 𝔼|u|2 ≤ C𝔼(|x| + σY + 󵄨󵄨󵄨∫ σ(Y(s))dB(s)󵄨󵄨󵄨) ≤ C(1 + |x|) , 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 and the moments of higher order of u and v can be bounded similarly. For |u2 − v2 |, in the same way as in (2.170), (2.171), and (2.172), we have 󵄨󵄨 T 󵄨󵄨 󵄨󵄨 T 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 |u − v| ≤ C(1 + |x| + 󵄨󵄨󵄨∫ σ(Y n (s))dB(s)󵄨󵄨󵄨)|σY − σY,n | + C 󵄨󵄨󵄨∫(σ(Y(s)) − σ(Y n (s)))dB(s)󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 󵄨0 󵄨 (2.180)

230 | 2 Rate of convergence of asset and option prices Additionally, |σY − σY,n | ≤

2 |σY2 − σY,n |

σY + σY,n

󵄨 2 󵄨󵄨 ≤ C 󵄨󵄨󵄨σY2 − σY,n 󵄨󵄨.

Hence, applying the Hölder, Minkowski, and Burkholder–Davis–Gundy inequalities together with condition (J2)(iii), we get from (2.180) 1/8

(𝔼|u − v|8 )

8 1/8

2 ≤ C(1 + |x|)(𝔼(σY2 − σY,n ) ) T

n

16

1/16

+ (𝔼(∫ σ(Y (s))dB(s)) ) 0

T

16 1/16

2 (𝔼(σY2 − σY,n ) ) 8

n

1/8

+ C(𝔼(∫(σ(Y(s)) − σ(Y (s)))dB(s)) ) 0

16 1/16

2 ≤ C(1 + |x|)(𝔼(σY2 − σY,n ) ) T

2

n

4

1/8

+ C(𝔼(∫(σ(Y(s)) − σ(Y (s))) ds) )

(2.181)

.

0

Proceeding as in (2.176), we get (𝔼(σY2

−

16 1/16 2 σY,n ) )

T

2

2

8

n

1/16

≤ (𝔼(∫(σ (Y(s)) − σ (Y (s)))ds) )

≤ Cn−rH

0

and T

n

2

4

1/8

(𝔼(∫(σ(Y(s)) − σ(Y (s))) ds) )

≤ Cn−rH .

0

Now we continue, preserving for the moment the notations u and v and taking into account (2.179)–(2.181): 󵄨󵄨 u 󵄨󵄨 󵄨 󵄨󵄨 󵄨 J2n (x) = 𝔼(󵄨󵄨󵄨 󵄨󵄨󵄨󵄨󵄨󵄨exp{−u2 } − exp{−v2 }󵄨󵄨󵄨) 󵄨󵄨 σY 󵄨󵄨 󵄨 󵄨 2 1/2 ≤ ϰ −1 (𝔼|u|2 𝔼(󵄨󵄨󵄨exp{−u2 } − exp{−v2 }󵄨󵄨󵄨) ) 󵄨 󵄨 2 1/2 ≤ C(1 + |x|)(𝔼(󵄨󵄨󵄨exp{−u2 } − exp{−v2 }󵄨󵄨󵄨) ) 󵄨 󵄨 4 1/4 ≤ C(1 + |x|)(𝔼(󵄨󵄨󵄨exp{−u2 } + exp{−v2 }󵄨󵄨󵄨) ) 8 1/8

× (𝔼(|u − v|) )

8 1/8

(𝔼(|u| + |v|) ) 3 󵄨 󵄨 4 1/4 ≤ C(1 + |x|) n−rH (𝔼(󵄨󵄨󵄨exp{−u2 } + exp{−v2 }󵄨󵄨󵄨) ) .

(2.182)

2.5 Fractional stochastic volatility | 231

Together with (2.182) this implies that 3

∫ G(x)J2 (x)dx ≤ Cn−rH ∫(1 + |x|) G(x) ℝ

ℝ

2

4

1

4 (x − mY,n ) (x − mY )2 × (𝔼(exp{− } + exp{− }) ) dx. 2 2μ2 σY2 2μ2 σY,n

(2.183)

The fact that the integral in the right-hand side of (2.183) is bounded in n can be esn (x)dx tablished via the same approach as applied to the integral ∫ℝ G(x)(1 + |x|)Eexp in (2.178). Without any doubt, the form of density is much simplified in the case ρ = 0, i. e., processes W and B are independent, because in this case the option price can be presented as a functional of σY2 only. The great advantage of this situation is that we can discretize just the trajectories of Y. And although this case is perhaps more particular and not so common, we still prefer to consider it, since similar methods can be applied also in the case of a weak dependence between W and B. In particular, if ρ = 0, we have σY2 T

CX,Z = (

T ), σZ2

1 mY = X(0) + bT − σY2 , 2

1 2 mY,n = X(0) + bT − σY,n , 2

and (2.168) transforms into 󵄨󵄨 󵄨󵄨 (x − mY,n ) (x − mY,n )2 󵄨󵄨 󵄨󵄨 −1 −rH exp{− })dx 󵄨󵄨𝔼g(X(T)) − (2π) 2 ∫ G(x)𝔼( 󵄨󵄨 ≤ Cn . 3 2 󵄨󵄨 󵄨󵄨 2σY,n σY,n ℝ

2.5.5 Option price in terms of the density of the integrated stochastic volatility Consider for simplicity the case when ρ = 0. Applying Theorem 2.79 and equality (2.165), we clearly see that the option price depends on the random variable σY2 = T

∫0 σ 2 (Y(s))ds. Therefore it is natural to derive the density of this random variable. Since σY2 depends on the whole trajectory of the fBm BH on [0, T], we apply Malliavin calculus in an attempt to find the density. First, we establish some auxiliary results. For any ε > 0 and δ > 0 we introduce the stopping times τε = inf{t > 0 : |BH (t)| ≥ ε} and νδ = inf{t > 0 : |Y(t) − Y(0)| ≥ δ}. Lemma 2.81. For any l > 0 the negative moment is well defined: 𝔼(νδ )−l < ∞.

232 | 2 Rate of convergence of asset and option prices Proof. For any δ > 0 choose ε = ε(δ) in such a way that ε(2 + α|Y(0)|) < δ. Then we get from the representation (2.141) that for 0 ≤ t ≤ τε ∧ ε t

󵄨 󵄨 H 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 −αt αs 󵄨 H −αt 󵄨󵄨Y(t) − Y(0)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨Y(0)󵄨󵄨󵄨(1 − e ) + 󵄨󵄨󵄨B (t)󵄨󵄨󵄨 + αe ∫ e 󵄨󵄨󵄨B (s)󵄨󵄨󵄨ds 0

󵄨 󵄨 󵄨 󵄨 ≤ 󵄨󵄨󵄨Y(0)󵄨󵄨󵄨αε + ε + εe−αt (eαt − 1) ≤ ε(2 + α󵄨󵄨󵄨Y(0)󵄨󵄨󵄨) < δ. δ Therefore, for ε < 2+α|Y(0)| we have νδ > τε ∧ ε. So, it is sufficient to prove that for any ε > 0 and any l > 0,

𝔼(τε ∧ ε)−l < ∞.

(2.184)

Now, for v < ε, 󵄨 󵄨 ℙ{τε ∧ ε < v} = ℙ{τε < v} = ℙ{ sup 󵄨󵄨󵄨BH (t)󵄨󵄨󵄨 ≥ ε}. 0≤t≤v

Furthermore, it follows from the self-similarity and symmetry of the fBm that ε 󵄨 󵄨 ℙ{ sup 󵄨󵄨󵄨BH (t)󵄨󵄨󵄨 ≥ ε} ≤ 2ℙ{ sup BH (t) ≥ H }. v 0≤t≤v 0≤t≤1 Let us denote ϑ = 𝔼 sup0≤t≤1 BH (t). Then, according to inequality (2.2) from [163], for v < ε such that additionally vεH > ϑ, we have ℙ{ sup BH (t) ≥ 0≤t≤v

( vεH − ϑ)2 ε (ε − ϑvH )2 } ≤ exp{− } = exp{− }, 2 vH 2v2H

and (2.184) follows since ∞

∞

𝔼(τε ∧ ε) = ∫ ℙ{(τε ∧ ε) > u}du = ∫ ℙ{τε ∧ ε < −l

−l

0 ∞

=∫ 0

0

1 }du u

1 ℙ{τε ∧ ε < v}du < ∞. v2

Remark 2.82. Exponential bounds for the distribution of τε allow to prove that 𝔼(τε ∧ ε ∧ a)−l < ∞ for any a, l > 0. Now we introduce additional assumptions on σ. (J5) The function σ ∈ C (2) (ℝ), its derivative σ ′ is strictly non-negative, σ ′ (x) > 0, x ∈ ℝ, and σ ′ , σ ′′ are of polynomial growth.

2.5 Fractional stochastic volatility | 233

Lemma 2.83. Under assumptions (J2) and (J5) the stochastic process DBt σY2 DB σY2 = { , t ∈ [0, T]} ‖DB σY2 ‖2H ‖DB σY2 ‖2H belongs to the domain Dom δ of the Skorokhod integral δ. Proof. As follows from Proposition 2.1.1 and Exercise 2.1.1 in [132], it is sufficient to show that σY2 ∈ D2,4

(2.185)

and 𝔼(‖DB σY2 ‖H )

−8

< ∞.

(2.186)

Recall that κ(x) = σ(x)σ ′ (x). It follows from conditions (J2) and (J5) that κ and κ ′ are functions of polynomial growth, κ(x) > 0. Recall the notation s

l(u, s) = cH e

−αs

∫ eαv vH−1/2 (v − u)H−3/2 dv. u

Taking into account (2.142) and (2.138), we write the stochastic derivative as DBu (σY2 )

=

T B Du (∫ σ 2 (Y(s))ds) 0

1/2−H

= 2cH u

= 2 ∫ κ(Y(s))DBu Y(s)ds

T

∫ κ(Y(s))e

T

u

T

0

−αs

s

∫ eαv vH−1/2 (v − u)H−3/2 dvds u

= 2u1/2−H ∫ κ(Y(s))l(u, s)ds. u

Therefore, the iterated derivative equals T

DBz (DBu (σY2 )) = 2u1/2−H z 1/2−H ∫ κ ′ (Y(s))l(z, s)l(u, s)ds.

(2.187)

u∨z

Since the right-hand side of (2.187) is in H ⊗ H and the corresponding integral has moments of any order due to the polynomial growth of κ ′ , (2.185) follows. To prove (2.186), note that DBu (σY2 )

T

≥ C ∫ κ(Y(s))(s − u)H−1/2 ds, u

234 | 2 Rate of convergence of asset and option prices whence ‖DB σY2 ‖2H

T

=

2 ∫(DBu σY2 ) du

T

T

H−1/2

≥ C ∫ du(∫ κ(Y(s))(s − u)

0

u

0

2

ds) .

Now, let σ ′ (Y(0)) = σ0 > 0. Choose δ > 0 so that for y ∈ [Y(0) − δ, Y(0) + δ] we have σ σ ′ (y) > 20 . Then choose ε = ε(δ) as in the proof of Lemma 2.81, and take ζ = τε ∧ ε ∧ T2 . Then T

T

H−1/2

∫ du(∫ κ(Y(s))(s − u) 0

u

1 ζ 3

2

ζ

H−1/2

ds) ≥ C ∫ du( ∫ κ(Y(s))(s − u) 0

2

ds)

2 ζ 3

1 ζ 3

ζ

H−1/2

1 ≥ C ∫ du( ∫ σmin σ0 ( ζ ) 3 0

2

ds) = Cζ 2+2H .

2 ζ 3

It follows immediately from Lemma 2.81 and Remark 2.82 that 𝔼(‖DB σY2 ‖H )

−8

≤ C𝔼ζ −8−8H ≤ C𝔼(τε ∧ ε ∧

T ) 2

−8−8H

< ∞.

s

Denote η = (‖DB σY2 ‖2H )−1 , l(u, s) = cH e−αs ∫u eαv vH−1/2 (v − u)H−3/2 dv, κ(y) = σ(y)σ ′ (y). Theorem 2.84. (i) The density pσ2 of the random variable σY2 is bounded, continuous, and given by the Y following formulas: pσ2 (u) = 𝔼[1σ2 >u δ( Y

Y

DB σY2 )], ‖DB σY2 ‖2H

(2.188)

where the Skorokhod integral is in fact reduced to a Wiener integral, δ(

T

s

0

0

DB σY2 ) = 2η ∫ κ(Y(s))(∫ u1/2−H l(u, s)dB(u))ds ‖DB σY2 ‖2H T

− ∫ DBu ηDBu (σY2 )du. 0

(ii) The option price 𝔼g(X(T)) can be represented as the integral with respect to the density pσ2 (u) defined by (2.188) as follows: Y

1

𝔼g(X(T)) = (2π)− 2 T ∫ G(x) ∫ ℝ

× exp{−

ℝ

(x + u/2 − X(0) − bT) u3

(x + u/2 − X(0) − bT)2 }pσ2 (u)du. Y 2u2

2.5 Fractional stochastic volatility | 235

Proof. From Lemma 2.83 and Proposition 2.1.1 in [132] we get the first part of equality (2.188): pσ2 (u) = 𝔼[1σ2 >u δ( Y

Y

DB σY2 )]. ‖DσY2 ‖2H

To get the second part, note that η := (‖DσY2 ‖H )−2 admits a stochastic derivative and, according to Proposition 1.3.3 from [132], the following holds: T

DB σ 2 δ( B 2Y 2 ) = ∫ ηDBu (σY2 )dB(u) ‖D σY ‖H 0

T

T

= η ∫ DBu (σY2 )dB(u) − ∫ DBu ηDBu (σY2 )du 0

T

1

0

T

T

= 2η ∫ u 2 −H ∫ κ(Y(s))l(u, s)dsdB(u) − ∫ DBu ηDBu (σY2 )du. 0

u

0

According to Lemma 2.10 from [106], we can apply the Fubini theorem for the Skorokhod integral. We have T

∫u

1 −H 2

T

s

1

∫ κ(Y(s))l(u, s)dsdB(u) = ∫ κ(Y(s))(∫ u 2 −H l(u, s)dB(u))ds, u

0

T 0

0

where the interior integral is a Wiener one. Finally, taking into account that mY = X(0) + bT − 21 σY2 , we get 𝔼

(x − mY ) (x − mY )2 exp{− } σY2 σY3 √2π =∫ ℝ

(x + u/2 − X(0) − bT)2 (x + u/2 − X(0) − bT) exp{− }pσ2 (u)du. Y u2 u3 √2π

Combining this with (2.165), we get the proof.

2.5.6 Simulations In this subsection we use the discretization schemes proposed in Sections 2.5.3 and 2.5.4 to simulate the option price. We treat double and single discretization, respectively, and we also compare it to the direct Monte Carlo average. Note that in Theorems 2.77 and 2.80 we prove convergence of the expectation; in other words, we prove that the bias is going to zero as the partition size n increases. The tables

236 | 2 Rate of convergence of asset and option prices below suggest that the variance of the estimate expectedly remain on about the same level. The values of b, α, and T are the same in all simulations, i. e., b = 0.2, α = 0.6, T = 1. We present here the results of simulations for different functions σ and f . Tables 2.8–2.13 demonstrate results of simulations for different values of n. All numbers have six digits after the decimal point. The tables include the outcomes of computations based on (2.144) (the double discretization, “dd” in the tables), (2.165) (the single discretization based on conditioning, “sd” in the tables), and the direct averaging (“da” in the tables). The mean squared errors are also included. For the single discretization the values in the mean squared errors column (column 5) are obtained by taking the same expression as in (2.165) and replace the mean of the sample by the variance of the sample. In Tables 2.8 and 2.9, f has a linear growth and σ takes moderate values (note that the larger values σ(Y(t)) takes, the higher the variance of S(T) is going to be). With f (s) = (s − 1)+ + 1s>1 , G is given by G(x) = 1x>0 (exp(x) − 1). We see that the single discretization performs better than the other methods considered here, while the double discretization results in a higher mean squared error than the direct averaging. Table 2.8: f (s) = (s − 1)+ + 1s>1 , σ(y) = √|y| + 0.55, H = 0.6. n

dd

dd_error

sd

sd_error

da

da_error

100 300 900 2700 8100 24,300

0.951019 0.973156 0.982417 0.935197 1.006177 0.953743

14.363830 15.927797 21.924333 19.155798 19.731601 16.835284

0.956867 0.956734 0.956843 0.957068 0.956925 0.957012

0.008173 0.007935 0.007999 0.007875 0.007998 0.007977

0.956300 0.965695 0.970280 0.938319 0.987641 0.949628

6.128996 6.652094 7.991714 7.233949 7.904191 6.776577

Table 2.9: f (s) = (s − 1)+ + 1s>1 , σ(y) = cos(8y) + 1.2, H = 0.6. n

dd

dd_error

sd

sd_error

da

da_error

100 300 900 2700 8100 24,300

1.005087 0.967322 1.014731 0.973777 0.967524 0.954076

39.331162 24.129696 38.291297 26.510487 26.988813 24.880125

0.973939 0.973903 0.973755 0.973658 0.973172 0.973564

0.010322 0.010282 0.010448 0.010284 0.010471 0.010226

0.987789 0.972769 0.984578 0.969705 0.963961 0.969705

11.421569 8.368354 10.110878 9.600445 9.561456 9.872354

2.5 Fractional stochastic volatility | 237

In Table 2.10 the results of the single discretization computations for various functions σ are given. Note that the choices of σ result in σ(Y(t)) taking higher values (on average) and therefore in a high variance for S(t). Because of that, both the double discretization and the direct averaging do not work well under these circumstances. It is therefore noteworthy to point out that the single discretization shows a very robust performance here. Table 2.10: The single discretization only, f (s) = (s − 1)+ + 1s>1 , H = 0.4. n

σ(y) = 2|y| + 2.2

error

σ(y) = 5 sin(y) + 5.05

error

100 300 900 2700 8100 24,300

1.196685 1.197129 1.197123 1.197186 1.197294 1.197131

0.000532 0.000511 0.000508 0.000508 0.000510 0.000512

1.221111 1.221111 1.221124 1.221130 1.221140 1.221117

0.000004 0.000006 0.000005 0.000004 0.000004 0.000005

n

σ(y) = 10√y 2 + 1

error

100 300 900 2700 8100 24,300

1.219343 1.219582 1.219601 1.219290 1.219335 1.219385

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

In Tables 2.11 and 2.12, σ is as in Tables 2.8 and 2.9, respectively, while f is an indicator of an interval. With f (s) = 1s∈[1,2] we have G(x) = max(0, min(x, log(2))). The single discretization again performs better here. Unlike previously, using double discretization results in a smaller mean squared error than the direct averaging. Table 2.11: f (s) = 1s∈[1,2] , σ(y) = √|y| + 0.55, H = 0.8. n

dd

dd_error

sd

sd_error

da

da_error

100 300 900 2700 8100 24,300

0.173459 0.170377 0.172513 0.170567 0.172327 0.172714

0.085342 0.083396 0.084694 0.083248 0.084907 0.084648

0.172270 0.171784 0.172209 0.171828 0.171824 0.171822

0.003021 0.002989 0.002994 0.002964 0.002967 0.002933

0.171200 0.171875 0.171275 0.174650 0.168925 0.170300

0.141891 0.142334 0.141940 0.144147 0.140389 0.141298

238 | 2 Rate of convergence of asset and option prices Table 2.12: f (s) = 1s∈[1,2] , σ(y) = cos(8y) + 1.2, H = 0.8. n

dd

dd_error

sd

sd_error

da

da_error

100 300 900 2700 8100 24,300

0.162822 0.167634 0.165612 0.163298 0.164814 0.162855

0.191493 0.199874 0.196169 0.193913 0.193648 0.191648

0.163004 0.163181 0.163669 0.164235 0.164236 0.164496

0.018580 0.018253 0.018401 0.019251 0.018682 0.018850

0.164850 0.162775 0.163775 0.166050 0.165650 0.165525

0.137674 0.136279 0.136953 0.138477 0.138210 0.138126

Table 2.13: f (s) = 1s∈[1,2] , σ(y) = 2|y| + 2.2, H = 0.8. n

dd

dd_error

sd

sd_error

da

da_error

100 300 900 2700 8100 24,300

0.009435 0.008986 0.009631 0.009373 0.009703 0.009217

0.004724 0.004406 0.004674 0.004600 0.004775 0.004554

0.009434 0.009259 0.009296 0.009241 0.009247 0.009206

0.000074 0.000072 0.000072 0.000070 0.000071 0.000071

0.009125 0.009025 0.009525 0.009300 0.009250 0.009650

0.009042 0.008944 0.009434 0.009214 0.009164 0.009557

Table 2.14: The single discretization only, f (s) = ((s − 1)+ )3/2 + 1s>1 , H = 0.7. n

σ(y) = √(|y| + 0.55)

error

σ(y) = cos(8y) + 1.2

error

100 300 900 2700 8100 24,300

1.914334 1.923092 1.919237 1.917376 1.920846 1.921878

0.009626 0.009487 0.009389 0.009398 0.009454 0.009446

2.326036 2.327452 2.321790 2.318913 2.313767 2.317305

0.021973 0.022298 0.022594 0.022438 0.022488 0.022312

Table 2.13 confirms a good overall performance of the single discretization and that the single discretization works well in settings where the direct averaging and the double discretization would require a huge number of trials. Table 2.14 shows the results of simulations for when f grows faster than linearly: f (s) = ((s − 1)+ )3/2 + 1s>1 . In this case x

x + ∫0 (ez − 1)3/2 dz, G(x) = { 0,

x ≥ 0, x < 0.

(2.189)

The single discretization method again shows a robust performance here, while double discretization and direct averaging have high variance. The simulations are done in R. For each estimate 40,000 trials were used. To simulate the fBm, we used a modified version of the function from the package dvfBm R

2.5 Fractional stochastic volatility | 239

kindly provided by J.-F. Coeurjolly. For simulations related to the double discretization we take the average of the value under the expectation in the right-hand side of (2.144) over 40,000 trials. For simulations related to the single discretization, we replace the infinite interval of integration in the right-hand side of (2.165) with a finite one, making sure that the integral over the complement is small. To be more precise, assume, for technical simplicity, that f (y) ≤ yk for some k ∈ ℕ. Then, for example, the right-hand tail of the integral in (2.165) can be bounded from above in the following way: ∞

− 21

−1

(2π) 𝔼((σY )

∫ G((x + mY )σY ) A

1

∞

2

x − 2μx 2 e dx) μ2 x

≤ (2π)− 2 𝔼((σY )−1 ∫ ek(x+mY )σY e μ2

− −Ax2 4μ

dx)

A − 21

≤ (2π) 𝔼((σY )−1 ekmY σY (

A − μ−2 − kσy ) ). 4μ2 −1

We discretize the finite interval; the partition size varies, but is at least 2000. For each x from the partition, we take the average over the same 40,000 trials of the value under the expectation in (2.165). Figures 2.1, 2.2, and 2.3 display the integrand in (2.168) from various simulations. The pictures give plots for a single choice of n. It is worth pointing out that the resulting function changes very little with n as can be seen in Figure 2.4. Each curve

Figure 2.1: A plot of the integrand in (2.168), n = 900, H = 0.6, f (s) = (s − 1)+ + 1s>1 , σ(y) = 2|y| + 2.2, G(x) = 1x>0 (exp(x) − 1).

240 | 2 Rate of convergence of asset and option prices

Figure 2.2: A plot of the integrand in (2.168), n = 24,100, H = 0.4, f (s) = (s − 1)+ + 1s>1 , σ(y) = 5 sin(y) + 5.05, G(x) = 1x>0 (exp(x) − 1).

Figure 2.3: A plot of the integrand in (2.168), n = 24,100, H = 0.7, f (s) = 1s∈[1,2] , σ(y) = cos(8y) + 1.2, G(x) = max(0, min(x, log(2))).

almost coincides with previous ones, so the curve for n = 24,300 almost completely conceals the plots for the other values of n. In fact the absolute value of the difference between each two functions does not exceed 0.0023 in our simulations. This pattern is also observed for other choices of f and σ considered in this subsection.

2.5 Fractional stochastic volatility |

241

Figure 2.4: A plot of the integrands in (2.168) for different values of n, H = 0.8, f (s) = ((s − 1)+ )3/2 + 1s>1 , σ(y) = √(|y| + 0.55), where G is given by (2.189). Since the functions are very close to each other, their plots overlap.

As a conclusion, the single discretization method allows to achieve a higher precision. It keeps showing consistent results and a relatively small error in situations where the other considered methods suffer from high variance.

3 Limit theorems for markets with non-random time-varying coefficients The concept of this and the next chapter is different from that of the previous ones. Namely, in the previous chapters, we considered limit theorems in the context of time discretization and applied to financial markets. In this chapter, the time in which the market operates is not discretized; it remains continuous. We consider a Black– Scholes market model with non-random but varying in time coefficients in the scheme of series that is marked by parameter n, and we discover what happens with asset and option prices when the time-varying coefficients tend to some limits as n → ∞. The chapter is organized as follows. Section 3.1 contains the Black–Scholes equation and the Black–Scholes formula for prices of the vanilla options written on assets with varying parameters. Also, we formulate there the conditions of robustness of asset and European option prices. This means that we consider the coefficients in the scheme of series and investigate what kind of convergence of these time-varying coefficients entails the convergence of stock prices and options. Moreover, we prove the convergence of prices in two ways, through the Black–Scholes formula and directly, through the convergence of the corresponding random variables. The second method is better because it requires weaker conditions. Section 3.2 is devoted to the barrier European options written still on the assets in the Black–Scholes model with non-random timevarying coefficients. As an example, we consider “up-and-out” barrier call options. First, we study the same problem of robustness and solve it by a direct probabilistic method. Then we formulate a boundary value problem for this barrier option price. We cannot produce its exact analytic solution; however, it turns out that we can establish a limit theorem for the sequence of solutions using our probabilistic approach. In Section 3.3 we return to the problem of time discretization and estimate the rate of convergence of the discretized option price to the true one. Due to the more complicated form of barrier options in comparison to vanilla options, the rate of convergence becomes slower, O(log n/√n) instead of O(1/√n) or even O(1/n); however, the elegant proof of this result is of independent interest and this partially compensates the indicated disadvantage. Section 3.4 was born from the following simple question. Everybody knows that the price of vanilla and barrier options is differentiable in standard parameters: asset price, time, etc. What about differentiability of the barrier price in barrier? The answer is positive; it is received with the help of Malliavin calculus.

3.1 Convergence of European option prices in the Black–Scholes model with time-varying parameters This section is devoted to solving the Black–Scholes equation for European call and put options with time-varying coefficients in the Black–Scholes market model. First, https://doi.org/10.1515/9783110654240-003

244 | 3 Limit theorems for markets with non-random time-varying coefficients we present an explicit form of the fair option price, and, as the next step, the conditions of convergence of asset and option prices with respect to the parameter n of series are established in two ways: using the explicit form of the option price and by the probabilistic method, without using the explicit form. Also, the conditions of convergence are compared.

3.1.1 Model Let (Ω, ℱ , 𝔽, ℙ) be a filtered probability space with filtration 𝔽 = {ℱt , t ∈ [0, T]} satisfying the standard conditions of the right continuity and completeness. We consider the Black–Scholes model, i. e., the continuous-time model of a financial market with one risky asset (a stock with price S(t) at time t) and one risk-free asset (a bond with price B(t) at time t). Price B(t) is defined as t

B(t) = exp{∫ r(s)ds} 0

for t ≥ 0, where r = r(t), t ≥ 0, is a non-negative locally Lebesgue integrable function. The equation for the stock price in differential form is the following: dS(t) = S(t)(μ(t)dt + σ(t)dW(t)),

(3.1)

where μ(t) and σ(t) are non-random functions and W is a standard Wiener process with respect to measure ℙ. We assume that the condition t

2

∫ σ (s)ds < ∞, 0

t

󵄨 󵄨 ∫󵄨󵄨󵄨μ(s)󵄨󵄨󵄨ds < ∞

(3.2)

0

holds for any t ≥ 0. In what follows we restrict our model to the interval [0, T], where T is the maturity date. Under additional condition T μ(s)−r(s) (N) ∫0 ( σ(s) )2 ds < ∞ the market will be arbitrage-free and complete, and the unique martingale measure ℙ∗ has the Radon–Nikodym derivative T

T

0

0

2

μ(s) − r(s) μ(s) − r(s) dℙ∗ 󵄨󵄨󵄨󵄨 1 dW(s) − ∫( ) ds}. 󵄨󵄨 = exp{− ∫ 󵄨 dℙ 󵄨ℱT σ(s) 2 σ(s) Consider the fair price π call (S, t) of the European call option as a function of the current stock price S and the current time t. It is very well known (see, e. g., [40, 62, 91, 158])

3.1 Black–Scholes with time-varying parameters | 245

that π call (S, t) satisfies the boundary value problem of the form 𝜕2 π call (S, t) 𝜕π call (S, t) 1 2 𝜕π call (S, t) + σ (t)S2 − r(t)π call (S, t) = 0 + r(t)S 2 𝜕t 2 𝜕S 𝜕S

(3.3)

on (S, t) ∈ R+ × [0, T] with boundary condition π call (S, T) = max(S − K, 0),

(3.4)

where K is a strike price. Remark 3.1. Note that the solution of such a problem satisfies the condition π call (0, t) = 0. The Black–Scholes equation for the fair price π put (S, t) of the European put option is the same as for the European call option: 𝜕π put (S, t) 1 2 𝜕π put (S, t) 𝜕2 π put (S, t) + r(t)S + σ (t)S2 − r(t)π put (S, t) = 0, 𝜕t 2 𝜕S 𝜕S2

(3.5)

where (S, t) ∈ R+ × [0, T] and the boundary condition has the form π put (S, T) = max(K − S, 0).

(3.6)

Remark 3.2. In this case, the solution of such a problem satisfies the condition π put (0, t) = K.

3.1.2 Explicit form of call and put option prices with time-varying parameters Theorem 3.3. Let the functions r(t) and σ(t) be continuous on [0, T], σ(t) > 0 for all t ∈ [0, T]. Then the solution of equation (3.3) with boundary condition (3.4) has the following form: π

call

T

(S, t) = SΦ(d1 ) − K exp{− ∫ r(s)ds}Φ(d2 ),

(3.7)

t

where T

d1 =

log KS + ∫t (r(s) + 21 σ 2 (s))ds √∫T σ 2 (s)ds t

T

,

d2 =

log KS + ∫t (r(s) − 21 σ 2 (s))ds

and Φ(x) is a standard normal distribution function.

√∫T σ 2 (s)ds t

,

246 | 3 Limit theorems for markets with non-random time-varying coefficients Proof. Equation (3.3) can be solved by changing the variables, as suggested in [171]. Namely, let S = Kex , π call = Kv, and also t = T − t ′ , t ′ ∈ [0, T]. Then K𝜕v 𝜕π call =− ′ , 𝜕t 𝜕t call 𝜕π 𝜕π call 𝜕x K𝜕v 1 1 𝜕v = = = , 𝜕S 𝜕x 𝜕S 𝜕x Kex ex 𝜕x call

call

𝜕π 𝜕π 𝜕2 π call 𝜕( 𝜕S ) 𝜕( 𝜕S ) 𝜕x 1 𝜕v 1 𝜕2 v 1 = = (− + = ) 𝜕S 𝜕x 𝜕S ex 𝜕x ex 𝜕x 2 Kex 𝜕S2 1 𝜕v 𝜕2 v = (− + 2 ). 2x 𝜕x 𝜕x Ke

After replacing and substituting the derivatives, the Black–Scholes equation will be as follows: −K

𝜕v 𝜕2 v 1 𝜕v 𝜕v 1 σ 2 (T − t ′ )S2 + (− + 2 ) + r(T − t ′ )S x − r(T − t ′ )π call = 0. ′ 2x 𝜕t 2 𝜕x 𝜕x e 𝜕x Ke

Substituting the expressions for S and π call , we get −K

K 2 e2x 𝜕v 1 2 K 2 e2x 𝜕2 v 𝜕v 1 2 − σ (T − t ′ ) 2x + σ (T − t ′ ) 2x ′ 𝜕t 2 Ke 𝜕x 2 Ke 𝜕x 2 x Ke 𝜕v + r(T − t ′ ) x − r(T − t ′ )Kv = 0. e 𝜕x

After reduction we get the equation 𝜕v 1 𝜕2 v 𝜕v 1 = σ 2 (T − t ′ ) 2 + (r(T − t ′ ) − σ 2 (T − t ′ )) − r(T − t ′ )v. ′ 𝜕t 2 2 𝜕x 𝜕x

(3.8)

Let us introduce a new variable τ̂ such that 21 σ 2 (T − t ′ )dt ′ = dτ,̂ i. e., t′

1 ̂ ) = ∫ σ 2 (T − s)ds = z, τ(t 2 ′

0

T

1 z ∈ [0; ∫ σ 2 (u)du]. 2 0

Then t ′ = τ̂ −1 (z) and r(T − t ′ ) = r(T − τ̂ −1 (z)). Also, 𝜕v 𝜕v 𝜕τ̂ 1 𝜕v = = σ 2 (T − τ̂ −1 (z)) . 𝜕t ′ 𝜕τ̂ 𝜕t ′ 2 𝜕τ̂

(3.9)

Substituting the expression for τ̂ in (3.8), we obtain 𝜕v 𝜕v 𝜕2 v ̂ + a(τ)̂ = − b(τ)v, 𝜕τ̂ 𝜕x2 𝜕x

(3.10)

3.1 Black–Scholes with time-varying parameters | 247

where r(T − τ̂ −1 ) , 1 2 σ (T − τ̂ −1 ) 2

b(τ)̂ :=

a(τ)̂ := b(τ)̂ − 1.

The general solution of the equation in partial derivatives of the first order, derived 𝜕2 v 𝜕v 𝜕v ̂ has the form ̂ 𝜕x − b(τ)v, from equation (3.10) by omitting the term 𝜕x 2 , i. e., 𝜕τ̂ = a(τ) dA dB −B(τ)̂ ̂ ̂ dτ̂ = b(τ), ̂ and F(⋅) is an arbitrary function. v(x, τ)̂ = F(x + A(τ))e , where dτ̂ = a(τ), Indeed, put 𝜕v ̂ −B(τ)̂ , = F ′ (x + A(τ))e 𝜕x 𝜕v ̂ ′ (x + A(τ))e ̂ −B(τ)̂ − B′ (τ)F(x ̂ ̂ −B(τ)̂ . = A′ (τ)F + A(τ))e 𝜕τ̂ ̂ B′ (τ)̂ = b(τ)̂ and substituting the values of Taking into account that A′ (τ)̂ = a(τ), the function v and its derivatives in the equation, we obtain the identity. Now the general solution of equation (3.10) for v will be looked for as v(x,̂ τ)̂ = ̂ where x̂ = x − A(τ). ̂ Let us deduce an equation for V. We have e−B(τ)̂ V(x, τ), ̂ τ), ̂ v(x, τ)̂ = e−B(τ) V(x + A(τ), ̂

𝜕v ̂ τ)e ̂ −B(τ)̂ , = Vx′ (x + A(τ), 𝜕x 𝜕v ̂ τ′̂ (x + A(τ), ̂ τ)̂ + Vτ′̂ (x + A(τ), ̂ τ))e ̂ −B(τ)̂ = (A′ (τ)V 𝜕τ̂ ̂ ̂ τ)e ̂ −B(τ)̂ . − B′ (τ)V(x + A(τ), ̂ B′ (τ)̂ = b(τ)̂ and substituting the values of the Having in mind that A′ (τ)̂ = a(τ), function v and its derivatives in the equation, after simplification we get the simplest ′′ heat equation Vxx = Vτ′̂ . After replacing, the boundary condition π call (S, T) = max(S − K, 0) will become Kv(x, 0) = max(Kex − K, 0), i. e., v(x, 0) = max(ex − 1, 0). Then v(x, 0) = e−B(0) V(x + A(0), 0) = max(ex − 1, 0), that is, V(x + A(0), 0) = eB(0) max(ex − 1, 0), whence we obtain V(x, 0) = eB(0) max(ex−A(0) − 1, 0). Finally, we get a heat equation 𝜕2 V 𝜕V = 𝜕τ̂ 𝜕x2

248 | 3 Limit theorems for markets with non-random time-varying coefficients with a boundary condition V0 (x) = V(x − A(0), 0) = eB(0) max(ex−A(0) − 1, 0). The solution of this boundary value problem has the form (x − s)2 1 }ds. V(x, τ)̂ = ∫ V0 (s) exp{− 4τ̂ 2√π τ̂ ∞

−∞

Let us transform this integral. As the first step, let us replace x ′ = V(x, τ)̂ =

=

∞

√2τ̂

∫ V0 (x + x′ √2τ)̂ exp{−

2√π τ̂ −∞ 1 √2π

s−x . √2τ̂

Then

x′ 2 }dx′ 2

∞

∫ exp{B(0) − A(0) + x + x ′ √2τ̂ − − x−A(0) √ ̂

x′ 2 }dx ′ 2

2τ

1 − √2π

∞

∫ exp{B(0) − − x−A(0) √ ̂

x′ 2 }dx ′ 2

2τ

= I1 − I2 . We calculate I1 and I2 separately. We start as follows: 1 I1 = √2π

∞

∫ exp{B(0) − A(0) + x + x ′ √2τ̂ − − x−A(0) √ ̂

x′ 2 }dx ′ 2

2τ

exp{B(0) − A(0) + x} = √2π

∞

1 ̂ ′ )}dx ′ ∫ exp{− (x ′ 2 − 2√2τx 2

− x−A(0) √ ̂ 2τ

=

exp{B(0) − A(0) + x} √2π

∞

1 1 2 ′ ̂ ∫ exp{− (x ′ − √2τ)̂ + 2τ}dx 2 2

− x−A(0) √ ̂ 2τ

=

exp{B(0) − A(0) + x + τ}̂ √2π

∞

∫

exp{−w2 /2}dw

− x−A(0) −√2τ̂ √ ̂ 2τ

̂ = exp{B(0) − A(0) + x + τ}Φ(d 1 ), where d1 = √2τ̂ +

x−A(0) √2τ̂

I2 =

1 √2π

and Φ(⋅) is a standard normal distribution function. Now, ∞

∫ exp{B(0) − − x−A(0) √ ̂ 2τ

x′ 2 }dx ′ 2

3.1 Black–Scholes with time-varying parameters | 249

=

exp{B(0)} √2π

∞

∫ exp{− − x−A(0) √ ̂

x′ 2 }dx′ = exp{B(0)}Φ(d2 ), 2

2τ

where d2 = So,

x−A(0) . √2τ̂

̂ V(x, τ)̂ = exp{B(0) − A(0) + x + τ}Φ(d 1 ) − exp{B(0)}Φ(d2 ) ̂ = exp{B(0)}(exp{−A(0) + x + τ}Φ(d 1 ) − Φ(d2 )).

Then ̂ v(x,̂ τ)̂ = e−B(τ) V(x, τ)̂ = e−(B(τ)−B(0)) (exp{−A(0) + x + τ}Φ(d 1 ) − Φ(d2 )) ̂

̂

and ̂ v(x, τ)̂ = e−(B(τ)−B(0)) (exp{−A(0) + A(τ)̂ + x + τ}Φ(d 1 ) − Φ(d2 )) ̂

τ̂

τ̂

0

0

̂ = exp{− ∫ b(s)ds}(exp{∫ a(s)ds + x + τ}Φ(d 1 ) − Φ(d2 )). Hence τ̂

τ̂

0

0

π call (S, τ)̂ = K exp{− ∫ b(s)ds}(exp{∫ a(s)ds + log

S ̂ + τ}Φ(d 1 ) − Φ(d2 )). K

We consider ̂ ′) τ(t

̂ ′) τ(t

∫ b(s)ds = ∫ 0

r(T − τ̂ −1 (s)) ds. − τ̂ −1 (s))

1 2 σ (T 2

0

̂ Let us make an inverse transformation: τ̂ −1 (s) = x, s = τ(x), ds = 21 σ 2 (T − x)dx. Then ̂ ′) τ(t

t′

∫ b(s)ds = ∫

0 2

0

T

T

T−t ′

t

r(T − x) 1 2 σ (T − x)dx = ∫ r(u)du = ∫ r(u)du, 1 2 σ (T − x) 2

and consequently π

call

τ̂

τ̂

̂ (S, τ)̂ = S exp{− ∫ b(s)ds + ∫ a(s)ds + τ}Φ(d 1) 0

0 τ̂

− K exp{− ∫ b(s)ds}Φ(d2 ) 0

250 | 3 Limit theorems for markets with non-random time-varying coefficients T

= SΦ(d1 ) − K exp{− ∫ r(u)du}Φ(d2 ), t

where d1 =

d2 =

=

√2τ̂

√2τ̂

log KS + ∫t r(s)ds + ∫t

√2 ∫T 1 σ 2 (s)ds t 2 T 1 2 σ (s)ds 2

T

τ̂

log KS + ∫0 a(s)ds

T 1 2 σ (s)ds 2

T

τ̂

log KS + ∫0 a(s)ds + 2τ̂

=

log KS + ∫0 r(s)ds − ∫t

√2 ∫T 1 σ 2 (s)ds t 2

,

.

As a result, π

call

T

(S, t) = SΦ(d1 ) − K exp{− ∫ r(s)ds}Φ(d2 ), d1 =

log

S K

+

T ∫t (r(s)

t

+

1 2 σ (s))ds 2

√∫T σ 2 (s)ds t

T

,

d2 =

log KS + ∫t (r(s) − 21 σ 2 (s))ds √∫T σ 2 (s)ds

.

t

Theorem 3.4. Let the functions r(t) and σ(t) be continuous on [0, T], σ(t) > 0 for all t ∈ [0, T]. Then the solution of equation (3.5) with boundary conditions (3.6) looks as π

put

T

(S, t) = K exp{− ∫ r(s)ds}Φ(d3 ) − SΦ(d4 ), t

where T

d3 = −

log KS + ∫t (r(s) − 21 σ 2 (s))ds √∫T σ 2 (s)ds t

T

,

d4 =

log KS + ∫t (r(s) + 21 σ 2 (s))ds √∫T σ 2 (s)ds

.

(3.11)

t

Proof. The proof of the theorem repeats the proof of the result for the European call option, because the problems differ only in the boundary condition. That is, the problem is reduced to a parabolic diffusion equation with the same type of boundary conditions. 3.1.3 Robustness of asset and European option prices In real markets, the values of parameters μ, σ, r are measured with some inaccuracy. Now our goal is to establish stability (robustness) of the European option prices regarding a change or inaccuracy of measurement parameters that define the model.

3.1 Black–Scholes with time-varying parameters | 251

In other words, our goal is to establish conditions of convergence of coefficients that imply convergence of asset and option prices. Consider a family of financial markets with continuous time, described by sequences of non-random measurable functions σn , μn , rn , n ≥ 0. More precisely, we consider the same filtered probability space (Ω, ℱ , 𝔽, ℙ) and Black–Scholes model with one risky asset (a stock with price Sn (t) at time t) and one risk-free asset (a bond with price Bn (t) at time t). Price Bn (t) is defined as t

Bn (t) = exp{∫ rn (s)ds}

(3.12)

0

for t ≥ 0, where rn = rn (t), t ≥ 0, for any n ≥ 0 is a non-negative locally Lebesgue integrable function. The dynamics of stock prices is given by the equation t

t

0

0

1 Sn (t) = Sn (0) exp{∫(μn (s) − σn2 (s))ds + ∫ σn (s)dW(s)}, 2

(3.13)

T

T

where W is a Wiener process and the conditions ∫0 σn2 (s)ds < ∞, ∫0 |μn (s)|ds < ∞ are satisfied for each n ≥ 0, where T is the maturity date. Recall that under condition T μ (s)−r (s) (Nn ) ∫0 ( n σ (s)n )2 ds < ∞ n

all these market models are arbitrage-free and complete. Namely, t

Xn (t) = Sn (t) exp{− ∫ rn (s)ds} t

0

t

σ 2 (s) = Sn (0) exp{∫(μn (s) − rn (s) − n )ds + ∫ σn (s)dW(s)} 2 0

t

= Sn (0) exp{− ∫ 0

(n,∗)

ℙ

σn2 (s) ds + ∫ σn (s)dWn∗ (s)}, 2

s μn (u)−rn (u) du σn (u)

where Wn∗ (s) = W(s) + ∫0

0

t

(3.14)

0

is a Wiener process with respect to the measure

, having the Radon–Nikodym derivative t

t

0

0

2

μ (s) − rn (s) μ (s) − rn (s) dℙ(n,∗) 󵄨󵄨󵄨󵄨 1 dW(s) − ∫( n ) ds}. 󵄨 = exp{− ∫ n dℙ 󵄨󵄨󵄨ℱt σn (s) 2 σn (s) Functions μn , σn , rn are usually called parameters of equation (3.14), so we say that n ≥ 0 is a series parameter.

252 | 3 Limit theorems for markets with non-random time-varying coefficients The European option price considered at moment t and with risky asset price S̆n at moment t equals πncall (S̆n , t)

T

= S̆n Φ(d1n ) − K exp{− ∫ rn (s)ds}Φ(d2n ),

(3.15)

t

where d1n

=

log

S̆n K

T

+ ∫t (rn (s) + 21 σn2 (s))ds √∫T σn2 (s)ds t

,

d2n

=

log

S̆n K

T

+ ∫t (rn (s) − 21 σn2 (s))ds √∫T σn2 (s)ds

.

t

The next result is evident. Theorem 3.5. Let the following conditions be met: (i) Sn (0) → S0 (0) as n → ∞, (ii) σn → σ0 in ℒ2 ([0, T]) as n → ∞, (iii) μn → μ0 in ℒ1 ([0, T]) as n → ∞. Then the stock prices Sn (t) given by formula (3.13) converge in probability to the corresponding stock price S0 (t) at each moment t ∈ [0, T]. Let us find out the conditions for the convergence of the call option prices. Theorem 3.6. Suppose the following conditions hold: (i) rn → r0 in ℒ1 ([0, T]), T (ii) σn → σ0 in ℒ2 ([0, T]) and ∫t σ02 (t) > 0, t ∈ [0, T], (iii) the sequence S̆n from formula (3.15) converges, i. e., S̆n → S̆0 , (iv) for all n ≥ 0 condition (Nn ) is fulfilled. Then all the markets are arbitrage-free and complete, and the fair option prices specified by formula (3.15) converge to the corresponding fair option price if we replace n by 0 and consider any time t ∈ [0, T]. Proof. First, we prove the statement analytically. It follows from conditions (i)–(iii) that dni → d0i , i = 1, 2; therefore, the option prices converge, because the function Φ is continuous. Second, we can also prove the convergence of the fair price of the European call option using probability methods, and not using its explicit form. To start with, the fair price of the European call option in the pre-limit model at moment t has the following form: πncall (S̆n , t)

T

=

𝔼∗n ((Sn (T)

󵄨󵄨 − K) exp{− ∫ rn (s)ds} 󵄨󵄨󵄨 Sn (t) = S̆n ), 󵄨 +

t

3.1 Black–Scholes with time-varying parameters | 253

and in the limit it equals T

󵄨󵄨 + π call (S0̆ , t) = 𝔼∗ ((S0 (T) − K) exp{− ∫ r0 (s)ds} 󵄨󵄨󵄨 S0 (t) = S̆0 ). 󵄨 t

According to (3.13), T

T

t

t

1 πncall (S̆n , t) = 𝔼∗n ((S̆n exp{∫(μn (s) − σn2 (s))ds + ∫ σn (s)dW(s)} − K) 2 T

+

× exp{− ∫ rn (s)ds}). t

Introduce the notation T

Xn (t) := Sn (t) exp{− ∫ rn (s)ds},

n ≥ 0.

t

Then πncall (S̆n , t) − π call (S,̆ t)

T

T

+

+

= 𝔼∗n (Xn (t) − K exp{− ∫ rn (s)ds}) − 𝔼∗ (X0 (t) − K exp{− ∫ r0 (s)ds}) t

T

t

T

T

+

1 = 𝔼(S̆n exp{∫ σn (s)dW(s) − ∫ σn2 (s)ds} − K exp{− ∫ rn (s)ds}) 2 t

0

T

t

T

T

0

t

+

1 − 𝔼(S̆0 exp{∫ σ0 (s)dW(s) − ∫ σ 2 (s)ds} − K exp{− ∫ r0 (s)ds}) , 2 t

where the expectation is calculated according to the probability measure with respect to which W is a Wiener process. Due to conditions (i)–(iii), there is a convergence T T T T ∫t rn (s)ds → ∫t r(s)ds and ∫t σn2 (s)ds → ∫t σ 2 (s)ds as well as S̆n → S̆0 . Convergence of σn to σ in ℒ2 [0, T] implies the convergence of Gaussian integrals in probability: T

ℙ

T

→ ∫ σ(s)dW(s), ∫ σn (s)dW(s) 󳨀 t

t

whence the corresponding pointwise convergence of processes follows: T

T

1 Yn (t) := (S̆n exp{∫ σn (s)dW(s) − ∫ σn2 (s)ds} 2 t

t

T

− K exp{− ∫ rn (s)ds})

+

t

254 | 3 Limit theorems for markets with non-random time-varying coefficients T

T

1 󳨀 → Y(t) := (S̆0 exp{∫ σ0 (s)dW(s) − ∫ σ02 (s)ds} 2 ℙ

t

t

T

+

− K exp{− ∫ r0 (s)ds}) . t

Uniform integrability of Yn is obvious, because due to the conditions of the theorem they have uniformly bounded moments of all orders. Remark 3.7. (i) For the European put option, the terms of fair price convergence have the same form. (ii) Convergence itself takes place without condition (Nn ). However, without this condition it is doubtful if we can interpret it as the convergence of the unique arbitrage-free prices. (iii) Considering analytical proof, we implicitly used continuity of integrands rn , σn in time. In the probabilistic proof this is not necessary. (iv) We can assume that the strike prices vary, namely, they equal Kn and Kn → K0 . Then the result of Theorem 3.6 still holds.

3.2 Convergence of barrier option prices with time-varying parameters In Section 3.2.1 we establish the continuity in the series parameter of the fair prices of the “up-and-out” barrier call option with time-varying coefficients in the Black– Scholes model. Proving continuity, we use direct probabilistic methods, without using an explicit form of the barrier option prices. In Section 3.2.2 the boundary value problem is formulated, the solution of which is a fair price of the “up-and-out” barrier call option, as well as a limit theorem for solutions of the first boundary value problem for parabolic equations with time-dependent coefficients. More precisely, in Section 3.2.2 the price of the barrier option is considered as a solution to the boundary value problem and we establish a limit result for the sequence of solutions. To set the problem, we first formulate what is a barrier option. This is a type of option, the payment of which depends on whether it goes beyond some known barrier to the trajectory of the underlying asset. Barrier options are divided into entry (“knock-out”) and exit (“knock-in”), American and European. Payment on the entry option occurs if the asset price does not exceed the barrier, payment on the exit option occurs if the asset price exceeds the barrier. For example, the price of a European “up-and-out” barrier call option has the form π

barr

T

= 𝔼(exp{− ∫ r(t)dt}(S(T) − K) 1max0≤t≤T S(t) S(0) is a barrier, and 1A is an indicator of event A. The price of the European “down-and-in” barrier put option is defined as T

π̆ barr = 𝔼(exp{− ∫ r(t)dt}(K − S(T)) 1min0≤t≤T S(t)≤H ), +

0

where H < S(0) is a barrier. The prices of the remaining barrier options are determined similarly.

3.2.1 Robustness of the barrier option price Consider again the family of the models (3.12)–(3.13) together with supplying condiT T tions ∫0 σn2 (s)ds < ∞, ∫0 |μn (s)|ds < ∞. Additionally, assume that the sequences Hn , Kn , n ≥ 0, of non-negative numbers are given. It means that now both the strike prices and barriers vary. As in the Section 3.1.3 devoted to vanilla European options, our goal now is to establish stability (robustness) of the European barrier option price regarding changing or insufficiently accurate measurement of parameters that define the model. Remind that condition (Nn ) implies that the market model is complete and arbitrage-free. Measure ℙ(n,∗) , whose Radon–Nikodym derivative has the form t

t

0

0

2

μ (s) − rn (s) μ (s) − rn (s) dℙ(n,∗) 󵄨󵄨󵄨󵄨 1 dW(s) − ∫( n ) ds}, 󵄨󵄨 = exp{− ∫ n 󵄨 dℙ 󵄨ℱt σn (s) 2 σn (s) is a martingale measure. The fair option price is defined as an expectation with respect T to the measure ℙ(n,∗) , with discounting factor exp{− ∫0 rn (t)dt}, that is, a fair price of the European “up-and-out” barrier call option in the pre-limit model has the form T

πnbarr = 𝔼∗n (exp{− ∫ rn (t)dt}(Sn (T) − Kn ) 1max0≤t≤T Sn (t) Sn (0) is a barrier. A fair price of the corresponding option in the limit model has the form T

π barr = 𝔼∗ (exp{− ∫ r0 (t)dt}(S0 (T) − K0 ) 1max0≤t≤T S0 (t) S0 (0). We use this form to prove convergence of the option prices according to the series parameter n.

256 | 3 Limit theorems for markets with non-random time-varying coefficients Theorem 3.8. Let the following conditions be met: (i) numerical coefficients satisfy condition Sn (0) < Hn , n ≥ 0, and converge, i. e., Sn (0) → S0 (0), Kn → K0 , and Hn → H0 for n → ∞, (ii) σn converges to σ0 in ℒ2 [0, T] and rn converges to r0 in ℒ1 [0, T], (iii) for all n ≥ 0 condition (Nn ) holds. Then, if the parameter n → ∞, then the price πnbarr of the European “up-and-out” barrier call option in the pre-limit model, given by formula (3.17), converges to the corresponding price π barr , given by formula (3.18). Proof. Taking into account condition (iii) and using equations (3.17) and (3.18), we could write πnbarr − π barr =

T ∗ 𝔼n (exp{− ∫ rn (t)dt}(Sn (T) 0

− Kn ) 1max0≤t≤T Sn (t) T) δ ≤ |H − K| ∑ ∫ 󵄨󵄨 󵄨󵄨 2 󵄨󵄨 i H 󵄨󵄨 󵄨󵄨 󵄨󵄨 i=0 󵄨 0 T 󵄨󵄨 󵄨󵄨 m−1 󵄨󵄨 󵄨󵄨 ≤ C 󵄨󵄨󵄨exp{− ∫ r(t)dt + ∑ r(ti )Δ} − 1󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 i=0 󵄨 󵄨 0

For arbitrary x there is an inequality |ex − 1| ≤ |x|e|x| (see Lemma A.2(iii) in the appendix). Using this, we obtain 󵄨󵄨 T 󵄨󵄨 󵄨󵄨n−1 ti+1 󵄨󵄨 n−1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨δ2 󵄨󵄨 ≤ C 󵄨󵄨󵄨− ∫ r(t)dt + ∑ r(ti )Δ󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨 ∑ ∫ (r(t) − r(ti ))dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 i=0 i=0 󵄨 ti 0 n−1

ti+1

󵄨 󵄨 ≤ C ∑ ∫ 󵄨󵄨󵄨r(t) − r(ti )󵄨󵄨󵄨dt ≤ CΔ. i=0 t i

To bound δ1n , note that S is a solution of stochastic differential equation dS(t) = r(t)dt + σdW(t) and Snd is an Euler approximation for the value S(T) of this solution. Then, due to a well-known bound for the first moment of their difference (see Theorem A.127 in the appendix) and the boundedness of r, we obtain 󵄨󵄨 n 󵄨󵄨 󵄨 d󵄨 󵄨󵄨δ1 󵄨󵄨 ≤ C𝔼󵄨󵄨󵄨S(T) − Sn 󵄨󵄨󵄨 ≤ C √Δ. That is why we get 󵄨 󵄨󵄨 d,barr 󵄨 󵄨 − π barr 󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨𝔼((H − K)(1τHn >T − 1τH >T ))󵄨󵄨󵄨 + C √Δ 󵄨󵄨πn 󵄨 󵄨 = C 󵄨󵄨󵄨𝔼(1τn >T − 1τH >T )󵄨󵄨󵄨 + C √Δ. H

(3.29)

To estimate the first term in the right-hand side of (3.29), we use the Girsanov theorem. In this connection, we introduce the following notation. For the bounded meat surable function g: [0, T] → ℝ we define Wg (t) = W(t) + ∫0 g(s) ds, Brownian motion σ with drift g/σ, and let dℙg dℙ

T

= exp{− ∫ 0

T

1 g 2 (s) g(s) dW(s) − ∫ 2 ds} σ 2 σ 0

be the Radon–Nikodym derivative of the measure ℙg with respect to ℙ. Denote by 𝔼g (⋅) = 𝔼(

dℙg dℙ

⋅)

264 | 3 Limit theorems for markets with non-random time-varying coefficients expectation with respect to this measure. Finally, for process X define X ∗ = sup X(t), [0,T]

Xn∗ = max X(ti ), 1≤i≤n

and denote n−1

μn (t) = ∑ μ(ti )1t∈[ti ,ti+1 ) , i=0

L=

1 log(H/S(0)), σ

Snmax = max Sd (ti ), 0≤i≤n

where μ is defined in (3.27). dℙ Let us take into account that by the Girsanov theorem, Wg and ( dℙg )−1 have the same joint distribution with respect to ℙg as W and that

dℙ−g dℙ

with respect to ℙ. This implies

󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝔼(1τHn >T − 1τH >T )󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼(1S∗ ≥H ) − 𝔼(1Snmax ≥H )󵄨󵄨󵄨 󵄨󵄨 dℙμn −1 dℙμ −1 󵄨󵄨󵄨 󵄨 = 󵄨󵄨󵄨𝔼μ (( ) 1(Wμ )∗ ≥L ) − 𝔼μn (( ) 1(Wμn )∗n ≥L )󵄨󵄨󵄨 󵄨󵄨 dℙ dℙ 󵄨󵄨 󵄨󵄨 dℙ−μn dℙ−μ 󵄨󵄨󵄨 󵄨 )1W ∗ ≥L ) − 𝔼(( )1{Wn∗ ≥L} )󵄨󵄨󵄨 = 󵄨󵄨󵄨𝔼(( 󵄨󵄨 󵄨󵄨 dℙ dℙ 󵄨 󵄨󵄨 dℙ−μ dℙ−μn 󵄨󵄨 dℙ−μn 󵄨 )−( )󵄨󵄨󵄨1W ∗ ≥L ) + 𝔼(( )|1W ∗ ≥L − 1Wn∗ ≥L |) ≤ 𝔼(󵄨󵄨󵄨( 󵄨 󵄨󵄨 dℙ dℙ 󵄨 dℙ = ϵ1n + ϵ2n . We start by estimating the first term. Note that

dℙ−μ dℙ

is the value at the point T of the μ(t)

dℙ

−μn (t) solution of the stochastic differential equation dX(t) = σ dW(t), and dℙ , being μ a solution of the equation dX(t) = σn dW(t), is its Euler approximation; therefore, according to the abovementioned standard estimate (Theorem A.127), we obtain

󵄨󵄨 dℙ−μ dℙ−μ 󵄨󵄨 󵄨 n 󵄨󵄨 ϵ1n ≤ 𝔼(󵄨󵄨󵄨 − 󵄨) ≤ C √Δ. 󵄨󵄨 dℙ dℙ 󵄨󵄨󵄨 In order to estimate ϵ2n , note that 1W ∗ ≥L − IWn∗ ≥L = 1W ∗ ≥L,Wn∗ x} ∩ An )dx 0 ∞

≤ ℙ(An ) + C ∫ eCx ℙ(Ax )ℙ(An )dx 0 CZ

= ℙ(An )𝔼(e ) ≤ Cℙ(W ∗ ≥ L, Wn∗ < L). According to the asymptotic behavior of the uniform modulus of continuity of the Wiener process, stated by P. Lévy and having the form sup

󵄨󵄨 󵄨 󵄨󵄨W(t) − W(s)󵄨󵄨󵄨 = O(√h log(1/h)),

0≤s≤t≤T,|t−s|≤h

h→0

a. s.

(see, e. g., formula (9.26) in [89]), we can claim that ℙ(An ) ∼ C √Δ log(1/Δ) for n → ∞, whence ϵ2n ≤ C √Δ log(1/Δ) = C √ logn n . Finally, log n log n 󵄨󵄨 d,barr 󵄨 = O(√ ). − π barr 󵄨󵄨󵄨 ≤ C √ 󵄨󵄨πn n n Now we can prove Theorem 3.12 using the result of Theorem 3.13. Proof. Given the result of Theorem 3.13, it is sufficient to prove that πnd,barr − πnbarr = O(log n/√n), n → ∞. It is clear that 󵄨󵄨 n−1 + 󵄨󵄨 d,barr 󵄨󵄨󵄨 barr 󵄨󵄨 d π − π = 󵄨󵄨 n 󵄨󵄨𝔼(exp{− ∑ r(ti )Δ}(S (T) − K) 1{M d (T) (2K + 2)2 log2 n + x)dx 0 ∞

≤ C log2 n + ∫ ℙ(R > (K + 1) log n + x/2)dx 0 2

≤ C log n + Kn

∞

−λ

∫ e−λ√x/2 dx ≤ C log2 n. 0

Next, we assume without loss of generality that Kλ > 1/2. Since d

{ξi , i = 0, . . . , n − 1} = {ηi , i = 0, . . . , n − 1} and d

{Zi , i = 0, . . . , n − 1} = {ζi , i = 0, . . . , n − 1}, in order to estimate |πnd,barr − πnbarr |, we can assume that ξi = ηi and Zi = ζi , because it should not change expectation in (3.30). Taking this into account, we can write 󵄨󵄨 d,barr 󵄨 − πnbarr 󵄨󵄨󵄨 ≤ C(I1 + I2 ), 󵄨󵄨πn where 󵄨 󵄨 I1 = 󵄨󵄨󵄨𝔼(((Sd (T) − K)+ − (Sbin (T) − K)+ )1M bin (T) 0. Put τ = U ∧ min{u : W(u) =

x }. 4

Then U

x x ∫( max W(v) − ) du ≥ τ, 0≤v≤u 2 − 4 0

whence T

ℙ(∫ σ(r)hx (r) dr < ε) ≤ ℙ(τ < 0

4ε ). x

The latter probability, at least for small ε > 0, can be evaluated explicitly: ℙ(τ
0, the restriction of the distribution of the random variable MT to [x, +∞) has a bounded and continuous density. Proof. Choosing the function F depending on the first coordinate only, we have 𝔼F ′ (MT ) = 𝔼F(MT )Ξx ,

F ∈ C1 (ℝ),

󵄨 󵄨 sup󵄨󵄨󵄨F ′ (y)󵄨󵄨󵄨 < ∞, y

F|(−∞,x) ≡ 0.

Then the statement of the corollary can easily be derived with the help of the reasoning used in the proof of [132, Proposition 2.1.1]; see also Proposition A.78. Note that the result of the corollary can be improved; namely, the probability density function pMT of the random variable MT admits an integral representation similar to that in [132, Proposition 2.1.1]: pMT (y) = 𝔼1MT >y Ξx ,

y ≥ x.

3.4 The differentiability of a barrier option price as a function of the barrier

| 277

The difference between the latter representation and that of [132, Proposition 2.1.1] (see Proposition A.78) is that the integral factor Ξx depends on the interval where the representation is considered. Note that [132, Proposition 2.1.1] does not apply to the funcθ tional MT , since the random element σ 1[0,θ] [∫0 σ 2 (t) dt]−1 is not in general stochastically integrable. The latter property is confirmed by the fact that in the simplest case of f ≡ 0 and σ ≡ 1, the random variable MT = max0≤t≤T W(t) does not have a continuous probability density function. Now we are in a position to prove the main result of this subsection. t

Theorem 3.19. Let r ∈ ℒ1 ([0, T]), σ ∈ ℒ2 ([0, T]). Assume that ∫0 σ 2 (s) ds > 0 for all t > 0. Then the price π barr (H) of a European “up-and-out” barrier call option has the barr derivative dπ dH(H) at any point H ∈ (S, +∞), where S is an initial asset price. Moreover, this derivative admits the following integral representation: T

1 dπ barr (H) = exp(− ∫ r(s) ds) dH H 0

T

× [S𝔼 exp(Y(T) + f (T))1Y(T)+f (T)≥log K 1M S

− 𝔼(S exp(Y(T) + f (T)) − K)+ 1M

H T ε) = 0

k→∞

n→∞

i=1

for any T, ε > 0. Then ℙ(n) , ℙ(0) , d

S(ξn , μn , 0, T) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ S(ξ0 , μ0 , 0, T) for any T > 0 as n → ∞.

286 | 4 Convergence of stochastic integrals in application to financial markets 2. Let additional conditions hold: either (L2) (1) limC→∞ lim supn→∞ ℙ(n) (sup0≤t≤T |ξn (t)| ≥ C) = 0 for any T > 0, (2) limδ→0 lim supn→∞ ℙ(n) (ΔD (|μn |(⋅), 0, T, δ) > ε) = 0 for any ε > 0, T > 0, or kτ+t +1 (L3) limδ→0 lim supn→∞ supτ∈𝕋T (ℱ n ) ℙ(n) (sup0≤t≤δ ∑i=k |ξn (ti−1k )||Δik μn | > ε) = 0 for τ

any ε > 0, where ℙ(n) , ℙ(0) are the measures corresponding to the pre-limit and limit processes, respectively.

Then for any T > 0 the sequence of stochastic processes S(ξn , μn , 0, ⋅) = {S(ξn , μn , 0, t), t ∈ [0, T]} weakly converges in measure: ℙ(n) , ℙ(0)

S((ξn , μn , 0, t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (S(ξ0 , μ0 , 0, t), t ∈ [0, T]),

n → ∞.

Proof. 1. On the one hand, consider the following sequence of random variables: kT +1 ST,n,k = ∑i=1 ξn (tik )Δik μn . Note that if condition (L) is met, then T

ST,0,k → ∫ ξ0 dμ0

(4.6)

0

a. s. as k → ∞. On the other hand, it follows from condition (L1) that d

→ ST,0,k ST,n,k 󳨀

as n → ∞ for any k.

(4.7)

Denote kT +1

S+ (πk , ξn , μn ) = ∑

sup

ξn (t)(μn (tik ) − μn (ti−1 k ))

inf

ξn (t)(μn (tik ) − μn (ti−1 k )).

i=1 ti−1 k ≤t ε)

k→∞

n→∞

≤ lim lim sup ℙ(n) (S+ (πk , ξn , μn ) − S− (πk , ξn , μn ) > ε) k→∞

n→∞

kT +1

= lim lim ℙ(n) ( ∑ ωi k (ξn )ωi k (μn ) > ε) = 0. k→∞ n→∞

i=1

(4.9)

Now the proof of the first part of the theorem follows from (4.6)–(4.9) and Theorem A.86. 2. Suppose that condition (L2) holds. Then we have an obvious upper bound 󵄨 󵄨 ΔD (S(ξn , μn , 0, ⋅), 0, T, δ) ≤ sup 󵄨󵄨󵄨ξn (t)󵄨󵄨󵄨 ⋅ ΔD (|μn |(⋅), 0, T, δ), 0≤t≤b

whence we get the density of probability measures that correspond to stochastic integrals (see Theorem A.96). From the weak convergence of finite-dimensional distributions and the density of the corresponding probability measures, we have convergence in the Skorokhod topology (see Theorem A.95) of the sequence of stochastic processes S(ξn , μn , 0, ⋅) = {S(ξn , μn , 0, t), t ∈ [0, T]} for any T > 0. Assume that condition (L3) holds. Then it follows from Theorem A.101 that the sequence S(ξn , μn , 0, t), t ∈ [0, T] is weakly relatively compact in the Skorokhod topology. Then from the weak convergence of finite-dimensional distributions of the sequence of stochastic integrals we obtain their convergence in the Skorokhod topology; see Theorem A.95. Remark 4.18. Note that in Theorem 4.17, μ0 can be any process (possibly of unbounded variation), such that the limit S(ξ0 , μ0 , 0, T) with (4.5) exists a. s. for any T > 0. Below an example of the application of Theorem 4.17 is provided. It is rather from the field of actuarial than financial mathematics, but we will present it due to its simplicity and clarity. Example 4.19. Let the sequence of gain processes {Un (t), n ≥ 0} of an insurance company be defined on the same probability space (Ω, ℱ , ℙ) and let it have the form Nn (t)

Un (t) := An (t) − ∑ Xnk , k=1

288 | 4 Convergence of stochastic integrals in application to financial markets where An , n ≥ 0, is a sequence of processes with continuous and non-decreasing trajectories which can be interpreted as processes of insurance premiums, {Xnk , k ≥ 1} for any n ≥ 0 is a sequence of non-negative i. i. d. r. v. with distribution functions Fn , and Nn (t) is a sequence of point processes with càdlàg trajectories, Nn (0) = 0, ∑0k=1 = 0. Nn (t) k Processes ∑k=1 Xn could be interpreted as a sequence of insurance payments. Denote 1 by 0 < Tn (ω) < Tn2 (ω) < ⋅ ⋅ ⋅ the sequence of jumps of Nn , ΔNn (Tnk ) = 1. The process of reserving capital (risk reserve process) with “accumulator” φn is defined, according to [67], as follows: t

Rn (t) := φn (t)u + ∫ φn (t − s)dUn (s) 0 t

Nn (t)

0

k=1

= φn (t)u + ∫ φn (t − s)dAn (s) − ∑ φn (t − Tkn )Xnk , where u > 0 is an initial level of reserving and φn : ℝ+ → ℝ+ is some non-random continuous non-decreasing function (as usual φn (t) = exp{δt}, δ > 0). Consider any T > 0 and denote ⌊nt⌋

Xn (t) := ∑ Xnk , t ∈ [0, T]. k=1

The next result is a direct consequence of Theorem 4.17. Theorem 4.20. Suppose that the following conditions hold: 1) φn (t) → φ0 (t), n → ∞ pointwise, ℙ(n) , ℙ, fdd

N (t)

2) (An (t), Xn (t), nn , t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (A0 (t), X0 (t), N0 (t), t ∈ 𝒯T ), 3) integral S(φ0 (T − ⋅), X0 , 0, T) exists a. s., 4) for any ε > 0 Nn (T)

󵄨 󵄨 lim lim sup ℙ( ∑ Δln k φn (b − ⋅)󵄨󵄨󵄨Xni 󵄨󵄨󵄨 > ε) = 0, i

k→∞

n→∞

i=1

where lin is defined so that tln −1k ≤ Tni < tln k , i i 5) for any ε > 0 kT +1

lim lim sup ℙ( ∑ Δik φn (T − ⋅)Δik An > ε) = 0.

k→∞

n→∞

i=1

Then a weak convergence of the sequence of risk reserve processes holds: ℙ(n) , ℙ(0)

t

t

(Rn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (φ0 (b)u + ∫ φ(t − s)dA0 (s) − ∫ φ0 (t − s)dY0 (s), t ∈ [0, T]) 0

0

4.2 Functional limit theorems for the integrals with respect to semimartingales | 289

as n → ∞, where the limit process has the form Y0 (t) := X0 (N0 (t)), t ∈ [0, T], and ℙ(n) , ℙ(0) are the measures that correspond to the pre-limit and limit processes, respectively. 4.2.2 Weak convergence of stochastic integrals with respect to martingales As before, let (Ω(n) , ℱ (n) , 𝔽(n) = {ℱt(n) , t ≥ 0}, ℙ(n) ), n ≥ 0, be a sequence of stochastic bases. All processes under consideration are supposed to be adapted. Consider the sequence {Mn (t), ℱt(n) , t ≥ 0, n ≥ 0} of square-integrable martingales with càdlàg trajectories. Let μn (t) := ⟨Mn ⟩(t) be the quadratic characteristics of the above-described martingales, and we consider their càdlàg modifications. Obviously, they are nondecreasing. Also, consider a sequence {ξn (t), ℱt(n) , t ≥ 0, n ≥ 0} of ℱ⋅(n) -predictable processes. Theorem 4.21. Assume the following conditions hold: (i) convergence of finite-dimensional distributions holds: ℙ(n) , ℙ, fdd

(ξn (t), Mn (t), μn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), M0 (t), μ0 (t), t ∈ 𝒯T ) for any T > 0, t (ii) supn≥0 𝔼n ∫0 ξn2 (s)dμn (s) < ∞, t ≥ 0,

(iii) limC→∞ lim supn→∞ ℙ(n) (sup0≤t≤T |ξn (t)| ≥ C) = 0 for any T > 0, kT +1 (iv) limk→∞ lim supn→∞ 𝔼n ∑i=1 ωik (ξn )ωik (μn ) = 0 for any T > 0, (v) limδ→0 lim supn→∞ supσ∈𝕋T (ℱ n ) 𝔼n (μn (σ + δ) − μn (σ)) = 0. Then for any T > 0 the sequence of stochastic integrals weakly converges in the Skorokhod topology on [0, T], i. e., the measures corresponding to these integrals weakly converge: t

t

ℙ(n) , ℙ(0)

(∫ ξn (s)dMn (s), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (∫ ξ0 (s)dM0 (s), t ∈ [0, T]), 0

n → ∞,

0

and the sequence of their quadratic characteristics converges as well: t

(∫ ξn2 (s)dμn (s), 0

t

ℙ(n) , ℙ(0)

t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (∫ ξ02 (s)dμ0 (s), t ∈ [0, T]),

n → ∞.

0

Proof. From conditions (i) and (iv) and from Theorem 4.17 we get a convergence of finite-dimensional distributions of quadratic characteristics: t

(∫ ξn2 (u)dμn (u), 0

ℙ(n) , ℙ, fdd

t

t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (∫ ξ02 (u)dμ0 (u), t ∈ 𝒯T ), 0

n → ∞.

(4.10)

290 | 4 Convergence of stochastic integrals in application to financial markets Additionally, condition (i) implies that k

k

ℙ(n) , ℙ, fdd

∑ ξn (ti−1k )Δik Mn ,

1 ≤ k ≤ kT 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ ∑ ξ0 (ti−1k )Δik M0 ,

i=1

1 ≤ k ≤ kT ,

i=1

(4.11)

as n → ∞. Moreover, condition (ii) implies the existence of stochastic integrals with respect to respective martingales. Therefore we can conclude that for each n ≥ 0 T

kT +1

→ ∫ ξn (s)dMn (s), ∑ ξn (ti−1k )Δik Mn 󳨀 ℙ

i=1

k → ∞.

(4.12)

0

Let us denote φn (s) := ξn (s) − ξn (ti−1 k ), φCn (s)

:= ξn (s) ∧ C ∨ (−C) − ξn (ti−1 k ) ∧ C ∨ (−C),

s ∈ Δik .

Then for any ε > 0 T 󵄨󵄨kT +1 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim lim sup ℙ(n) (󵄨󵄨󵄨 ∑ ξn (ti−1 k )Δik Mn − ∫ ξn (s)dMn (s)󵄨󵄨󵄨 > ε) 󵄨󵄨 󵄨󵄨 k→∞ n→∞ 󵄨 i=1 󵄨 0

󵄨󵄨 T 󵄨󵄨 󵄨󵄨 󵄨󵄨 = lim lim sup ℙ (󵄨󵄨󵄨∫ φn (s)dMn (s)󵄨󵄨󵄨 > ε) 󵄨 󵄨󵄨 k→∞ n→∞ 󵄨󵄨 󵄨 0 (n)

󵄨 󵄨 ≤ lim lim sup ℙ(n) { sup 󵄨󵄨󵄨ξn (t)󵄨󵄨󵄨 ≥ C} C→∞

n→∞

0≤t≤T

T

+ε

−2

2

lim lim lim sup 𝔼n ∫(φCn (s)) dμn (s)

C→∞ k→∞

n→∞

0

kT +1

≤ ε−2 lim (C lim lim sup 𝔼n ∑ ωik (ξn )ωik (μn )) = 0. C→∞

k→∞

n→∞

i=1

(4.13)

Relations (4.11)–(4.13) being valid for any t along with Theorem A.86 imply weak convergence of finite-dimensional distributions: t

ℙ(n) , ℙ, fdd

t

(∫ ξn (s)dMn (s), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (∫ ξ0 (s)dM0 (s), t ∈ 𝒯T ), 0

n → ∞.

(4.14)

0

In addition, for any T > 0 it follows from condition (ii) and the Burkholder–Gundy inequality for martingales that for some constant C1 > 0 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim lim sup ℙ(n) (sup󵄨󵄨󵄨∫ ξn (u)dMn (u)󵄨󵄨󵄨 ≥ C) 󵄨󵄨 C→∞ n→∞ t≤T 󵄨󵄨󵄨 󵄨 0

2

󵄨󵄨 t 󵄨󵄨 1 󵄨󵄨 󵄨󵄨 ≤ lim 2 lim sup 𝔼n (sup󵄨󵄨󵄨∫ ξn (u)dMn (u)󵄨󵄨󵄨 ) 󵄨 󵄨󵄨 C→∞ C n→∞ t≤T 󵄨󵄨 󵄨 0

4.2 Functional limit theorems for the integrals with respect to semimartingales | 291 T

C ≤ lim lim sup 12 𝔼n ∫ ξn2 (u)dμn (u) = 0. C→∞ n→∞ C

(4.15)

0

Consider now the stochastic process t+σ

σ

0

0

Znσ (t, C) = ∫ ξnC (u)dMn (u) − ∫ ξnC (u)dMn (u), where t ≥ 0, σ ∈ 𝕋T (ℱ n ), ξnC (u) = ξn (u) ∧ C ∨ (−C). It is a martingale with respect (n) to the filtration ℱt+σ , t ≥ 0. Therefore it follows from conditions (iii) and (v) and the Burkholder–Gundy inequality (Theorem A.38) that for any ε > 0 σ 󵄨󵄨 t+σ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 lim lim sup sup ℙ (sup󵄨󵄨 ∫ ξn (u)dMn (u) − ∫ ξn (u)dMn (u)󵄨󵄨󵄨 ≥ ε) 󵄨 󵄨󵄨 δ→0 n→∞ σ∈𝕋T (ℱ n ) t≤δ 󵄨󵄨 󵄨 0 0 (n)

󵄨 󵄨 ≤ lim (lim sup ℙ(n) ( sup 󵄨󵄨󵄨ξn (t)󵄨󵄨󵄨 ≥ C) C→∞

+ =

n→∞

0≤t≤T

C2 2 lim lim sup sup 𝔼 (Z σ (δ, C)) ) ε2 δ→0 n→∞ σ∈𝕋T (ℱ n ) n n

C2 lim lim sup sup 𝔼 (μ (σ + δ) − μn (σ)) = 0. ε2 δ→0 n→∞ σ∈𝕋T (ℱ n ) n n

(4.16)

In the same way as done in (4.15) we could easily prove that for any T > 0 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim lim sup ℙ (sup󵄨󵄨󵄨∫ ξn2 (u)dμn (u)󵄨󵄨󵄨 ≥ C) = 0 󵄨 󵄨󵄨 C→∞ n→∞ t≤T 󵄨󵄨 󵄨 0 (n)

(4.17)

and lim lim sup

δ→0

n→∞

σ 󵄨󵄨 t+σ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ℙ(n) (sup󵄨󵄨󵄨 ∫ ξn2 (u)dμn (u) − ∫ ξn2 (u)dμn (u)󵄨󵄨󵄨 ≥ ε) = 0, 󵄨 󵄨󵄨 σ∈𝕋T (ℱ n ) t≤δ 󵄨󵄨 󵄨 0 0

sup

(4.18)

ε > 0. It follows from (4.15)–(4.18) and Theorem A.96 that the sequences ⋅

∫ ξn (u)dMn (u) and 0

⋅

∫ ξn2 (u)dμn (u) 0

are weakly relatively compact in the Skorokhod topology. Then the weak convergence of finite-dimensional distributions (relationships (4.10) and (4.14)), Prokhorov’s theorem (Theorem A.91), and Theorem A.95 imply a weak convergence of these sequences in the Skorokhod topology.

292 | 4 Convergence of stochastic integrals in application to financial markets Remark 4.22. Obviously, instead of condition (ii) it is possible to require the following condition: t

𝔼n ∫ ξn2 (s)dμn (s) < ∞,

t ≥ 0,

0

lim lim sup ℙ(n) (μn (t) ≥ C) = 0,

C→∞

n→∞

t > 0.

4.2.3 Weak convergence of stochastic integrals with respect to semimartingales Now Xn = {Xn (t), ℱt(n) , t ≥ 0, n ≥ 0} is a sequence of semimartingales on a respective stochastic basis (Ω(n) , 𝔽(n) = {ℱt(n) , t ≥ 0}, ℙ(n) ). We assume that they admit the following decompositions: Xn (t) = Xn0 + Mn (t) + An (t),

(4.19)

where {Mn (t), ℱt(n) , t ≥ 0, n ≥ 0} is a sequence of square-integrable martingales with càdlàg trajectories and {An (t), t ≥ 0, n ≥ 0} is a sequence of processes of bounded variation with càdlàg trajectories. Let μn (t) = ⟨Mn ⟩(t) be quadratic characteristics of respective martingales. Consider also {ξn (t), ℱt(n) , t ≥ 0, n ≥ 0}, a sequence of ℱ⋅(n) -predictable processes. We say that condition (M1) is satisfied if T T (M1) 𝔼n ∫0 ξn2 (t)dμn (t) < ∞, ∫0 |ξn (t)|d|An |(t) < ∞, ℙ(n) -a. s., T ≥ 0, n ≥ 0. T

T

Here 𝔼n ∫0 ξn2 (t)dμn (t) and ∫0 ξn (t)dAn (t) are assumed to exist as the Riemann– Stieltjes integrals for any of such decomposition of Xn . t t t Define ∫0 ξn (s)dXn (s) := ∫0 ξn (s)dMn (s) + ∫0 ξn (s)dAn (s). Note that this definition is correct because the right-hand side is invariant with respect to all such decompositions (see for instance [110], p. 93). Theorem 4.23. Let {Xn (t), ℱt(n) , t ≥ 0, n ≥ 0} be a sequence of semimartingales with decomposition (4.19). Suppose that condition (M1) and additionally the following conditions are satisfied: (i) (ii)

ℙ(n) , ℙ, fdd

(ξn (t), Mn (t), An (t), μn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), M0 (t), A0 (t), μ0 (t), t ∈ 𝒯T ) for any T > 0, t supn≥0 𝔼n ∫0 ξn2 (s)dμn (s) < ∞, t ≥ 0,

(iii) limC→∞ lim supn→∞ ℙ(n) (sup0≤t≤T |ξn (t)| ≥ C) = 0 for any T > 0, kT +1 (iv) limn→∞ lim supn→∞ 𝔼n ∑i=1 ωik (ξn )ωik (An ) = 0 for any T > 0, (v) limδ→0 lim supn→∞ ℙ(n) (ΔD (|An |(⋅), 0, T, δ) > ε) = 0 for any ε > 0, T > 0, kT +1 (vi) limk→∞ lim supn→∞ 𝔼n ∑i=1 ωik (ξn )ωik (μn ) = 0 for any T > 0, (vii) limδ→0 lim supn→∞ supσ∈𝕋T (ℱ n ) 𝔼n (μn (σ + δ) − μn (σ)) = 0, (viii) lim supn→∞ ℙ(n) (supt∈[0,T] |ΔAn (t)| > ε) = 0 for any ε > 0 and T > 0.

4.2 Functional limit theorems for the integrals with respect to semimartingales | 293

Then for any T > 0 weak convergence of stochastic integrals holds: t

ℙ(n) , ℙ

t

(∫ ξn (s)dXn (s), t ∈ [0, T]) 󳨐󳨐󳨐󳨐⇒ (∫ ξ0 (s)dX0 (s), t ∈ [0, T]) 0

0

as n → ∞ for any T > 0. Proof. First, we prove the convergence of finite-dimensional distributions of integrals with respect to the processes of bounded variation and with respect to the martingales, similarly to Theorems 4.17 and 4.20. It follows from condition (i) that k

k

(∑ ξn (ti−1k )Δik Mn + ∑ ξn (ti−1k )Δik An , 1 ≤ k ≤ kT ) i=1

i=1

k

k

i=1

i=1

ℙ , ℙ, fdd (n)

(4.20)

󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (∑ ξ0 (ti−1k )Δik M0 + ∑ ξ0 (ti−1k )Δik A0 , 1 ≤ k ≤ kT ) as n → ∞. Condition (M1) implies the existence of stochastic integrals, and hence for each n ≥ 0 kT +1

kT +1

∑ ξn (ti−1k )Δik Mn + ∑ ξn (ti−1k )Δik An i=1

i=1

T

T

󳨀 → ∫ ξn (s)dMn (s) + ∫ ξn (s)dAn (s), ℙ

k → ∞.

(4.21)

0

0

Next, for any ε > 0 it follows from relationships (4.9) and (4.13) and conditions (iii), (iv), and (vi) that 󵄨󵄨kT +1 kT +1 󵄨󵄨 lim lim sup ℙ(n) (󵄨󵄨󵄨 ∑ ξn (ti−1k )Δik Mn + ∑ ξn (ti−1k )Δik An 󵄨󵄨 k→∞ n→∞ i=1 󵄨 i=1 T T 󵄨󵄨󵄨 󵄨 − ∫ ξn (s)dMn (s) − ∫ ξn (s)dAn (s)󵄨󵄨󵄨 > ε) 󵄨󵄨 󵄨 0 0

T 󵄨󵄨kT +1 󵄨󵄨 󵄨󵄨 󵄨󵄨 ε 󵄨 ≤ lim lim sup ℙ (󵄨󵄨 ∑ ξn (ti−1k )Δik Mn − ∫ ξn (s)dMn (s)󵄨󵄨󵄨 > ) 󵄨󵄨 󵄨󵄨 2 k→∞ n→∞ 󵄨 i=1 󵄨 0 (n)

T 󵄨󵄨kT +1 󵄨󵄨 󵄨󵄨 󵄨󵄨 ε + lim lim sup ℙ(n) (󵄨󵄨󵄨 ∑ ξn (ti−1k )Δik An − ∫ ξn (s)dAn (s)󵄨󵄨󵄨 > ) 󵄨󵄨 󵄨󵄨 2 k→∞ n→∞ 󵄨 i=1 󵄨 0

= 0.

(4.22)

294 | 4 Convergence of stochastic integrals in application to financial markets Relations (4.21)–(4.22) and Theorem 4.2 [18] imply weak convergence of finitedimensional distributions of the integrals: t

t

t

∫ ξn (s)dXn (s) = ∫ ξn (s)dAn (s) + ∫ ξn (s)dMn (s) 0 ℙ , ℙ, fdd (n)

t

0

t

0

t

󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ ∫ ξ0 (s)dA0 (s) + ∫ ξ0 (s)dM0 (s) = ∫ ξ0 (s)dX0 (s) 0

0

(4.23)

0

on any interval [0, T] as n → ∞. Furthermore, it follows from conditions (v) and (vii), together with Theorems 4.17 and 4.21, respectively, that the measures corresponding t t to the sequences ∫0 ξn (s)dAn (s) and ∫0 ξn (s)dMn (s) are tight. t

Denote the process I(t) = ∫0 ξn (s)dAn (s) and denote by ΔI(t) = I(t) − I(t−) its jump. It follows from conditions (iii) and (viii) that for any ε > 0 and T > 0 󵄨 󵄨 lim sup ℙ(n) ( sup 󵄨󵄨󵄨ΔI(t)󵄨󵄨󵄨 > ε) = 0. n→∞ t∈[0,T]

(4.24)

t

Tightness of the sequence ∫0 ξn (s)dAn (s) and (4.24) imply that this sequence of integrals is C-tight (see Proposition A.103). t Note that C-tightness of the sequence ∫0 ξn (s)dAn (s) and the tightness of the t

sequence ∫0 ξn (s)dMn (s) imply tightness of the sum of corresponding sequences t

∫0 ξn (s)dXn (s) (see Proposition A.104). Finally, the weak convergence of finitedimensional distributions (4.23), tightness of the sequence of probability measures t generated by the processes ∫0 ξn (s)dXn (s), and Theorem A.95 imply convergence of the corresponding probability measures in the Skorokhod space. Now we generalize Theorem 4.23 to the multi-dimensional case. Assume that Xn = {Xn (t) = (Xn1 (t), Xn2 (t), . . . , Xnd (t)), ℱt(n) , t ≥ 0, n ≥ 0} is a sequence of d-dimensional semimartingales whose components allow the following decompositions: Xnj (t) = Xnj (0) + Mnj (t) + Ajn (t),

1 ≤ j ≤ d,

(4.25)

where {Mnj (t), ℱt(n) , t ≥ 0, n ≥ 0} is a sequence of square-integrable martingales with càdlàg trajectories and {Ajn (t), t ≥ 0, n ≥ 0} is a sequence of processes of bounded variation with càdlàg trajectories. Let μjn (t) := ⟨Mnj ⟩(t) be the quadratic characteristics of the corresponding martingales, and let {ξnj (t), ℱt(n) , t ≥ 0, n ≥ 0, 1 ≤ j ≤ d} be a sequence of 𝔽(n) -predictable processes. We say that condition (M2) holds if

4.2 Functional limit theorems for the integrals with respect to semimartingales | 295

T

T

(M2) 𝔼n ∫0 (ξnj )2 (t)dμjn (t) < ∞, ∫0 |ξnj (t)|d|Ajn |(t) < ∞, ℙ(n) -a. s., T > 0, n ≥ 0, 1 ≤ j ≤ d. Denote t

d

t

∫ ξn (s) ⋅ dXn (s) = ∑ ∫ ξnj (s)dXnj (s). j=1 0

0

Theorem 4.24. Suppose that {Xn (t), ℱt(n) , t ≥ 0, n ≥ 0} is a sequence of semimartingales with decomposition (4.25). Consider the case where condition (M2) holds along with the following conditions: (i)

ℙ(n) , ℙ(0) , fdd

(ξn (t), Mn (t), An (t), μn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), M0 (t), A0 (t), μ0 (t), t ∈ 𝒯T ) for any T > 0,

and for all 1 ≤ j ≤ d: t (ii) supn≥0 𝔼n ∫0 (ξnj )2 (s)dμjn (s) < ∞, t ≥ 0,

(iii) limC→∞ lim supn→∞ ℙ(n) (sup0≤t≤T |ξnj (t)| ≥ C) = 0 for any T > 0, kT +1 (iv) limk→∞ lim supn→∞ 𝔼n ∑i=1 ωik (ξnj )ωik (Ajn ) = 0 for any T > 0, (v) limδ→0 lim supn→∞ ℙ(n) (ΔD (|Ajn |(⋅), 0, T, δ) > ε) = 0 for any ε > 0, T > 0, kT +1 (vi) limk→∞ lim supn→∞ 𝔼n ∑i=1 ωik (ξnj )ωik (μjn ) = 0 for any T > 0, (vii) limδ→0 lim supn→∞ supσ∈𝕋T (ℱ n ) 𝔼n (μjn (σ + δ) − μjn (σ)) = 0, (viii) lim supn→∞ ℙ(n) (supt∈[0,T] |ΔAjn (t)| > ε) = 0 for any ε > 0, T > 0, (ix) lim supn→∞ ℙ(n) (supt∈[0,T] |ΔMnj (t)| > ε) = 0 for any ε > 0, T > 0. Then for any T > 0 the sequence of stochastic integrals weakly converges: t

ℙ(n) , ℙ(0)

t

(∫ ξn (s) ⋅ dXn (s), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (∫ ξ0 (s) ⋅ dX0 (s), t ∈ [0, T]) 0

0

as n → ∞ for any T > 0. The proof is carried out according to the same scheme as the proof of Theorem 4.23. 4.2.4 Weak convergence of integrands under the condition of convergence of stochastic integrals In this section we formulate the conditions for integrators and integrands for the multi-dimensional case that provide weak convergence of the latter under the condition of convergence of integrals. It means that we solve, in a sense, an “inverse” problem of weak convergence. It arises in financial mathematics when having convergence of capitals, and it is required to investigate the limiting behavior of a certain

296 | 4 Convergence of stochastic integrals in application to financial markets class of strategies. Specific examples of the application of the obtained results will be given in the next section. Let (Ω, ℱ , 𝔽 = {ℱt , t ≥ 0}, ℙ) be a stochastic basis, and let the process X = {X(t), ℱt , t ≥ 0} = {(X 1 (t), X 2 (t), . . . , X d (t)), ℱt , t ≥ 0} be a square-integrable semimartingale, with components admitting a decomposition X j (t) = Aj (t) + M j (t).

(4.26)

Here for all 1 ≤ j ≤ d, M j is a square-integrable martingale with quadratic characteristic μj and Aj is a process of bounded variation. Assume for simplicity that the filtration 𝔽 is generated by M and A. Let ξ (t) = (ξ 1 (t), . . . , ξ d (t)) ∈ ℝd be 𝔽-predictable processes for which T

j

2

2

T

j

𝔼 ∫(ξ (s)) dμ (s) < ∞, 0

󵄨 󵄨 󵄨 󵄨 𝔼(∫󵄨󵄨󵄨ξ j (s)󵄨󵄨󵄨d󵄨󵄨󵄨Aj 󵄨󵄨󵄨(s)) < ∞,

1 ≤ j ≤ d.

(4.27)

0

Lemma 4.25. Let X be a semimartingale with decomposition (4.26), let the stochastic process ξ satisfy conditions (4.27), and let for some processes ζ , Y, Z t

d

(ξ (t), X(t), ∫ ξ (s) ⋅ dX(s), t ∈ [0, T]) = (ζ (t), Y(t), Z(t), t ∈ [0, T]). 0 t

Then Y is a semimartingale and for all t ∈ [0, T] we have Z(t) = ∫0 ζ (s) ⋅ dY(s). Proof. Note that the semimartingale with respect to the filtration 𝔽 = {ℱt , t ∈ [0, T]} is an 𝔽-adapted process Z, say, such that for any sequence {Hn = Hn (t), t ∈ [0, T], n ≥ 1} of stepwise 𝔽-predictable processes that are uniformly convergent to H in probability, we have convergence of the integrals T

T

→ ∫ H(t)dZ(t) ∫ Hn (t)dZ(t) 󳨀 ℙ

0

0

(see [140]). Therefore, the property of being a semimartingale with respect to a natural filtration depends only on the distribution of the process. That is why Y is a semimartingale. Furthermore, if the processes ξ and X satisfy the above conditions, then the intet gral ∫0 ξ (s) ⋅ dX(s) exists as a limit in probability of integral sums n

n n S(πn , ξ , X) = ∑ ξ (tk−1 ) ⋅ (X(tkn ) − X(tk−1 )) k=1

4.2 Functional limit theorems for the integrals with respect to semimartingales | 297

under any sequence of partitions πn = {0 = t0n < t1n < ⋅ ⋅ ⋅ < tnn = t} with vanishing t

diameter. But the joint probability distribution of these sums and ∫0 ξ (s) ⋅ dX(s) is the same as the joint probability distribution of the sums S(πn , ζ , Y) and Z(t). Therefore t

S(πn , ζ , Y) 󳨀 → Z(t), |πn | → 0, and consequently, Z(t) = ∫0 ζ (s) ⋅ dY(s). ℙ

Definition 4.26. We say that the components X j , 1 ≤ j ≤ d, of the semimartingale X are linearly independent in the sense that for any 𝔽-predictable process ζ ∈ ℝd , from the fact that d

t

∑ ∫ ζ j (s)dX j (s) = 0 j=1 0

for all t ∈ [0, T] it follows that (ζ 1 (t), . . . , ζ d (t)) = 0 for all t ∈ [0, T] a. s. As before, t t denote ∫0 ξ (s) ⋅ dX(s) := ∑dj=1 ∫0 ξ j (s)dX j (s). Theorem 4.27. Let ξn = {ξn (t), t ≥ 0, n ≥ 0} = {(ξn1 (t), . . . , ξnd (t)), t ≥ 0, n ≥ 0} be a sequence of predictable d-dimensional processes satisfying condition (4.27). Also, let X be a semimartingale with decomposition (4.26), let stochastic processes ξn satisfy conditions (4.27), let X have linearly independent components, and for some T > 0 let the following assumptions hold: t

t

→ ∫0 ξ0 (s) ⋅ dX(s) as n → ∞, (i) for all 0 ≤ t ≤ T, ∫0 ξn (s) ⋅ dX(s) 󳨀 ℙ

(ii) the sequence of measures ℙ(n) corresponding to the processes {ξn , n ≥ 0} is weakly relatively compact in the Skorokhod topology. Then there is a weak convergence ℙ(n) , ℙ(0)

(ξn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), t ∈ [0, T]),

n → ∞.

Proof. It is sufficient to prove that from any subsequence {ξn , n ≥ 0} it is possible to choose a subsequence that is weakly convergent in the Skorokhod topology to ξ0 . So, let {ξnk , k ≥ 0} be the subsequence of {ξn , n ≥ 0}. It follows from the weak relative compactness that a certain subsequence of the sequence {ξnk , k ≥ 0} is weakly convergent. Furthermore, for simplicity, we assume that the sequence {ξn , n ≥ 0} is weakly convergent in the Skorokhod topology. Moreover, we will assume that the sequence of vectors ⋅

(ξ0 (⋅), ∫ ξ0 (s) ⋅ dX(s), ξn (⋅), X(⋅)) 0

is weakly convergent (weak relative compactness of {ξn , n ≥ 0} implies weak relative compactness of this sequence). Denote its limit by (ζ0 (⋅), Z(⋅), ζ (⋅), Y(⋅)). From

298 | 4 Convergence of stochastic integrals in application to financial markets t

d

Lemma 4.25 we have Z(t) = ∫0 ζ0 (s) ⋅ dY(s). Now our goal is to prove that ξ0 = ζ . According to Theorem A.105, we have the following weak convergence in the Skorokhod topology: t

t

(ξ0 (t), ∫ ξ0 (s) ⋅ dX(s), ξn (t), X(t), ∫ ξn (s) ⋅ dX(s), t ∈ [0, T]) 0 (n)

0

t

(0)

ℙ ,ℙ

t

󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ζ0 (t), ∫ ζ0 (s) ⋅ dY(s), ζ (t), Y(t), ∫ ζ (s) ⋅ dY(s), t ∈ [0, T]). 0

(4.28)

0

In fact, Theorem A.105 states only joint convergence of the last three components, namely, integrands, integrators, and integrals. However, we can say that the extended vectors converge because additional components can be added to the integrands, and the integrator can be supplemented by zeros. Then the stochastic integral will have the same value. Now, on the one hand, convergence in probability in condition (i) means that for any ε > 0 t 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 ℙ(󵄨󵄨󵄨∫ ξn (s) ⋅ dX(s) − ∫ ξ0 (s) ⋅ dX(s)󵄨󵄨󵄨 > ε) → 0, 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 0

n → ∞.

On the other hand, it follows from the weak convergence of (4.28) that t 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 ℙ(󵄨󵄨∫ ζ (s) ⋅ dY(s) − ∫ ζ0 (s) ⋅ dY(s)󵄨󵄨󵄨 > ε) = 0. 󵄨󵄨 󵄨󵄨 󵄨0 󵄨 0

Thus, for all t > 0 we have t

t

∫ ζ (s) ⋅ dY(s) = ∫ ζ0 (s) ⋅ dY(s). 0

0

Note that the condition of “linear independence” of the processes X j is formulated in terms of 𝔽-predictable processes, but the filtration 𝔽 = {ℱt , t ∈ [0, t]} is generated by the process X. Therefore this condition is preserved when going to the process Y with the same distribution. In particular, it follows from the last equality that for all d

t ∈ [0, T], ζ (t) = ζ0 (t) a. s. Hence ξ0 = ζ and therefore, due to the weak convergence (n)

(0)

ℙ ,ℙ

(ξn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ζ (t), t ∈ [0, T]), n → ∞, the theorem is proved. Remark 4.28. Convergence in probability in condition (i) of Theorem 4.27 cannot be replaced by weak convergence. Indeed, if we put d = 1, X = W, a Wiener process,

4.3 Stochastic integrals and financial investment | 299

ξn ≡ −1, and ξ0 ≡ 1, then t

d

t

{∫ ξn (t)dX(t), t ∈ [0, T]} = {∫ ξ0 (t)dX(t), t ∈ [0, T]}, 0

0

where the equality is in distribution, but there is no convergence of ξn to ξ0 .

4.3 Application of functional limit theorems for stochastic integrals to financial investment This section discusses the convergence of capitals of self-financing strategies constructed on the d-dimensional financial market with continuous time, provided that the strategies themselves converge. In proving the basic theorems, the results of the previous section concerning weak convergence of stochastic integrals with respect to semimartingales are used. The “inverse” problem is also considered, namely, the convergence of strategies that minimize quadratic and local quadratic risk is studied under the condition of convergence of the sequence of corresponding contingent claims.

4.3.1 Weak convergence of capitals of self-financing strategies Let, as usual, (Ω(n) , ℱ (n) , 𝔽(n) = {ℱt(n) , t ≥ 0}, ℙ(n) ), n ≥ 0, be a sequence of filtered probability spaces. Also, let Xn = {Xn (t), ℱt(n) , t ≥ 0, n ≥ 0} = {(Xn1 (t), . . . , Xnd (t)), ℱt(n) , t ≥ 0, n ≥ 0} be a sequence of d-dimensional semimartingales, whose components allow the following decompositions: Xnj (t) = Xnj (0) + Mnj (t) + Ajn (t),

1 ≤ j ≤ d,

(4.29)

where {Mnj (t), ℱt(n) , t ≥ 0, n ≥ 0} is a sequence of square-integrable martingales, Mnj (0) = 0, {Ajn (t), t ≥ 0, n ∈ ℤ+ } is a sequence of processes of bounded variations, and Ajn (0) = 0. Let also μjn (t) := ⟨Mnj ⟩(t) be quadratic characteristics of respective martingales. Also, we shall fix some countable set 𝒯 and for any T > 0 we create 𝒯T according to Definition 4.16. Theorem 4.29. Let {Xn (t), ℱt(n) , t ≥ 0, n ≥ 0} be a sequence of d-dimensional semimartingales, whose components allow decomposition (4.29) and for any T > 0 fulfill the following conditions: ℙ(n) , ℙ(0) , fdd

(i) (ξn (t), Xn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), X0 (t), t ∈ 𝒯T ) as n → ∞, and

300 | 4 Convergence of stochastic integrals in application to financial markets for all 1 ≤ j ≤ d and ε > 0: (ii) limC→∞ lim supn→∞ ℙ(n) {sup0≤t≤T |ξnj (t)| ≥ C} = 0, (iii) limC→∞ lim supn→∞ ℙ(n) {sup0≤t≤T |Xnj (t)| ≥ C} = 0, (iv) limδ→0 lim supn→∞ supσ∈𝕋T (ℱ n ) ℙ(n) (supt≤δ |ξnj (t + σ) − ξnj (σ)| ≥ ε) = 0, (v) limδ→0 lim supn→∞ supσ∈𝕋T (ℱ n ) ℙ(n) (supt≤δ |Xnj (t + σ) − Xnj (σ)| ≥ ε) = 0, or (vi) strategies {ξnj (t), ℱt(n) , t ≥ 0, n ≥ 0, 1 ≤ j ≤ d} are self-financing and satisfy, together with Xn , Mn , and An , conditions (i)–(ix) of Theorem 4.24 and Vn (0) → V(0); then for any T > 0 ℙ(n) , ℙ(0)

(Vn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (V0 (t), t ∈ [0, T]),

n → ∞.

Proof. 1. First, let conditions (i)–(vi) hold. Condition (i) implies a weak convergence of finite-dimensional distributions: ℙ(n) , ℙ(0) , fdd

(Vn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (V0 (t), t ∈ 𝒯T ),

n → ∞.

(4.30)

Now our aim is to prove the tightness of the sequence of processes Vn using conditions (i)–(iv). In this connection, note that for any ε > 0 lim lim sup

δ→0

n→∞

sup

σ∈𝕋T (ℱ n )

󵄨 󵄨 ℙ(n) (sup󵄨󵄨󵄨Vn (t + σ) − Vn (σ)󵄨󵄨󵄨 ≥ η) t≤δ

󵄨󵄨 󵄨󵄨 ℙ(n) (sup󵄨󵄨󵄨ξn0 (t + σ) − ξn0 (σ) δ→0 n→∞ σ∈𝕋T (ℱ n ) t≤δ 󵄨󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 + ∑ ξnj (t + σ)Xnj (t + σ) − ∑ ξnj (σ)Xnj (σ)󵄨󵄨󵄨 ≥ ε) 󵄨󵄨󵄨 j=1 j=1

= lim lim sup

≤ lim lim sup δ→0

n→∞

sup

sup

σ∈𝕋T (ℱ n )

󵄨 󵄨 ℙ(n) (sup󵄨󵄨󵄨ξn0 (t + σ) − ξn0 (σ)󵄨󵄨󵄨 ≥ t≤δ

ε ) 2d + 1

d

󵄨 󵄨 + ∑ lim (lim sup ℙ(n) ( sup 󵄨󵄨󵄨ξnj (t)󵄨󵄨󵄨 ≥ C) j=1

C→∞

n→∞

+ C lim lim sup δ→0

n→∞

0≤t≤T

sup

σ∈𝕋T (ℱ n )

󵄨 󵄨 ℙ(n) (sup󵄨󵄨󵄨Xnj (t + σ) − Xnj (σ)󵄨󵄨󵄨 ≥ t≤δ

ε ) 2d + 1

󵄨 󵄨 + lim sup ℙ(n) ( sup 󵄨󵄨󵄨Xnj (t)󵄨󵄨󵄨 ≥ C) n→∞

0≤t≤T

+ C lim lim sup δ→0

= 0.

n→∞

sup

σ∈𝕋T

(ℱ n )

󵄨 󵄨 ℙ(n) (sup󵄨󵄨󵄨ξnj (t + σ) − ξnj (σ)󵄨󵄨󵄨 ≥ t≤δ

ε )) 2d + 1 (4.31)

4.3 Stochastic integrals and financial investment | 301

Then it follows from Theorem A.101 that the sequence Vn is weakly relatively compact in the Skorokhod topology. From the weak convergence of the finite-dimensional distributions of the sequence of capitals and Theorem A.95 we obtain their convergence in the Skorokhod topology. 2. Now let condition (vi) hold. Since the strategies {ξnj (t), ℱt(n) , t ≥ 0, n ≥ 0, 1 ≤ j ≤ t

d} are self-financing, respective capital has the form Vn (t) = Vn (0) + ∫0 ξn (s) ⋅ dXn (s), where C is a constant. The application of Theorem 4.24 completes the proof.

4.3.2 Convergence of risk minimizing strategies In this section, we examine the convergence of a particular class of strategies, provided that the prices of contingent claims generating by these strategies converge. In order to prove the main results, we apply Theorem 4.27. Let contingent claim ℍ be a random variable on the financial market constructed in Section 4.1 and managed by the semimartingale price process of the risky assets defined by (4.1). We consider ℍ at time T, so that ℍ ∈ ℒ2 (Ω, ℱT , ℙ). Suppose also that we have a sequence of contingent claims {ℍn , n ≥ 0}, which allow the decomposition T

ℍn = H̆ n0 + ∫ ξn (s) ⋅ dX(s),

ℙ-a. s.,

(4.32)

0

where H̆ n0 are some constants and ξn are 𝔽(n) -predictable processes, satisfying conditions (K1)–(K4). Then, according to Proposition 4.11, the strategy {ξnj , 1 ≤ j ≤ d} and t ξ 0 = Vn − ξn ⋅ X, where Vn (t) := H̆ 0 + ∫ ξn (s) ⋅ dX(s), 0 ≤ t ≤ T, is ℍ-admissible and n

n

0

self-financing and minimizes the mean squared risk.

Lemma 4.30. If for any 𝔽-predictable process ζ ∈ ℝd and 𝔽-adapted process ζ 0 from the fact that ∑dj=0 ζ j (t)X j (t) = 0 it follows that (ζ 0 (t), ζ 1 (t), . . . , ζ d (t)) = 0, ℙ-a. s., for all

t ∈ [0, T], then the process of discounted risky asset prices X = {X j , 1 ≤ j ≤ d} has linearly independent components. t

Proof. Let ∑dj=1 ∫0 ξ j (s)dX j (s) = 0. Process ξ 0 could be defined in a way such that the strategy (ξ 0 , ξ ) becomes self-financing. This is equivalent to the constant value of the cost process from (4.2): C(t) = C, ℙ-a. s., where C is some constant. To achieve this property of the cost process, it is enough to put 0

d

t

d

d

j=1

j=1

ξ (t) = C + ∑ ∫ ξ j (s)dX j (s) − ∑ ξ j (t)X j (t) = C − ∑ ξ j (t)X j (t). j=1 0

302 | 4 Convergence of stochastic integrals in application to financial markets Then we immediately deduce that d

(ξ 0 (t) − C) + ∑ ξ j (t)X j (t) = 0.

(4.33)

j=1

Recalling that X 0 (t) = 1, from (4.33) we obtain d

(ξ 0 (t) − C)X 0 (t) + ∑ ξ j (t)X j (t) = 0. j=1

According to the condition of our lemma, we obtain that ξ j (t) = 0 for all 1 ≤ j ≤ d, whence the proof follows. Theorem 4.31. Let the discounted price process of the risky assets have the form (4.1), and let the following conditions be satisfied: (i) contingent claims {ℍn , n ≥ 0} allow decomposition (4.32), (ii) for all 0 ≤ t ≤ T t

t

→ ∫ ξ0 (s) ⋅ dX(s) ∫ ξn (s) ⋅ dX(s) 󳨀 ℙ

0

0

for n → ∞, (iii) the discounted price process of the risky assets {X j , 1 ≤ j ≤ d} has linearly independent components, (iv) the sequence of measures corresponding to the processes {ξn (t), t ∈ [0, T], n ≥ 0} is weakly relatively compact in the Skorokhod topology. Then for each 0 ≤ j ≤ d there is a weak convergence of minimizing risk strategies: ℙ(n) , ℙ(0)

j

(ξnj (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), t ∈ [0, T]),

n → ∞.

Proof. Recall that condition (i) and Proposition 4.11 imply that the strategies {(Vn (t) − ξn (t) ⋅ X, ξn1 (t), ξn2 (t), . . . , ξnd (t)), n ≥ 0}, t

where Vn (t) := H̆ n0 + ∫0 ξn (s) ⋅ dX(s), are self-financing and minimize the mean squared risk. Furthermore, conditions (ii)–(iv), together with Theorem 4.27, supply weak convergence: ℙ(n) , ℙ(0)

j

(ξnj (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), t ∈ [0, T]),

n → ∞ for any 1 ≤ j ≤ d.

(4.34)

Weak convergence for ξn0 (t) = Vn (t) − ξn (t) ⋅ X(t) follows from (4.34) and condition (ii) of this theorem.

4.3 Stochastic integrals and financial investment | 303

Let us now consider the martingale case, that is, we assume that the process X is a square-integrable martingale with respect to measure ℙ. Then any squareintegrable contingent claim ℍ admits the Kunita–Watanabe decomposition (see Proposition 4.12): T

ℍ = H̆ 0 + ∫ ξℍ (t) ⋅ dX(t) + Lℍ (T),

ℙ-a. s.,

(4.35)

0

where ξℍ satisfies conditions (K1)–(K4), process Lℍ is a square-integrable martingale, orthogonal to X, with Lℍ (0) = 0, and H̆ 0 is a constant. This decomposition is unique in the sense that in the case of equality, T

T

̂ (t) ⋅ dX(t)) + L̂ (T), ℍ = H̆ 0 + ∫ ξℍ (t) ⋅ dX(t) + Lℍ (T) = Ĥ 0 + ∫ ξℍ ℍ 0

0

T T ̂ we have H̆ 0 = Ĥ 0 , ℙ-a. s., ∫0 ξℍ (t) ⋅ dX(t) = ∫0 ξℍ (t) ⋅ dX(t), ℙ-a. s., and Lℍ (T) = L̂ ℍ (T), ℙ-a. s. Similarly to (4.32), decomposition (4.35) defines strategies that minimize R(ξ 0 ,ξ ) (t), namely, j

ξ j = ξℍ ,

1 ≤ j ≤ d, t

ξ 0 = V̆ − ξ ⋅ X,

̆ := H̆ 0 + ∫ ξℍ (s) ⋅ dX(s) + Lℍ (t), V(t)

0 ≤ t ≤ T.

0

Obviously, such strategies are mean self-financing. Consider now a sequence of square-integrable contingent claims {ℍn , n ≥ 0}, which allow the following decomposition: T

ℍn = H̆ n + ∫ ξn (t) ⋅ dX(t) + Ln (T),

ℙ-a. s.,

(4.36)

0

whose components have the same properties as the components of the decomposition (4.35) of ℍ. Then the following statements about the convergence of integrands in the components of such decomposition (i. e., convergence of strategies that minimize the mean squared risk) hold. Theorem 4.32. Suppose the process X of discounted risky asset prices is a martingale with respect to the measure ℙ and the following conditions are satisfied: (i) 𝔼(ℍn − ℍ0 )2 → 0, for n → ∞, (ii) the processes {X j , 1 ≤ j ≤ d} of discounted risky asset prices are pairwise orthogonal martingales.

304 | 4 Convergence of stochastic integrals in application to financial markets Then for each 1 ≤ j ≤ d and 0 ≤ t ≤ T there is a following convergence of strategies that minimize the mean squared risk: t

2

j

𝔼(∫(ξnj (s) − ξ0 (s)) d⟨X j ⟩(s)) → 0

as n → ∞.

0

Proof. Transform the value 𝔼(ℍn − ℍ0 )2 as follows: T

2

2

T

𝔼(ℍn − ℍ0 ) = 𝔼((H̆ n − H̆ 0 ) + (∫ ξn (t) ⋅ dX(t) − ∫ ξ0 (t) ⋅ dX(t)) + (Ln (T) − L0 (T))) 0

0

T

2

2

= (H̆ n − H̆ 0 )2 + 𝔼(∫(ξn (t) − ξ0 (t)) ⋅ dX(t)) + 𝔼(Ln (T) − L0 (T)) 0

+ I1,n + I2,n + I3,n , where T

I1,n = 𝔼((H̆ n − H̆ 0 ) ∫(ξn (t) − ξ0 (t)) ⋅ dX(t)) 0 T

= (H̆ n − H̆ 0 )𝔼(∫(ξn (t) − ξ0 (t)) ⋅ dX(t)) = 0, 0

for any n ≥ 1, because the expression under the sign of expectation is a martingale starting from zero. Furthermore, I2,n = 𝔼((H̆ n − H̆ 0 )(Ln (T) − L0 (T))) = (H̆ n − H̆ 0 )𝔼(Ln (T) − L0 (T)) = 0,

for any n ≥ 1,

since, again, the expression under the sign of expectation is a martingale starting from zero. Finally, T

I3,n = 𝔼((∫(ξn (t) − ξ0 (t)) ⋅ dX(t))(Ln (T) − L0 (T))) = 0,

for any n ≥ 1,

0

due to the orthogonality of the martingale X to Ln , n ≥ 0. Thus, from condition (i) we obtain the following relations: 𝔼(H̆ n − H̆ 0 )2 → 0,

2

𝔼(Ln (T) − L0 (T)) → 0,

4.3 Stochastic integrals and financial investment | 305

T

and 𝔼(∫0 (ξn (t) − ξ0 (t)) ⋅ dX(t))2 → 0, for n → ∞. Furthermore, 2

T

𝔼(∫(ξn (t) − ξ0 (t)) ⋅ dX(t)) 0

T

d

=

∑ 𝔼(∫(ξnj (t) j=1 0

−

2 j j ξ0 (t))dX (t))

T

+ ∑

1≤i=j≤d ̸

d

T

j=1

0

𝔼(∫(ξni (t)

−

0

T i i ξ0 (t))dX (t) ∫(ξnj (t) 0

2

j

j

− ξ0 (t))dX j (t))

= ∑ 𝔼(∫(ξnj (t) − ξ0 (t))dX j (t)) , since the processes {X j , 1 ≤ j ≤ d} are pairwise orthogonal martingales. Thus, for each 1 ≤ j ≤ d the following convergence holds: T

𝔼(∫(ξnj (t) 0

−

2 j j ξ0 (t))dX (t))

→ 0,

as n → ∞. As a result, for each 1 ≤ j ≤ d and 0 ≤ t ≤ T t

𝔼(∫(ξnj (s) 0

−

2 j ξ0 (s)) d⟨X j ⟩(s))

t

=

𝔼(∫(ξnj (s) 0

=

−

2 j j ξ0 (s))dX (s))

T

𝔼(𝔼(∫(ξnj (s) 0

−

j ξ0 (s))dX j (s)

2

󵄨󵄨 󵄨󵄨 ℱt )) 󵄨󵄨 2

T

󵄨󵄨 j ≤ 𝔼(𝔼((∫(ξnj (s) − ξ0 (s))dX j (s)) 󵄨󵄨󵄨 ℱt )) 󵄨 T

=

0

𝔼(∫(ξnj (s) 0

−

2 j j ξ0 (s))dX (s))

→ 0,

as n → ∞,

and the theorem is proved. Theorem 4.33. Let the process X of discounted risky asset prices be a martingale with respect to the measure ℙ and let the following conditions be satisfied for any T > 0: (i) 𝔼(ℍn − ℍ0 )2 → 0 as n → ∞, (ii) the components {X j , 1 ≤ j ≤ d} of the process of discounted risky asset prices are linearly independent, (iii) the sequence of measures corresponding to processes {ξn (t), t ∈ [0, T], n ≥ 0} is weakly relatively compact in the Skorokhod topology.

306 | 4 Convergence of stochastic integrals in application to financial markets Then for each 0 ≤ j ≤ d there is a weak convergence of strategies that minimize the mean squared risk: ℙ(n) , ℙ(0)

j

(ξnj (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), t ∈ [0, T]),

n → ∞.

Proof. The fact that 𝔼(ℍn −ℍ0 )2 → 0 as n → ∞ and the orthogonality of the decomposition components (4.36), similarly to the previous theorem, imply that (H̆ n − H̆ 0 )2 → 0, 𝔼(Ln (T) − L0 (T))2 → 0, and 2

T

𝔼(∫(ξn (t) − ξ0 (t)) ⋅ dX(t)) → 0, 0

as n → ∞. In addition, the fact that stochastic integrals as well as the processes Ln , n ≥ 0, are martingales, implies that for all 0 ≤ t ≤ T 2

t

𝔼(∫(ξn (s) − ξ0 (s)) ⋅ dX(s)) → 0,

2

𝔼(Ln (t) − L0 (t)) → 0,

as n → ∞,

0

and, consequently, t

t

∫ ξn (s) ⋅ dX(s) → ∫ ξ0 (s) ⋅ dX(s), ℙ

0

Ln (t) → L0 (t), ℙ

as n → ∞.

(4.37)

0

Conditions (i)–(ii), together with equality (4.37) and Theorem 4.27, imply weak convergence: ℙ(n) , ℙ(0)

j

(ξnj (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), t ∈ [0, T]),

n → ∞ for any 1 ≤ j ≤ d.

(4.38)

Weak convergence of ξn0 = Vn − ξn ⋅ X, where t

Vn (t) = H̆ n + ∫ ξn (s) ⋅ dX(s) + Ln (t), 0

follows from equations (4.37)–(4.38). The theorem is proved. Consider the general semimartingale case when the process of discounted risky asset prices equals X j (t) = X j (0) + Aj (t) + M j (t),

(4.39)

where M j is a square-integrable 𝔽-martingale with M j (0) = 0 and Aj is an 𝔽-predictable process of bounded variation with Aj (0) = 0.

4.4 Barrier options, stochastic drift and volatility | 307

Suppose now that the minimal martingale measure ℙ̂ exists and {ℍn , n ≥ 0} creates a sequence of contingent claims that allow the following decomposition (see Proposition 4.13): T

ℍn = H̆ n + ∫ ξn (t) ⋅ dX(t) + Ln (T),

ℙ-a. s.

(4.40)

0

Then the following statement concerning the convergence of integrands of the components of strategies minimizing the mean squared risk holds. Theorem 4.34. Assume the following conditions hold: (i) 𝔼ℙ̂ [ℍn − ℍ0 ]2 → 0 for n → ∞, (ii) the process {X j , 1 ≤ j ≤ d} of discounted risky asset prices has linearly independent components, (iii) the sequence of measures corresponding to processes {ξn , n ≥ 0} is weakly relatively compact in the Skorokhod topology. ℙ(n) , ℙ(0)

Then for each 1 ≤ j ≤ d we have the weak convergence (ξnj (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ j (ξ0 (t), t ∈ [0, T]), n → ∞. Proof. Since ℙ̂ is the minimal martingale measure, the components of the decomposition (4.40) have the following property. The process Ln is a martingale with respect to the measure ℙ,̂ orthogonal to each component of the vector process X (see Proposition 4.15). Thus, (4.40) is the Kunita–Watanabe decomposition with respect to the measure ℙ.̂ The application of Theorem 4.32 completes the proof. ℙ 2 Remark 4.35. Since 𝔼( ddℙ ) < ∞, condition (i) of Theorem 4.34 could be replaced by the following condition: 𝔼(ℍn − ℍ0 )4 → 0 as n → ∞. It is also evident that the conℙ̂ ditions 𝔼(ℍn − ℍ0 )2 → 0 when n → ∞ and ddℙ ≤ C, ℙ-a. s., for some C > 0, are also sufficient to satisfy condition (i) of Theorem 4.34. ̂

4.4 Limit behavior of capitals and barrier option prices in the Black–Scholes model with stochastic drift and volatility This subsection is devoted to a generalized model of the Black–Scholes market with stochastic drift and volatility. The convergence of the capitals of self-financing strategies in such a market is investigated under the condition of convergence of finite-dimensional distributions of the sequence of the strategies. The question of convergence of the price of a barrier option to the limit value is studied as well. Among the conditions that ensure such convergence is the weak convergence of finitedimensional distributions of model parameters (drift and volatility). Thus, the proof

308 | 4 Convergence of stochastic integrals in application to financial markets of the main results of the section is based on the theorems on the weak convergence of stochastic integrals on semimartingales from Section 4.2. 4.4.1 Description of the model Consider, for the technical simplicity, the unique for all n filtered probability space (Ω, ℱ , 𝔽 = {ℱt , t ∈ [0, T]}, ℙ) and the following processes of risky asset prices on it: Snj (t)

=

t j S0 exp{∫ μjn (s)ds 0

t

+ ∫ σnj (s)dW j (s)},

t

Sn0 (t) = exp{∫ rn (s)ds},

1 ≤ j ≤ d,

0

n ≥ 0,

t ∈ [0, T],

(4.41)

0

j

where W = (W 1 , W 2 , . . . , W d ) is a d-dimensional Wiener process, S0 are constants, and μn = μn (t), σn = σn (t), rn = rn (t) are 𝔽-adapted stochastic processes satisfying the following conditions: process σn is predictable and for all t ∈ [0, T] rn (t) ≥ 0,

σn (t) > 0

ℙ-a. s., and all integrals from equality (4.41) are correctly defined because we assume that T

T

∫ rn (s)ds < ∞,

󵄨 󵄨 ∫󵄨󵄨󵄨μn (s)󵄨󵄨󵄨ds < ∞,

0

T

0

∫ σn2 (s)ds < ∞

ℙ-a. s.

0

Assume also that the filtration 𝔽 is generated by the processes rn , μn , σn , W. The discounted process of prices of risky assets equals Xnj (t)

=

μ̂ jn (s) =

t t j j S0 exp{∫ μ̂ n (s)ds + ∫ σnj (s)dW j (s)}, 0 0 j μn (s) − rn (s), n ≥ 0.

1 ≤ j ≤ d, where (4.42)

4.4.2 Weak convergence of capitals in the generalized Black–Scholes model Recall that 𝒯 is some countable dense in ℝ+ set, and for any T > 0, 𝒯T = 𝒯 ∩ [0, T]. Theorem 4.36. Let the discounted processes of risky asset prices be given by formulas (4.42), and let the following conditions be satisfied: ℙ(n) , ℙ(0) , fdd

(i) for any T > 0, (ξn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (ξ0 (t), t ∈ 𝒯T ), and

4.4 Barrier options, stochastic drift and volatility | 309

for all 1 ≤ j ≤ d: T

T

j j (ii) ∫0 |μ̂ jn (s) − μ̂ 0 (s)|ds → 0, 𝔼 ∫0 (σnj (s) − σ0 (s))2 ds → 0, n → ∞, ℙ

δ+τ

|μ̂ in (s)|ds ≥ ε) = 0, δ+τ i limδ→0 lim supn→∞ supτ∈𝕋T (ℱ ) ℙ(∫τ (σn (s))2 ds ≥ ε) = 0, limδ→0 lim supn→∞ supτ∈𝕋T (ℱ ) ℙ(supt≤δ |ξni (t + τ) − ξni (τ)| limC→∞ lim supn→∞ ℙ{sup0≤t≤T |ξni (t)| ≥ C} = 0.

(iii) limδ→0 lim supn→∞ supτ∈𝕋T (ℱ ) ℙ(∫τ

(iv)

≥ ε) = 0 for any ε > 0,

Then the finite-dimensional distributions of capitals converge for any T > 0, namely, ℙ(n) , ℙ(0) , fdd

(Vn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (V0 (t), t ∈ 𝒯T ),

n → ∞.

Proof. It follows from condition (ii) that for any 1 ≤ j ≤ d t

t

j

∫ μ̂ jn (s)ds → ∫ μ̂ 0 (s)ds, ℙ

0

0

t

ℙ ∫ σnj (s)dW j (s) → 0

t

j

∫ σ0 (s)dW j (s),

n → ∞,

t ∈ 𝒯T .

(4.43)

0

Furthermore, it follows from conditions (i) and (ii) and equality (4.43) that t

t

(ξn (t), {∫ μ̂ jn (s)ds, ∫ σnj (s)dW j (s)} 0

0

(n)

ℙ ,ℙ

(0)

, fdd

󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒

1≤j≤d

, t ∈ 𝒯T )

t

t j j (ξ0 (t), {∫ μ̂ 0 (s)ds, ∫ σ0 (s)dW j (s)} ,t 1≤j≤d 0 0

∈ 𝒯T ).

In addition, conditions (i)–(iv) provide implementation of the conditions of Theoℙ(n) , ℙ(0) , fdd

rem 4.24; therefore (Vn (t), t ∈ 𝒯T ) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (V0 (t), t ∈ 𝒯T ). Now we are in a position to establish the tightness of the measures corresponding to Vn : 󵄨 󵄨 lim lim sup sup ℙ(sup󵄨󵄨󵄨Vn (t + τ) − Vn (τ)󵄨󵄨󵄨 ≥ ε) n→∞

δ→0

τ∈𝕋T (ℱ )

t≤δ

󵄨󵄨 󵄨󵄨 = lim lim sup sup ℙ(sup󵄨󵄨󵄨ξn0 (t + τ) − ξn0 (τ) δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ 󵄨󵄨󵄨 󵄨󵄨 d d 󵄨󵄨 + ∑ ξnj (t + τ)Xnj (t + τ) − ∑ ξnj (τ)Xnj (τ)󵄨󵄨󵄨 ≥ ε) 󵄨󵄨 j=1 j=1 󵄨

ε 󵄨 󵄨 ≤ lim lim sup sup ℙ(sup󵄨󵄨󵄨ξn0 (t + τ) − ξn0 (τ)󵄨󵄨󵄨 ≥ ) 2d +1 δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ

310 | 4 Convergence of stochastic integrals in application to financial markets d

󵄨 󵄨 + ∑ lim (lim sup ℙ( sup 󵄨󵄨󵄨ξnj (t)󵄨󵄨󵄨 ≥ C) j=1

C→∞

n→∞

0≤t≤T

ε 󵄨 󵄨 ) + C lim lim sup sup ℙ(sup󵄨󵄨󵄨Xnj (t + τ) − Xnj (τ)󵄨󵄨󵄨 ≥ 2d +1 δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ 󵄨 󵄨 + lim sup ℙ( sup 󵄨󵄨󵄨Xnj (t)󵄨󵄨󵄨 ≥ C) n→∞

0≤t≤T

ε 󵄨 󵄨 + C lim lim sup sup ℙ(sup󵄨󵄨󵄨ξnj (t + τ) − ξnj (τ)󵄨󵄨󵄨 ≥ )) 2d +1 δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ d ε 󵄨 󵄨 = ∑ lim (C lim lim sup sup ℙ(sup󵄨󵄨󵄨Xnj (t + τ) − Xnj (τ)󵄨󵄨󵄨 ≥ ) C→∞ 2d +1 δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ j=1

󵄨 󵄨 + lim sup ℙ( sup 󵄨󵄨󵄨Xnj (t)󵄨󵄨󵄨 ≥ C)). n→∞

0≤t≤T

Furthermore, for all 1 ≤ j ≤ d, it follows from condition (ii) that 󵄨 󵄨 lim lim sup ℙ( sup 󵄨󵄨󵄨Xnj (t)󵄨󵄨󵄨 ≥ C)

C→∞

n→∞

0≤t≤T

j

t

t

0

0

= lim lim sup ℙ( sup S0 exp{∫ μ̂ jn (s)ds + ∫ σnj (s)dW j (s)} ≥ C) C→∞

n→∞

0≤t≤T

t 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 j 󵄨󵄨 C 󵄨 = lim lim sup ℙ( sup 󵄨󵄨∫ μ̂ n (s)ds + ∫ σnj (s)dW j (s)󵄨󵄨󵄨 ≥ log( j )) 󵄨 󵄨 C→∞ n→∞ 󵄨󵄨 0≤t≤T 󵄨󵄨 S0 0 0

󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 C ≤ lim lim sup(ℙ( sup 󵄨󵄨󵄨∫ μ̂ jn (s)ds󵄨󵄨󵄨 ≥ log( j )) 󵄨󵄨 2 C→∞ n→∞ 0≤t≤T 󵄨󵄨󵄨 S0 󵄨 0 󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨󵄨 1 C + ℙ( sup 󵄨󵄨󵄨∫ σnj (s)dW j (s)󵄨󵄨󵄨 ≥ log( j ))) 󵄨 󵄨 󵄨󵄨 2 0≤t≤T 󵄨󵄨 S0 0 T

t

0

0

󵄨 󵄨 ̂ + lim lim sup 1 ∫ 𝔼(σ j (s))2 ds = 0 ≤ lim lim sup ℙ(∫󵄨󵄨󵄨μ̂ jn (s)󵄨󵄨󵄨ds ≥ C) n ̃ ̃ ̂ n→∞ n→∞ C C→∞ C→∞ and 󵄨 󵄨 lim lim sup sup ℙ(sup󵄨󵄨󵄨Xnj (t + τ) − Xnj (τ)󵄨󵄨󵄨 ≥ ε) n→∞

δ→0

τ∈𝕋T (ℱ )

t≤δ

τ τ 󵄨󵄨 󵄨󵄨 j = lim lim sup sup ℙ(sup󵄨󵄨󵄨S0 exp{∫ μ̂ jn (s)ds + ∫ σnj (s)dW j (s)} δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ 󵄨󵄨󵄨 0 0

−

t+τ j S0 exp{ ∫ μ̂ jn (s)ds 0

t+τ

󵄨󵄨 󵄨󵄨 + ∫ σnj (s)dW j (s)}󵄨󵄨󵄨 ≥ ε) 󵄨󵄨 󵄨 0

(4.44)

4.4 Barrier options, stochastic drift and volatility | 311

≤

t j lim (lim sup ℙ( sup S0 exp{∫ μ̂ jn (s)ds C→∞ n→∞ 0≤t≤T 0

t

+ ∫ σnj (s)dW j (s)} ≥ C) 0

t+τ t+τ 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ε + lim lim sup sup ℙ(sup󵄨󵄨󵄨exp{ ∫ μ̂ jn (s)ds + ∫ σnj (s)dW j (s)} − 1󵄨󵄨󵄨 ≥ )) 󵄨󵄨 C δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ 󵄨󵄨󵄨 󵄨 τ τ 󵄨󵄨 t+τ 󵄨󵄨 = lim (lim lim sup sup ℙ(sup󵄨󵄨󵄨 ∫ μ̂ jn (s)ds C→∞ δ→0 n→∞ τ∈𝕋 (ℱ ) t≤δ 󵄨󵄨󵄨 T τ t+τ t+τ 󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨󵄨 t+τ 󵄨󵄨 ε 󵄨󵄨 󵄨󵄨 j j j + ∫ σn (s)dW (s)󵄨󵄨 exp{󵄨󵄨 ∫ μ̂ n (s)ds + ∫ σnj (s)dW j (s)󵄨󵄨󵄨} ≥ )), (4.45) 󵄨󵄨 󵄨󵄨 󵄨󵄨 C 󵄨 󵄨τ 󵄨 τ τ

where we took into account (4.44) and equated to zero the term t t 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 j lim lim sup ℙ( sup 󵄨󵄨󵄨S0 exp{∫ μ̂ jn (s)ds + ∫ σnj (s)dW j (s)}󵄨󵄨󵄨 ≥ C) 󵄨󵄨 C→∞ n→∞ 0≤t≤T 󵄨󵄨󵄨 󵄨 0 0

in the middle part of (4.45). Now, again considering (4.44), it is enough to prove equality,

t+τ 󵄨󵄨 t+τ 󵄨󵄨 󵄨󵄨 󵄨󵄨 Ψ := lim lim sup sup ℙ(sup󵄨󵄨󵄨 ∫ μ̂ jn (s)ds + ∫ σnj (s)dW j (s)󵄨󵄨󵄨 ≥ ε) = 0, 󵄨 󵄨󵄨 δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ 󵄨󵄨 󵄨 τ τ

for all ε > 0. But, due to condition (iii) and the Lenglart inequality (Theorem A.33) we have

󵄨󵄨 t+τ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ε Ψ ≤ lim lim sup sup ℙ(sup󵄨󵄨󵄨 ∫ μ̂ jn (s)ds󵄨󵄨󵄨 ≥ ) 󵄨 󵄨󵄨 2 δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ 󵄨󵄨 󵄨 τ 󵄨󵄨 t+τ 󵄨󵄨 󵄨󵄨 󵄨󵄨 ε + lim lim sup sup ℙ(sup󵄨󵄨󵄨 ∫ σnj (s)dW j (s)󵄨󵄨󵄨 ≥ ) 󵄨󵄨 2 δ→0 n→∞ τ∈𝕋T (ℱ ) t≤δ 󵄨󵄨󵄨 󵄨 τ = 0.

Then it follows from Theorem A.101 that the sequence Vn is weakly relatively com-

pact in the Skorokhod topology. From the weak convergence of finite-dimensional distributions of the sequence of stochastic integrals and Theorem A.95 we obtain the

weak convergence in the Skorokhod topology (and hence in the uniform topology).

312 | 4 Convergence of stochastic integrals in application to financial markets 4.4.3 Weak convergence of European barrier option prices in the generalized Black–Scholes model Let us now examine the asymptotic behavior of the fair prices of barrier options written on the asset prices given by equations (4.41) and (4.42), but in the one-dimensional case, i. e., when d = 1. Consider the European “up-and-out” call barrier option; see Section 3.2 for the main definitions. For each market in the abovementioned family, the payment on such contingent claim at time T is equal to (Sn (T) − Kn ) 1max0≤t≤T Sn (t) Sn (0) is the barrier. Contrary to initial values H̆ n of contingent claims (see, e. g., formula (4.36)), the barriers are denoted simply by Hn . Consider the measure ℙ(n,∗) for which the Radon–Nikodym derivative has the form t

t

0

0

dℙ(n,∗) 󵄨󵄨󵄨󵄨 1 2 󵄨 = exp{∫ ζn (s)dW(s) − ∫ ζn (s)ds}, dℙ 󵄨󵄨󵄨ℱt 2 μ (s)−r (s)

where ζn (s) = − n σ (s)n − 21 σn (s). n If Novikov’s condition (see Theorem A.70) t

1 𝔼 exp{ ∫ ζn2 (s)ds} < ∞ 2

(4.46)

0

is met, then the probability measure ℙ(n,∗) is the unique martingale measure for the discounted price process of the risky asset. In this case, the fair price of any contingent claim is calculated as an expectation with respect to this measure. Thus, the fair price of the European “up-and-out” call barrier option equals πnbarr

=

T ∗ 𝔼n (exp{− ∫ rn (t)dt}(Sn (T) 0

− Kn ) 1max0≤t≤T Sn (t) 0, X (n,C) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ X C as ℚC , ℚ

n → ∞ and X C 󳨐󳨐󳨐󳨐⇒ X as C → ∞. Suppose furthermore that for any ϵ > 0 󵄨 󵄨 lim lim sup ℙ{ sup 󵄨󵄨󵄨X (n,C) (t) − X (n) (t)󵄨󵄨󵄨 ≥ ϵ} = 0.

C→∞

n→∞

0≤t≤T

ℚ(n) , ℚ

Then X (n) 󳨐󳨐󳨐󳨐󳨐⇒ X, n → ∞. Definition A.88. The family {ℙα , α ∈ 𝒜} of probability measures defined on (𝕊, ℬ(𝕊)) is tight if for any ε ∈ (0, 1) there exists a compact set K such that ℙα (K) > 1 − ε for any α ∈ 𝒜. Remark A.89. We say that a family of stochastic processes {Xα , α ∈ 𝒜} is tight if the respective family of probability measures is tight. Definition A.90. The family {ℙα , α ∈ 𝒜} of probability measures defined on (𝕊, ℬ(𝕊)) is weakly relatively compact if any subsequence {ℙαn } contains a weakly convergent subsubsequence {ℙαn(k) } (that converges, generally speaking, not obligatorily to the element of this family).

342 | A Essentials of calculus, probability, and stochastic processes Theorem A.91 (Prokhorov’s theorem). If the family of probability measures defined on some metric space (𝕊, ρ) is tight, then it is weakly relatively compact. If a metric space (𝕊, ρ) is separable and complete, then the weakly relatively compact family of probability measures is tight. Remark A.92. Note that both spaces, C([0, T]) and D([0, T]), are separable and complete in the uniform topology and Skorokhod topology, respectively. Therefore, in these spaces tightness and weak relative compactness of the families of probability measures are equivalent. Theorem A.93 (see [18]). Let h: 𝕊 → 𝕊1 be a measurable mapping from the metric space 𝕊 to the space 𝕊1 , stochastic processes Xn and X admit their values in 𝕊. Let Dh be the ℙ(n) , ℙ, d

ℙ(n) , ℙ, d

set of its discontinuities. If Xn 󳨐󳨐󳨐󳨐󳨐󳨐⇒ X and ℙ{X ∈ Dh } = 0, then h(Xn ) 󳨐󳨐󳨐󳨐󳨐󳨐⇒ h(X) as n → ∞. Notation A.94. Weak convergence of probability measures in the Skorokhod space D(𝕋) (see Definition A.12), that is, weak convergence of stochastic processes that are considered as the stochastic elements in D(𝕋), supplied with the Skorokhod metrics, ℙ(n) , ℙ, U

ℙ(n) , ℙ

will be denoted by 󳨐󳨐󳨐󳨐⇒. We reserve the notation 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ for the weak convergence of the probability measures in the uniform topology, that is, the weak convergence of stochastic processes that are considered as the stochastic elements in C(𝕋), supplied with the uniform metrics. Now we formulate sufficient conditions of the weak convergence of probability measures corresponding to the stochastic processes with continuous (càdlàg) trajectories, or, in other words, sitting in C([a, b]) (D([a, b])). Theorem A.95 ([18]). Let 𝕆 ⊂ [a, b] be some countable dense set containing a and b. If the sequence {Xn , n ≥ 1} of stochastic processes with continuous (càdlàg) trajectories satisfies the condition ℙ(n) , ℙ, fdd

(Xn (t), t ∈ 𝕆) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ 𝕆),

n → ∞,

(A.23)

and the sequence of probability measures ℙ(n) corresponding to these processes is tight in C([a, b]) (D([a, b])), then ℙ(n) , ℙ, U

ℙ(n) , ℙ

(Xn (t), t ∈ [a, b]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (󳨐󳨐󳨐󳨐⇒)(X(t), t ∈ [a, b]),

n → ∞.

Let us introduce the following moduli of continuity in the space C([a, b]) equipped with uniform topology and in the space D([a, b]) equipped with Skorokhod topology. Let for x ∈ C([a, b]), 󵄨 󵄨 ΔC (x, a, b, δ) = sup 󵄨󵄨󵄨x(s) − x(t)󵄨󵄨󵄨, s,t∈[a,b] |s−t|≤δ

A.3 Essentials of stochastic processes | 343

and for x ∈ D([a, b]), ΔD (x, a, b, δ) =

sup

t,t ,t ∈[a,b] t−δ≤t ′ ≤t≤t ′′ ≤t+δ ′ ′′

󵄨 󵄨󵄨 󵄨 min(󵄨󵄨󵄨x(t ′ ) − x(t)󵄨󵄨󵄨, 󵄨󵄨󵄨x(t ′′ ) − x(t)󵄨󵄨󵄨)

󵄨 󵄨 󵄨 󵄨 + sup 󵄨󵄨󵄨x(t) − x(a)󵄨󵄨󵄨 + sup 󵄨󵄨󵄨x(t) − x(b)󵄨󵄨󵄨. b−δ≤t≤b

a≤t≤a+δ

(A.24)

Theorem A.96. Let 𝕆 ⊂ [a, b] be some countable dense set containing a and b. 1. Let the sequence {Xn , n ≥ 1} of stochastic processes with continuous trajectories satisfy the condition (A.23), and for any ε > 0, lim sup ℙ(n) (ΔC (Xn , a, b, δ) > ε) = 0.

δ→0

n

Then ℙ(n) , ℙ, U

(Xn (t), t ∈ [a, b]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [a, b]),

n → ∞.

2. Let the sequence {Xn , n ≥ 1} of stochastic processes with càdlàg trajectories satisfy the condition (A.23), and for any ε > 0, lim sup ℙ(n) (ΔD (Xn , a, b, δ) > ε) = 0.

δ→0

n

Then ℙ(n) , ℙ

(Xn (t), t ∈ [a, b]) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [a, b]),

n → ∞.

Theorem A.97 ([18, Theorem 13.5]). Assume that a sequence {Xn , n ≥ 1} of stochastic processes with càdlàg trajectories satisfies the conditions ℙ(n) , ℙ, fdd

(Xn (t), t ∈ [a, b]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [a, b]), ℙ

X(b) − X(b − δ) 󳨀 → 0,

as n → ∞,

as δ → 0,

and for a ≤ r ≤ s ≤ t ≤ b, n ≥ 1, and λ > 0, 1 2α 󵄨 󵄨 󵄨 󵄨 ℙ(󵄨󵄨󵄨Xn (s) − Xn (r)󵄨󵄨󵄨 ∧ 󵄨󵄨󵄨Xn (t) − Xn (s)󵄨󵄨󵄨 ≥ λ) ≤ 4β (F(t) − F(r)) , λ

(A.25)

where β ≥ 0, α > 21 , and F is a non-decreasing continuous function on [a, b]. Then ℙ(n) , ℙ

(Xn (t), t ∈ [a, b]) 󳨐󳨐󳨐󳨐⇒ (X(t), t ∈ [a, b]),

n → ∞.

Remark A.98. There is a more restrictive version of (A.25) involving moments, namely 2α 󵄨 󵄨2β 󵄨 󵄨2β 𝔼(󵄨󵄨󵄨Xn (s) − Xn (r)󵄨󵄨󵄨 󵄨󵄨󵄨Xn (t) − Xn (s)󵄨󵄨󵄨 ) ≤ (F(t) − F(r)) .

344 | A Essentials of calculus, probability, and stochastic processes Moreover, it is useful to formulate the following simple sufficient condition of weak convergence. Theorem A.99. Let (Ω(n) , ℱ (n) , ℙ(n) ) be a sequence of probability spaces and let X n = {Xtn , t ∈ [a, b]} be a sequence of càdlàg stochastic processes on these probability spaces. Denote by ℚn the measures corresponding to these stochastic processes. Assume the following conditions hold: (i) Finite-dimensional distributions of X n weakly converge to the corresponding distributions of X. (ii) There exist constants C > 0, α > 0, β > 0 such that for any n ≥ 1, a ≤ t1 ≤ t2 ≤ t3 ≤ b, 󵄨 󵄨α 󵄨 󵄨α 𝔼n 󵄨󵄨󵄨Xtn3 − Xtn2 󵄨󵄨󵄨 󵄨󵄨󵄨Xtn2 − Xtn1 󵄨󵄨󵄨 ≤ C|t2 − t1 |1+β . ℚn , ℚ

Then X n 󳨐󳨐󳨐󳨐⇒ X. Remark A.100. Condition (ii) ensures weak compactness of the sequence ℚn . If X n are the processes with the independent increments, the following condition is sufficient for (ii): There exist constants C > 0, α > 0, β > 0 such that for any n ≥ 1, a ≤ t1 ≤ t2 ≤ b, 1 󵄨 󵄨α 𝔼n 󵄨󵄨󵄨Xtn2 − Xtn1 󵄨󵄨󵄨 ≤ C|t2 − t1 | 2 +β .

Convergence in the uniform topology C(𝕋) or Skorokhod topology D(𝕋) implies the weak convergence of the functionals f : C(𝕋) → ℝ or f : D(𝕋) → ℝ, respectively, that are continuous in the respective topology. As a rule, supremum, infimum, and some integrals are considered as such functionals. Now we formulate conditions of weak convergence in terms of stopping times. Theorem A.101 ([110]). Let 𝕋 = ℝ+ . Also, let (Ω(n) , ℱ (n) , 𝔽(n) , ℙ(n) ) be the sequence of filtered probability spaces and let {Xn (t), n ≥ 1, t ≥ 0} be the sequence of càdlàg processes satisfying the following conditions: (i) For any T > 0, 󵄨 󵄨 lim lim sup ℙ(n) ( sup 󵄨󵄨󵄨Xn (t)󵄨󵄨󵄨 ≥ ε) = 0.

ε→0

n→∞

0≤t≤T

(ii) For any T > 0 and ε > 0, 󵄨 󵄨 lim lim sup sup ℙ(n) (sup󵄨󵄨󵄨Xn (τ + t) − Xn (τ)󵄨󵄨󵄨 ≥ ε) = 0,

δ→0

n→∞

τ∈𝒯T (ℱ n )

t≤δ

where 𝒯T (ℱ (n) ) consists of 𝔽(n) -stopping times τ such that τ ≤ T. Then for any T > 0 the sequence {ℙ(n) T , n ≥ 1} of probability measures, generated by the processes {Xn (t), n ≥ 1, t ∈ [0, T]}, is weakly relatively compact.

A.3 Essentials of stochastic processes | 345

Definition A.102. A sequence {Xn , n ≥ 1} of stochastic processes is called C-tight if it is tight and any of its limit points is a distribution of continuous process. Proposition A.103 (see [83, v.1, p. 488]). The following statements are equivalent: (i) The sequence of processes {Xn , n ≥ 1} is C-tight. (ii) The sequence of processes {Xn , n ≥ 1} is tight and for any T ≥ 0, ε > 0 the following equality holds: 󵄨 󵄨 lim ℙ(n) ( sup 󵄨󵄨󵄨ΔXn (t)󵄨󵄨󵄨 > ε) = 0.

n→∞

0≤t≤T

Proposition A.104 (see [83, v.1, p. 490]). Let {Yn , n ≥ 1} be a C-tight sequence of d-dimensional stochastic processes and let {Zn , n ≥ 1} be a sequence of tight (C-tight) d1 -dimensional stochastic processes. (i) If d = d1 , then the sequence {Yn + Zn , n ≥ 1} is tight (C-tight). (ii) The sequence {(Yn , Zn ), n ≥ 1} of d + d1 -dimensional stochastic processes is tight (C-tight). Theorem A.105 (see [102, p. 1039]). Let (Ω, ℱ , 𝔽, ℙ) be a filtered probability space and let {Xn (t) = (Xn11 (t), . . . , Xn1d (t), . . . , Xnmd (t)), n ≥ 1, t ≥ 0} be a sequence of 𝔽-adapted stochastic processes on (Ω, ℱ , 𝔽, ℙ) such that for any 1 ≤ i ≤ m, 1 ≤ j ≤ d the trajectories of components {Xnij (t), n ≥ 1} are càdlàg. Additionally, let {Y(t) = (Y 1 (t), . . . , Y d (t)), t ≥ 0} be a d-dimensional semimartingale such that all its components are semimartingales with respect to the filtration 𝔽. Denote by ℚ(n) (ℚ) the measures corresponding ̃ (n) (ℚ) ̃ the measures corresponding to the proto the processes (Xn , Y) ((X, Y)) and by ℚ cesses (Xn , Y, ∫ Xn dY) ((X, Y, ∫ XdY)). Then the following statements hold: (i) If for any T > 0 ℚ(n) , ℚ

((Xn (t), Y(t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐󳨐⇒ ((X(t), Y(t)), t ∈ [0, T]), then for any T > 0 t

((Xn (t), Y(t), ∫ Xn dY), t ∈ [0, T]) 0 ̃ (n)

̃ ℚ ,ℚ

t

󳨐󳨐󳨐󳨐󳨐⇒ ((X(t), Y(t), ∫ XdY), t ∈ [0, T]). 0 ℙ

(ii) If for any s, t > 0 (Xn (t), Y(s)) 󳨀 → (X(t), Y(s)), then for any r, s, t > 0 (Xn (t), Y(s), r

ℙ

r

→ (X(t), Y(s), ∫0 XdY). ∫0 Xn dY) 󳨀

The next result is a version of [83, Theorem 6.26, p. 384].

346 | A Essentials of calculus, probability, and stochastic processes Theorem A.106. Assume that the sequence Xn = {Xn (t), ℱtn , t ∈ [0, T]}, n ≥ 1, of stochasℙ(n) , ℙ

tic processes weakly converges to a continuous semimartingale X, Xn 󳨐󳨐󳨐󳨐⇒ X. Assume that {Xn , n ≥ 1} are stepwise and consequently admit quadratic variations [Xn ]. Then ℙ(n) , ℙ

([Xn ], Xn ) 󳨐󳨐󳨐󳨐⇒ ([X], X). Let us formulate a simplified version of Theorem 6.22 from [83], p. 383. Let us have a sequence of stepwise adapted stochastic processes: Hn (t) =

∑ 1≤k≤⌊ nt ⌋ T

Hnk 1 kT ≤t< (k+1)T n

n

and Xn (t) =

∑ 1≤k≤⌊ nt ⌋ T

Xnk 1 kT ≤t< (k+1)T . n

n

Create the sum of the form (Hn ⋅ Xn )(t) =

∑ 1≤k≤⌊ nt ⌋ T

Hnk ΔXnk

and denote by ℙ(n) and ℙ the measures corresponding to (Xn , Hn ) and (W, H), respec̃(n) and ℙ ̃ stand for the measures corresponding to (Xn , Hn , Hn ⋅ Xn ) and tively, while ℙ (W, H, H ⋅ W), respectively. Theorem A.107. Let the following conditions hold: (i) For any T > 0 the sequence ℙ(n) , ℙ

(Xn (t), Hn (t), t ∈ [0, T]) 󳨐󳨐󳨐󳨐⇒ (W(t), H(t), t ∈ [0, T]), where W is a Wiener process. (ii) There exists q ≥ 1 such that 󵄨 󵄨q sup sup 𝔼󵄨󵄨󵄨Xn (t)󵄨󵄨󵄨 < ∞. n≥1 t∈[0,T]

Then any T > 0 (Xn (t), Hn (t), (Hn ⋅ Xn )(t), t ∈ [0, T]) ̃ (n) , ℙ ̃ ℙ

󳨐󳨐󳨐󳨐⇒ (W(t), H(t), (H ⋅ W)(t)), t ∈ [0, T]),

n → ∞.

The following lemma contains the well-known result of A. V. Skorokhod, namely Skorokhod’s convergent subsequence principle (see [154, Chapter I, § 6]).

A.3 Essentials of stochastic processes | 347

Lemma A.108 (Skorokhod’s representation theorem). Let T > 0, let the sequence of stochastic processes ξ = {ξn (t), t ∈ [0, T], n ≥ 0} be defined on a corresponding sequence of probability spaces (Ω(n) , ℱ (n) , ℙ(n) ), n ≥ 0, and let ℙ(n) , ℙ(0)

ξn 󳨐󳨐󳨐󳨐󳨐󳨐⇒ ξ0 . ̃ ℱ ̃, ℙ), ̃ Then we can choose a subsequence nk → ∞, a unique probability space (Ω, ̃ ̃ and stochastic processes ξnk and ξ defined on this space such that the finite-dimensional distributions of the processes ξ̃ coincide with the finite-dimensional distributions of the processes ξnk and, moreover,

nk

̃

ℙ ξ̃nk (t) 󳨀→ ξ̃ (t),

as nk → +∞, for all t ∈ [0, T]. A.3.7 Weak convergence to Wiener process with a drift Consider the sequence (Ω(n) , ℱ (n) , ℚ(n) ) of probability spaces. Denote by 𝔼n and Varn the expectation and the variance with respect to ℚ(n) . Recall the following notations: a Gaussian random variable with mean a and variance σ 2 is denoted as 𝒩 (a, σ 2 ), weak ℚn , ℚ, d

convergence in distribution is denoted as 󳨐󳨐󳨐󳨐󳨐󳨐⇒, while weak convergence of finiteℚn , ℚ, fdd

dimensional distributions is denoted as 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒. Theorem A.109 ([120, Thm. A.10]). Let {ξnk , 1 ≤ k ≤ n, n ≥ 1} be the sequence of random variables in the scheme of series, defined on (Ω(n) , ℱ (n) , ℚ(n) ), and let the random variables {ξnk , 1 ≤ k ≤ n} be mutually independent in any series. Assume that there exist a ∈ ℝ and σ > 0 such that for any 0 ≤ t1 ≤ t2 ≤ 1 [nt2 ]

∑ k=[nt1 ]+1

ℚn , ℚ, d

ξnk 󳨐󳨐󳨐󳨐󳨐󳨐⇒ 𝒩 (a(t2 − t1 ), σ(t2 − t1 )).

Then the weak convergence of finite-dimensional distributions of the stochastic proξ k holds: cesses with continuous time ξn (t) = ∑[nt] k=1 n ℚn , ℚ, fdd

(ξn (t), t ∈ [0, 1]) 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ (at + σW(t), t ∈ [0, 1]),

(A.26)

where W = {W(t), t ∈ [0, 1]} is a Wiener process. A.3.8 Central limit theorems in the scheme of series First, we formulate one of the possible versions of the CLT in the scheme of series. A proof can be deduced from the general necessary and sufficient conditions of con-

348 | A Essentials of calculus, probability, and stochastic processes vergence to Gaussian distribution; see [136], Theorem 15, p. 92 and Theorem A.41 from [58]. Consider a sequence of probability spaces (Ω(n) , ℱ (n) , ℚ(n) ). Theorem A.110. Let for any n ≥ 1 the collection of random variables {ξnk , 1 ≤ k ≤ n} be defined on (Ω(n) , ℱ (n) , ℚ(n) ), be mutually independent, and satisfy the following conditions: (i) for any n ≥ 1 there exist constants cn such that cn → 0 as n → ∞ and |ξnk | ≤ cn , 1 ≤ k ≤ n, (ii) ∑nk=1 𝔼ℚ(n) (ξnk ) → α ∈ ℝ, (iii) ∑nk=1 Varℚ(n) (ξnk ) → σ 2 > 0. ℚ(n) , ℚ, d

Then ∑nk=1 ξnk 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ ξ , where the random variable ξ has 𝒩 (α, σ 2 )-distribution with respect to the measure ℚ. The next theorem gives a more general condition for convergence of a scheme of series to a normal distribution. For the proof see, e. g., [17, Theorem 27.2]. Theorem A.111 (Lindeberg’s theorem). Let for any n ≥ 1 the collection of random variables {ξnk , 1 ≤ k ≤ rn } defined on (Ω(n) , ℱ (n) , ℚ(n) ) be mutually independent. Suppose that 𝔼ℚ(n) ξnk = 0,

Varℚ(n) ξnk < ∞.

Assume that rn

σn2 := ∑ Varℚ(n) ξnk > 0 k=1

for large n and that the following Lindeberg condition holds: r

1 n 2 ∑ 𝔼ℚ(n) ((ξnk ) 1|ξ k |≥ϵσn ) = 0 n n→∞ σ 2 n k=1 lim

r

ℚ(n) , ℚ, d

n for all ε > 0. Then ∑k=1 ξnk 󳨐󳨐󳨐󳨐󳨐󳨐󳨐⇒ ξ , where the random variable ξ has 𝒩 (0, 1)-distribution with respect to the measure ℚ.

A.3.9 The rate of convergence in the central limit theorem Consider a sequence of independent random variables {Xn ; n = 1, 2, . . . } with the same distribution function F(x), x ∈ ℝ. Let us introduce the following notations and assumptions: 𝔼X1 = 0,

𝔼X12 = σ 2 > 0,

f (t) = 𝔼eitX1 ,

Fn (x) = ℙ(

1 n ∑ Xi < x). σ√n i=1

A.3 Essentials of stochastic processes | 349

Considering the general theorem, introduce additionally the following notations. Assume that we consider the random variable having the moments up to order k and the characteristic function f (t), t ∈ ℝ. Then its cumulants γl , 1 ≤ l ≤ k, are determined from the relation that is true as t → 0: k

log f (t) = ∑ l=1

γl (it)l + o(|t|k ). l!

Furthermore, let the functions Qν (x), ν ≥ 1, x ∈ ℝ, be defined as follows: Qν (x) = −

k

ν m γm+2 1 1 −x2 /2 ) . e ( ∑ Hν+2s−1 (x) ∏ m+2 √2π m=1 km ! (m + 2)!σ

The sum is taken over all integer non-negative solutions (k1 , k2 , . . . , kν ) of the equation k1 + 2k2 ⋅ ⋅ ⋅ + νkν = ν, s = k1 + k2 ⋅ ⋅ ⋅ + kν , Hν+2s−1 (x) are the Chebyshev–Hermite polynomials, and γm+2 is the cumulant of order m + 2 of the random variable X1 . For more details concerning Chebyshev–Hermite polynomials, see [136]. Theorem A.112 ([136, pp. 158–171]). If 𝔼|X1 |k < ∞ for a certain integer k ≥ 3, then, for all x and n, 󵄨󵄨 󵄨󵄨 k−2 󵄨󵄨 󵄨 󵄨󵄨Fn (x) − Φ(x) − ∑ Qν (x) 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 ν/2 󵄨󵄨 󵄨󵄨 ν=1 n ≤ c(k){σ −k n−(k−2)/2 (1 + |x|)

−k

∫

|y|k dF(y)

|y|≥σ√n(1+|x|)

+ σ −k−1 n−(k−1)/2 (1 + |x|)

−k−1

n

∫

|y|k+1 dF(y)

|y|≤σ√n(1+|x|)

−k−1 󵄨 󵄨 1 + (sup󵄨󵄨󵄨f (t)󵄨󵄨󵄨 + ) nk(k+1)/2 (1 + |x|) }, 2n |t|≥δ 2

σ where δ = 12𝔼|X 3 and c(k) is a positive constant depending only on k. The functions 1| Qν (x) are defined as follows:

Qν (x) = −

k

ν m γm+2 1 −x2 /2 1 e ( ) . ∑ Hν+2s−1 (x) ∏ m+2 √2π m=1 km ! (m + 2)!σ

The sum is taken over all integer non-negative solutions (k1 , k2 , . . . , kν ) of the equation k1 +2k2 ⋅ ⋅ ⋅+νkν = ν, s = k1 +k2 ⋅ ⋅ ⋅+kν , Hν+2s−1 (x) are the Chebyshev–Hermite polynomials, and γm+2 is the cumulant of order m + 2 of the random variable Xm . In what follows, we use a corollary of this theorem.

350 | A Essentials of calculus, probability, and stochastic processes Theorem A.113 ([136, p. 168]). If 󵄨 󵄨 lim sup 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 < 1 |t|→∞

(A.27)

and 𝔼|X1 |k < ∞ for some integer k ≥ 3, then 󵄨󵄨 󵄨 k−2 Q (x) 󵄨󵄨󵄨 1 k 󵄨󵄨 (1 + |x|) 󵄨󵄨󵄨Fn (x) − Φ(x) − ∑ νν/2 󵄨󵄨󵄨 = o( (k−2)/2 ) 󵄨󵄨 󵄨 n 󵄨󵄨 ν=1 n 󵄨

(A.28)

uniformly with respect to x ∈ ℝ. Now we formulate the partial result that follows from Theorem 7, p. 175 of the book [136] and gives the rate of convergence to Gaussian distribution for the sums of non-identically distributed independent random variables. Let {Xj }, j ≥ 1, be the sequence of independent random variables, σj2 = 𝔼Xj2 < ∞. Denote the distribution function as Vj (x) = ℙ(Xj < x) and the characteristic functions υj (t) = ∫ℝ eitx dVj (x). −1

Also, let Bn = ∑nj=1 σj2 , Φn (x) = ℙ(Bn 2 ∑nj=1 Xj < x) and let Φ(x) be the function of the standard Gaussian distribution. Theorem A.114 ([136]). Let the following conditions hold: B (i) lim infn nn > 0 and lim supn n1 ∑nj=1 𝔼|Xj |3 < ∞, (ii) for some number τ < 21 , n1 ∑nj=1 ∫|x|>nτ |x|3 dVj (x) → 0, n → ∞, 1

(iii) for any ε > 0 n 2 ∫|t|>ε |t|−1 ∏ni=1 |υj (t)|dt → 0, n → ∞.

Then the following asymptotical relation holds: Φn (x) = Φ(x) +

Q1n (x) 1 + o( ), √n √n

where Q1n (x) 1 − x22 x 2 − 1 n =− e ∑ 𝔼|Xj |3 . 3 √n √2π 6(Bn ) 2 j=1 Lemma A.115. The sequence {Xj }, j ≥ 1, satisfies condition (iii) of Theorem A.114 if its characteristic functions satisfy the inequality C 󵄨󵄨 󵄨 󵄨󵄨υj (t)󵄨󵄨󵄨 ≤ δ |t| for |t| ≥ R, where δ, R, C are some positive constants. A.3.10 The rate of convergence to the normal law in terms of pseudomoments The aim of this subsection is to prove the auxiliary theoretical results formulated in Section 2.3 and provide an example.

A.3 Essentials of stochastic processes | 351

A.3.10.1 Proofs of Theorem 2.33 and Corollary 2.34

t2

At first we prove two auxiliary results. Denote ω(t) = |f ( σt ) − e− 2 |. Lemma A.116. Let μk = 0, k = 3, . . . , m. Then for all t ∈ ℝ ω(t) ≤

|t|m+1 (1) 2|t|m (2) νn (m) + ν (m). (m + 1)! m! n

Proof. Recall that f (t) = ∫−∞ eitx dF(x). Therefore ∞

∞

∞

−∞

−∞

itx t f ( ) = ∫ e σ dF(x) = ∫ eitx dF(xσ). σ

According to the condition of the lemma, pseudomoments up to order m inclusive equal zero. Hence it is easy to deduce that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 ω(t) = 󵄨󵄨󵄨∫ eitx dF(xσ) − ∫ eitx dΦ(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨∫ eitx d(F(xσ) − Φ(x))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 ℝ

ℝ

ℝ

󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 m m 󵄨󵄨 󵄨󵄨 󵄨󵄨 (itx)k 󵄨󵄨󵄨󵄨󵄨󵄨 (itx)k 󵄨 )dH(x)󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨eitx − ∑ = 󵄨󵄨󵄨∫ (eitx − ∑ 󵄨󵄨󵄨󵄨dH(x)󵄨󵄨󵄨. 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k! k! k=0 k=0 󵄨 ℝ󵄨 󵄨 󵄨ℝ Take into account the inequality ([174], p. 372) 󵄨󵄨 (iα)m 󵄨󵄨󵄨󵄨 21−δ |α|m+δ 󵄨󵄨 iα , 󵄨≤ 󵄨󵄨e − 1 − ⋅ ⋅ ⋅ − 󵄨󵄨 m! 󵄨󵄨󵄨 m!(m + 1)δ m = 0, 1, . . . , δ ∈ [0, 1], and put δ = 1 and δ = 0 to obtain ω(t) ≤

≤

󵄨󵄨 󵄨󵄨 󵄨 m k 󵄨󵄨󵄨 󵄨󵄨 itx m (itx)k 󵄨󵄨󵄨󵄨 󵄨 󵄨󵄨e − ∑ 󵄨󵄨󵄨󵄨dH(x)󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨eitx − ∑ (itx) 󵄨󵄨󵄨󵄨󵄨󵄨dH(x)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 k! 󵄨󵄨󵄨󵄨󵄨 k! 󵄨󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 k=0 k=0 |x|≤σ√n |x|>σ√n ∫

∫ |x|≤σ√n

|tx|m+1 󵄨󵄨 󵄨 󵄨dH(x)󵄨󵄨󵄨 + (m + 1)! 󵄨

∫ |x|>σ√n

2|tx|m 󵄨󵄨 󵄨 󵄨dH(x)󵄨󵄨󵄨 m! 󵄨

2|t|m (2) |t|m+1 (1) νn (m) + ν (m). = (m + 1)! m! n The lemma is proved.

Now, denote T1 (n, m) = √−2 log(2eνn (m)). Then, in turn, we have νn (m) = exp{− 21 T12 (n, m)}. Note also that condition (ii) implies that T1 (n, m) ≥ 1.

1 × 2e

Lemma A.117. Suppose that condition (ii) of Theorem 2.33 holds. 1) For |t| ≤ T1 (n, m), the characteristic function allows the following bound: |f ( σt )| ≤ t2

e− 6 .

352 | A Essentials of calculus, probability, and stochastic processes 2) For |t| > T1 (n, m), the characteristic function allows the following bound: |f ( σt )| ≤ (2e + 83 )νn (m)|t|m+1 . t2

t2

t2

Proof. Evidently, |f ( σt )| = |f ( σt ) − e− 2 + e− 2 | ≤ e− 2 + ω(t). Now we consider two cases. 1) Let |t| ≤ T1 (n, m). Then we can deduce from Lemma A.116 that 󵄨󵄨 t 󵄨󵄨 2 2 t2 󵄨󵄨 󵄨 −t −t 󵄨󵄨f ( )󵄨󵄨󵄨 ≤ e 4 (e 4 + e 4 ω(t)) 󵄨󵄨 σ 󵄨󵄨 t2 t2 |t|m+1 (1) 2|t|m (2) νn (m) + ν (m))) ≤ e− 4 (1 + e 4 ( (m + 1)! m! n t2

≤ e− 4 (1 + e t2

T12 (n,m) 4

≤ e− 4 (1 + t 2 e t2

= e− 4 (1 +

t2(

T12 (n,m) 4

2T m−2 (n, m) (2) T1m−1 (n, m) (1) νn (m) + 1 νn (m))) (m + 1)! m!

νn (m)(

(A.29)

T1m−1 (n, m) 2T1m−2 (n, m) + )) (m + 1)! m!

2 T m−1 (n, m) 2T1m−2 (n, m) 1 2 − T1 (n,m) t e 4 ( 1 + )). 2e (m + 1)! m! 2

Consider the function f1 (x) = exp{− x4 }xm−1 . It attains its maximal value at the point x = √2(m − 1) and this value equals f1,max = exp{−

m−1 m−1 }(2(m − 1)) 2 . 2

Furthermore, m−1

exp{− m−1 }(2(m − 1)) 2 m − 1 (2(m − 1)) 2 2 exp{− } ≤ 2 (m + 1)! m(m + 1)√2π(m − 1)(m − 1)m−1 e−(m−1) m−1

2e =( ) m−1

m−1 2

1 1 1 ≤ . √2π(m − 1) m(m + 1) m(m + 1)

The last fraction attains its maximal value at the point m = 3. Therefore, exp{−

T12 (n, m) T1m−1 (n, m) 1 } ≤ . 4 (m + 1)! 12

exp{−

T12 (n, m) 2T1m−2 (n, m) 1 } ≤ . 4 m! 3

Similarly,

It follows from (A.29) together with the two last bounds that 󵄨󵄨 t 󵄨󵄨 2 2 1 2 1 1 1 2 󵄨󵄨 󵄨 −t −t 󵄨󵄨f ( )󵄨󵄨󵄨 ≤ e 4 (1 + t ( + )) ≤ e 4 (1 + t ), 󵄨󵄨 σ 󵄨󵄨 2e 12 3 12 t2

t2

t2

≤ e− 4 e 12 ≤ e− 6 , and the proof of the first statement follows.

A.3 Essentials of stochastic processes | 353

2) Now, let |t| > T1 (n, m). Then we obtain from Lemma A.116 󵄨󵄨 t 󵄨󵄨 2 󵄨 󵄨󵄨 −t 󵄨󵄨f ( )󵄨󵄨󵄨 ≤ e 2 + ω(t) 󵄨󵄨 σ 󵄨󵄨 ≤ e−

T12 (n,m) 2

+

|t|m+1 (1) 2|t|m (2) νn (m) + ν (m) (m + 1)! m! n

≤ νn (m)(2e +

(A.30)

2|t|m |t|m+1 + ). (m + 1)! m!

Recall that T1 (n, m) > 1. Then |t| > T1 (n, m) > 1, and we obtain from (A.30) 󵄨󵄨 t 󵄨󵄨 3 |t|m+1 2|t|m+1 󵄨󵄨 󵄨 m+1 + ) ≤ (2e + )νn (m)|t|m+1 , + 󵄨󵄨f ( )󵄨󵄨󵄨 ≤ νn (m)(2e|t| 󵄨󵄨 σ 󵄨󵄨 24 6 8 whence the proof follows. Proof of Theorem 2.33. Let F and G be two distribution functions with characteristic functions f and g, respectively, and suppose that G has a density function which we denote as G′ . We take from [111, p. 297] the inequality T

′ 󵄨 󵄨 2 󵄨 󵄨 dt 24 supx∈ℝ |G (x)| sup󵄨󵄨󵄨F(x) − G(x)󵄨󵄨󵄨 ≤ ∫󵄨󵄨󵄨f (t) − g(t)󵄨󵄨󵄨 + π t πT x∈ℝ 0

and put F(x) = Φn (x), G(x) = Φ(x). Then T 󵄨 t 2 󵄨󵄨 dt 24 t 󵄨󵄨 󵄨 󵄨󵄨 2 󵄨󵄨󵄨 n ) − e− 2 󵄨󵄨󵄨 + . ρn := sup󵄨󵄨Φn (x) − Φ(x)󵄨󵄨 ≤ ∫󵄨󵄨f ( 󵄨 󵄨 √ π t σ√n π 2πT x∈ℝ 󵄨 󵄨

(A.31)

0

Let n ≥ 2. Firstly, it follows from the elementary inequality n

󵄨󵄨 n n󵄨 k−1 n−k 󵄨󵄨u − v 󵄨󵄨󵄨 ≤ |u − v| ∑ |u| |v| k=1

and from Lemma A.116 that for t ≤ T1 (n, m)√n n 󵄨󵄨 󵄨󵄨 󵄨󵄨k−1 t2 t 2 󵄨󵄨 t t t 󵄨󵄨 n 󵄨 󵄨 󵄨 ) − e− 2 󵄨󵄨󵄨 ≤ ω( ) ∑ 󵄨󵄨󵄨f ( )󵄨󵄨󵄨 e− 2n (n−k) 󵄨󵄨f ( 󵄨󵄨 󵄨󵄨 √n k=1󵄨󵄨 σ√n 󵄨󵄨 σ√n

≤ ω( ≤ n(

n t2 t ) ∑ e− 6 √n k=1

|t|m+1

(m + 1)!n

= exp{−

m+1 2

n−1 n

≤ ω(

νn(1) (m)

+

t2 t )ne− 12 √n

2|t|m

m

m!n 2

t νn(2) (m)) exp{−

(A.32)

2

12

}

t2 |t|m+1 2|t|m (2) }( ν(1) (m) + ν (m)). m−1 n m−2 n 12 (m + 1)!n 2 m!n 2

354 | A Essentials of calculus, probability, and stochastic processes Secondly, we introduce the following notations: (1) Cm,n

12

=

2n

m−1 2

m−1 2

Γ( m+1 ) 2

(m + 1)!

(2) (1) Cm,n = 2Cm−1,n ,

,

1 . (1) (1) (2) (2) √2π(Cm,n νn (m) + Cm,n νn (m))

T2 (n, m) = Then

(1) (2) 24 (1) νn (m) (2) νn (m) = Cm + C . m−1 m−2 m π √2πT2 (n, m) n 2 n 2

(A.33)

Let T3 (n, m) = (T1 (n, m)√n) ∧ T2 (n, m). Then it follows from (A.31) and (A.33) that 2 ρn ≤ π

+

T3 (n,m)

T2 (n,m)

0

T3 (n,m)

󵄨󵄨 t 2 󵄨󵄨 dt 2 t 󵄨 󵄨 ) − e− 2 󵄨󵄨󵄨 + ∫ 󵄨󵄨󵄨f n ( 󵄨󵄨 󵄨 π σ√n 󵄨 t

2 π

T2 (n,m)

∫

󵄨󵄨 󵄨󵄨n dt t 󵄨 󵄨 )󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨f ( 󵄨󵄨 σ√n 󵄨󵄨 t

dt 24 + t π √2πT2 (n, m)

t2

e− 2

T3 (n,m)

(1) = I1 (n, m) + I2 (n, m) + I3 (n, m) + Cm

νn(1) (m) n

m−1 2

(2) + Cm

νn(2) (m) n

m−2 2

.

(A.34)

Since T3 (n, m) ≤ T1 (n, m)√n, we obtain from (A.32) 2 I1 (n, m) = π ≤

≤ =

2 π

T3 (n,m)

󵄨󵄨 t 2 󵄨󵄨 dt t 󵄨 󵄨 ) − e− 2 󵄨󵄨󵄨 ∫ 󵄨󵄨󵄨f n ( 󵄨󵄨 󵄨󵄨 t σ√n 0

T3 (n,m)

12

∫ ( 0

m+1 2

tm (m + 1)!n

Γ( m+1 ) 2

π(m + 1)!n

(1) (1) νn (m) Cm m−1

n

2

m−1 2

+

m−1 2

νn(1) (m) +

νn(1) (m) +

2 ⋅ 12

(2) (2) νn (m) Cm . m−2

n

m 2

2t m−1 m!n

t2

m−2 2

νn(2) (m))e− 12 dt

(A.35)

Γ( m2 ) (2) νn (m) m−2

πm!n

2

2

If T3 (n, m) = T2 (n, m), then I2 (n, m) = 0 and I3 (n, m) = 0. Therefore we consider the case T3 (n, m) = T1 (n, m)√n. Then 2 I2 (n, m) = π

T2 (n,m)

󵄨󵄨 󵄨󵄨n dt t 2 󵄨 󵄨 )󵄨󵄨󵄨 = ∫ 󵄨󵄨󵄨f ( π 󵄨󵄨 σ√n 󵄨󵄨 t

T3 (n,m)

T2 (n,m)/σ√n

∫ T1 (n,m)/σ

󵄨󵄨 󵄨n dt . 󵄨󵄨f (t)󵄨󵄨󵄨 t

A.3 Essentials of stochastic processes | 355

Now we apply the result of Statulyavichus [157]: if a random variable with characteristic function f (t) has a density p(x) ≤ d < ∞ and variance σ 2 , then for any t ∈ ℝ t2 󵄨 󵄨󵄨 }. 󵄨󵄨f (t)󵄨󵄨󵄨 ≤ exp{− 96d2 (2σ|t| + π)2

(A.36)

It follows from condition (i) that the density p(x) of any ξn can be obtained as the inA 1 ∞ 1 . verse Fourier transformation p(x) = 2π ∫ℝ e−itx f (t)dt and p(x) ≤ 2π ∫−∞ |f (t)|dt = 2π 2

t Besides, the function (2σt+π) 2 is increasing for t > 0. Therefore, when |t| ≥ T1 (n, m)/σ (recall that T1 (n, m) > 1),

π2 󵄨󵄨 󵄨 } =: b 󵄨󵄨f (t)󵄨󵄨󵄨 ≤ exp{− 24A2 σ 2 (2 + π)2 and 0 < b < 1. Then 2 I2 (n, m) = π

T2 (n,m)/σ√n

∫ T1 (n,m)/σ

∞

σA n−1 󵄨󵄨 󵄨n dt 2σ n−1 󵄨󵄨 󵄨 ≤ b ∫ 󵄨󵄨f (t)󵄨󵄨󵄨dt = b . 󵄨󵄨f (t)󵄨󵄨󵄨 t π π

(A.37)

0

At last, we bound I3 (n, m). Note that I3 (n, m) is non-zero only if T1 (n, m)√n < T2 (n, m). Therefore 2 I3 (n, m) ≤ π ≤

∞

e

∫

nT12 (n,m)

dt 2 e− 2 ≤ t π nT12 (n, m)

2

− t2

T1 (n,m)√n

2(2eνn (m))n 4eνn (m) 4e ⋅ e− n−1 ≤ (2eνn (m)) ≤ νn (m) πn πn πn 3

n−1 2

(A.38)

n

4e 2 e− 2 = νn (m) . π n

Relations (A.34)–(A.38) supply the proof of Theorem 2.33. Remark A.118. Let the following conditions hold: μk = 0, k = 3, . . . , m, m ≥ 3. Then 󵄨 󵄨 󵄨 󵄨 sup󵄨󵄨󵄨Φ1 (x) − Φ(x)󵄨󵄨󵄨 = sup󵄨󵄨󵄨F(xσ) − Φ(x)󵄨󵄨󵄨 x∈ℝ

x∈ℝ

≤(

1 2 6 + ) max(ν1 (m), (ν1 (m)) m+2 ). π(m + 1)! π √2π

Indeed, let n = 1. The theorem is obvious when ν1 (m) > 1. Let ν1 (m) ≤ 1. Put (A.31) 1 T = (ν1 (m))− m+2 . Then it follows from Lemma A.116 that 󵄨 󵄨 󵄨 󵄨 ρ1 = sup󵄨󵄨󵄨Φ1 (x) − Φ(x)󵄨󵄨󵄨 = sup󵄨󵄨󵄨F(xσ) − Φ(x)󵄨󵄨󵄨 x∈ℝ

T

≤

x∈ℝ

2 |t|m+1 (1) 2|t|m (2) dt 24 ν1 (m) + ν (m)) + ∫( √ π (m + 1)! m! 1 t π 2πT 0

356 | A Essentials of calculus, probability, and stochastic processes 1 2 T m+1 2T m (2) 24 ( ν1(1) (m) + ν1 (m)) + (ν1 (m)) m+2 π (m + 1) ⋅ (m + 1)! m ⋅ m! π √2π 1 3 24 2 ≤ (ν1 (m)) m+2 ( + ). π (m + 1)! π √2π

≤

Proof of Corollary 2.34. This proof is similar to the proof of Theorem 2.33. We apply t2 inequality (A.36) and mention again that the function (2σt+π) 2 is increasing for t > 0. Therefore, when |t| ≥ T1 (n, m)/σ (recall that T1 (n, m) > 1), 󵄨󵄨 󵄨 󵄨󵄨f (t)󵄨󵄨󵄨 ≤ exp{−

1

96A21 σ 2 (2

+ π)2

} =: b1

and 0 < b1 < 1. It follows from [56, p. 510] that ∫−∞ |f (t)|2 dt ≤ 2πA1 . Therefore ∞

2 I2 (n, m) = π

T2 (n,m)/σ√n

∫ T1 (n,m)/σ

∞

󵄨󵄨 󵄨n dt 2σ n−2 󵄨󵄨 󵄨2 ≤ b ∫ 󵄨󵄨f (t)󵄨󵄨󵄨 dt = 2σA1 bn−2 󵄨󵄨f (t)󵄨󵄨󵄨 1 . t π 1 0

Corollary 2.34 is proved. Remark A.119. For n = 2 we can get the estimate similar to those in Remark A.118. A.3.10.2 Example Below we exhibit a distribution function for which all assumptions of Corollary 2.34 hold. This examplle was proposed by P. Slyusarchuk, see [121]. Let a > 0 and θ > 1. We assume that a is fixed, while θ > 1 needs to be specified later. Consider the symmetric distribution function F(x) such that Φ(x), { { { F(x) = {Φ(a) + { { {1,

1−Φ(a) (x aθ−a

if x ∈ [0, a], − a), if a < x ≤ θa, if x > θa.

It is clear that the corresponding probability density is bounded. Since the function F(x) is symmetric, μ1 = 0 and μ3 = 0. To find μ2 , consider a

σ̃ 2 = ∫ x2 dΦ(x) +

x2

∫ a 1.

Thus there exists θ > 1 such that 31 (θ2 +θ+1) = γ, whence we conclude that θ necessarily is such that 1 3 θ = − + √3γ − . 2 4 For this value of θ, ∞

∫ x2 dΦ(x) − (1 − Φ(a)) a

a2 2 (θ + θ + 1) = 0, 3

and thus σ̃ 2 = 1. This, in turn, means that μ2 = 0. Now we study the behavior of γ as a function of a. Let ∞

y = ∫ x2 dΦ(x) − a2 (1 − Φ(a)),

y ≥ 0.

a

The derivative is equal to y′ = −a2 φ(a) − 2a(1 − Φ(a)) + a2 φ(a) = −2a(1 − Φ(a)) < 0, where φ(x), x ∈ ℝ, is the standard normal density. Therefore y decreases and approaches 0 as a → ∞. Moreover, the l’Hopital rule allows us to evaluate the limit lim

a→∞

∫a x2 dΦ(x) ∞

a2 (1 − Φ(a))

−a2 φ(a) = 1, a→∞ 2a(1 − Φ(a)) − a2 φ(a)

= lim

since lim

a→∞

−φ(a) a(1 − Φ(a)) (1 − Φ(a)) = lim = lim a→∞ a→∞ φ(a) + aφ′ (a) aφ(a) a2 φ(a) a2

= lim

a→∞

−e− 2 a2

a2

e− 2 − a2 e− 2

= 0.

Then γ → 1 as a → ∞, and thus 1 3 θ = − + √3γ − → 1, 2 4

a → ∞.

358 | A Essentials of calculus, probability, and stochastic processes Consider the pseudomoments ∞

∞

󵄨 󵄨 󵄨 󵄨 ν4 = ∫ x d󵄨󵄨󵄨F(x) − Φ(x)󵄨󵄨󵄨 = 2 ∫ x 4 d󵄨󵄨󵄨F(x) − Φ(x)󵄨󵄨󵄨 4

0

−∞ aθ

∞ 󵄨 − Φ(a) 1 − x22 󵄨󵄨󵄨󵄨 = 2 ∫ x 󵄨󵄨 e 󵄨󵄨dx + 2 ∫ x 4 dΦ(x) − 󵄨󵄨 a(θ − 1) √2π 󵄨󵄨 a 4 󵄨󵄨󵄨 1

aθ

∞

≤

2 1 − Φ(a) 5 5 2 a (θ − 1) + 2 ∫ x 4 dΦ(x) + φ(a)a5 (θ5 − 1). 5 a(θ − 1) 5 a

Hence ν4 → 0 as a → ∞ and we conclude that νn(1) (3, a) :=

󵄨 󵄨 ∫ x4 󵄨󵄨󵄨d(F(x) − Φ(x))󵄨󵄨󵄨 → 0,

a → ∞,

|x|≤√n 3

for all n ≥ 1. This implies that one can choose a0 > 0 such that νn(1) (3, a0 ) < 21 e− 2 for all n ≥ 1. It is obvious that νn(2) (3, a0 ) :=

󵄨 󵄨 ∫ |x|3 󵄨󵄨󵄨d(F(x) − Φ(x))󵄨󵄨󵄨 → 0,

n → ∞.

|x|>√n

Therefore 1 3 νn (3, a0 ) = max{νn(1) (3, a0 ), νn(2) (3, a0 )} < e− 2 , 2 and condition (ii) of Theorem 2.33 holds. Since the density is bounded, the distribution function F(x) satisfies conditions of Corollary 2.34 for m = 3 and n ≥ n0 . A.3.11 Stochastic differential equations and the approximations of solutions A.3.11.1 Existence and uniqueness of a strong solution Let (Ω, ℱ , 𝔽 = {ℱt , t ∈ [0, T]}, ℙ) be a stochastic basis and let W be a standard Wiener process on this basis. Assume that we are given deterministic functions a: [0, T] × ℝ → ℝ and b: [0, T] × ℝ → ℝ, which serve as coefficients for the equation, and an ℱ0 -measurable random variable X(0), serving as an initial condition for the equation. The corresponding stochastic differential equation is dX(t) = a(t, X(t))dt + b(t, X(t))dW(t),

t ∈ [0, T],

(A.39)

with initial condition X(0). The functions a and b are called the drift and diffusion coefficients, respectively.

A.3 Essentials of stochastic processes | 359

Definition A.120. A strong solution to equation (A.39) is a stochastic process {X(t), t ∈ [0, T]} such that its stochastic differential satisfies (A.39). In other words, it is an adapted process satisfying t

t

X(t) = X(0) + ∫ a(s, X(s))dt + ∫ σ(s, X(s))dW(s) 0

(A.40)

0

a. s. for any t ∈ [0, T]. Let us turn now to the question of solvability of stochastic differential equations. We will assume that the coefficients a, b are measurable and satisfy the following conditions with some non-random constant K > 0: – linear growth: for any t ∈ [0, T], x ∈ ℝ, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨a(t, x)󵄨󵄨󵄨 + 󵄨󵄨󵄨b(t, x)󵄨󵄨󵄨 ≤ K(1 + |x|), –

(A.41)

Lipschitz continuity: for any t ∈ [0, T], x, y ∈ ℝ, 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨a(t, x) − a(t, y)󵄨󵄨󵄨 + 󵄨󵄨󵄨b(t, x) − b(t, y)󵄨󵄨󵄨 ≤ K|x − y|.

(A.42)

Theorem A.121. Let the coefficients of equation (A.39) satisfy (A.41) and (A.42) and let X(0) be such that 𝔼(|X(0)|2 ) < ∞. Then the equation has a unique strong solution X. Moreover, 𝔼|X(t)|2 < ∞ for all t ∈ [0, T]. Let the functions a, b: ℝ → ℝ be continuous. Consider the following homogeneous stochastic differential equation: dX(t) = a(X(t))dt + b(X(t))dW(t),

(A.43)

where W = {W(t), t ≥ 0} is a Wiener process. Theorem A.122 (Yamada, Watanabe [81, p. 182]). Suppose that a(x) and b(x) are bounded. Assume furthermore that the following conditions are satisfied: (i) there exists a strictly increasing function ρ(u) on [0, ∞) such that ρ(0) = 0, ∫0+ ρ−2 (u)du = ∞, and |b(x) − b(y)| ≤ ρ(|x − y|) for all x, y ∈ ℝ, (ii) there exists an increasing and concave function k(u) on [0, ∞) such that k(0) = 0, ∫0+ k −1 (u)du = ∞, and |a(x) − a(y)| ≤ k(|x − y|) for all x, y ∈ ℝ. Then equation (A.43) has a unique strong solution. A.3.11.2 Existence and uniqueness of a weak solution Let a: [0, T] × ℝ → ℝ and b: [0, T] × ℝ → ℝ be some measurable functions, and let W be a standard Wiener process. Consider a stochastic differential equation dX(t) = a(t, X(t))dt + b(t, X(t))dW(t), with ℱ0 -measurable initial condition X(0).

t ∈ [0, T],

(A.44)

360 | A Essentials of calculus, probability, and stochastic processes Definition A.123. A weak solution to stochastic differential equation (A.44) is a triple, consisting of: – a stochastic basis (Ω′ , ℱ ′ , {ℱt′ , t ∈ [0, T]}, ℙ′ ); – a Wiener process W ′ on this basis; –

d

an adapted process {X ′ (t), t ∈ [0, T]} on this basis such that X ′ (0) = X(0) and dX ′ (t) = a(t, X ′ (t))dt + b(t, X ′ (t))dW ′ (t),

t ∈ [0, T].

The difference from the notion of strong solution is that the former is constructed for a given Wiener process and a strong solution is a function of an initial condition and the path of the underlying Wiener process. In contrast, a weak solution is constructed for some Wiener process and in general it is not measurable with respect to this Wiener process. We start with existence of a weak solution for a stochastic differential equation with continuous coefficients. Theorem A.124 (Skorokhod [154]). Suppose that for a stochastic differential equation (A.44) the coefficients a, b ∈ C([0, T] × ℝ) and the assumption (A.41) holds. Then equation (A.44) has a weak solution, which is continuous with probability 1. The next weak existence–uniqueness result for equations with bounded coefficients was established in [160]. Theorem A.125 (Stroock, Varadhan [160]). Suppose that for a stochastic differential equation (A.44) the coefficient a is measurable and bounded, the coefficient b is continuous and bounded, and b ≠ 0 at each point. Then equation (A.44) has a unique weak solution. In the following theorem from [95], the diffusion coefficient b need not be continuous. However, the statement deals with homogeneous equations only. Theorem A.126 (Krylov [95]). Suppose that for a homogeneous stochastic differential equation (A.43) the coefficient a is measurable and bounded, the coefficient b is measurable and bounded, and there exists a constant ε > 0 such that 󵄨󵄨 󵄨 󵄨󵄨b(x)󵄨󵄨󵄨 ≥ ε for all x ∈ ℝ. Then equation (A.43) has a unique weak solution. A.3.11.3 Approximations of solutions Assume that the coefficients a, b: [0, T] × ℝ → ℝ of the stochastic differential equation dX(t) = a(t, X(t))dt + b(t, X(t))dW(t), satisfy the conditions (A.41) and (A.42).

t ∈ [0, T],

(A.45)

A.3 Essentials of stochastic processes | 361

A solution of equation (A.45) admits a variety of discrete-time approximations. We deal with the so-called Euler–Maruyama scheme, also referred to as Euler scheme. For a fixed n ≥ 1, we put δn = T/n and consider the following uniform partition of the n interval [0, T]: tm = mδn , m = 0, . . . , n. Let the initial value of the scheme be fixed: (n) X (0) = X(0). The values of an approximate solution in other points of the partition are recursively defined by n n n n X (n) (tm ) = X (n) (tm−1 ) + a(tm−1 , X (n) (tm−1 ))δn

n n n n + b(tm−1 , X (n) (tm−1 ))(W(tm ) − W(tm−1 )),

m = 1, . . . , n.

(A.46)

The values between the points of the partition are defined by linear interpolation: n n n n X (n) (t) = X (n) (tm−1 ) + a(tm−1 , X (n) (tm−1 ))(t − tm−1 ) n n n + b(tm−1 , X (n) (tm−1 ))(W(t) − W(tm−1 )),

n n t ∈ [tm−1 , tm ],

m = 1, . . . , n.

We assume that, in addition to the conditions (A.41) and (A.42), the coefficients a and b of equation (A.45) satisfy the following Hölder condition: 󵄨󵄨 󵄨 󵄨 󵄨 1/2 󵄨󵄨a(t, x) − a(s, x)󵄨󵄨󵄨 + 󵄨󵄨󵄨b(t, x) − b(s, x)󵄨󵄨󵄨 ≤ K(1 + |x|)|t − s|

(A.47)

for all t, s ∈ [0, T] and x ∈ ℝd . Theorem A.127. Let the coefficients of equation (A.45) satisfy the conditions (A.41), (A.42), and (A.47), and let 𝔼(|X(0)|2 ) < ∞. Then 󵄨 󵄨2 󵄨 󵄨2 𝔼(󵄨󵄨󵄨X(t) − X (n) (t)󵄨󵄨󵄨 ) ≤ C(1 + 𝔼󵄨󵄨󵄨X(0)󵄨󵄨󵄨 )δn .

(A.48)

Definition A.128 (see [92]). Let X (n) (t) be a discretization scheme of the process X(t). We shall say that an approximating process X (n) (t) converges in the strong sense with order γ ∈ (0, ∞] to the true process X(t) if there exists a finite constant K such that 󵄨 󵄨 𝔼󵄨󵄨󵄨X(t) − X (n) (t)󵄨󵄨󵄨 ≤ Kn−γ . Evidently, the bound (A.48) implies that the Euler scheme converges strongly with order 0.5. Proposition A.129. For a diffusion process with additive noise, that is, b(t, x) = const, the Euler scheme converges strongly with order 1. Indeed, in the case of additive noise, the Euler scheme coincides with the Milstein scheme, which is of order 1 (see, e. g., [92, Theorem 10.6.3]).

362 | A Essentials of calculus, probability, and stochastic processes

A.4 Some algebra related to matrices in the Cholesky decomposition In this section we present main results of the paper [12], which are used in the proof of Theorem 1.86. Theorem A.130 ([12, Theorem 1]). Consider an n × n symmetric Toeplitz matrix T = (ti,j ) generated by a sequence {Xi , 0 ≤ i ≤ n}, i. e., ti,j = X|i−j| for 1 ≤ i, j ≤ n: X0 ( T=(

X1 X0

X2 X1 .. .

(symm.

... ... .. . X0

Xn Xn−1 .. ) . . ) X1 X0 )

Suppose the sequence X0 , X1 , . . . , Xn satisfies the following relations: (i) monotonicity and positivity: X0 ≥ X1 ≥ X2 ≥ ⋅ ⋅ ⋅ ≥ Xn ≥ 0, and (ii) convexity: X0 − X1 ≥ X1 − X2 ≥ ⋅ ⋅ ⋅ ≥ Xn−1 − Xn ≥ 0. Then the Cholesky decomposition of the matrix T, given by T = LL⊤ , where L = (li,j ) is lower-triangular, satisfies li,j ≥ 0 (1 ≤ j ≤ i ≤ n) and li,j ≥ li+1,j

(1 ≤ j ≤ i ≤ n − 1).

Corollary A.131 ([12, Corollary 1]). Under the assumptions of Theorem A.130 ljj lkj ≥ (Xk−j − Xk−(j−1) ) + Xk−1 (Xj−2 − Xj−1 ),

(A.49)

for k ≥ j. In particular, setting k = j in (A.49), we have a lower bound on the diagonal entries in L, ljj ≥ √(X0 − X1 ) + Xj−1 (Xj−2 − Xj−1 ).

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Index adapted process 323 admissible strategy 179 arbitrage 5 arbitrage-free market 5, 22 – in “global” sense 5 – in “local” sense 5 attainable – contingent claim 6 – payoff 6 (B, S)-market 19 binomial model 11 Black–Scholes formula 37 Black–Scholes–Merton model 28 Brownian motion see Wiener process – fractional 92, 332 – geometric 23, 28 Burkholder–Davis–Gundy inequality 326 C-tightness 345 càdlàg – function 322 – process 322 capital 4 central limit theorem 347 – rate of convergence 348 Chebyshev–Hermite polynomials 349 Cholesky decomposition 362 class – 𝒟 327 – M2 (𝕋) 328 – 𝒮 ̆ 337 complete market 6 conditional expectation 320 contingent claim 6 – attainable 6 – derivative 6 – discounted 6 – European 6 – hedgeable 6 continuous stochastic process 322 convergence – in probability 340 – weak 340 – in Skorokhod space 342 – of finite-dimensional distributions 341 – of measures 340

– of random variables 340 – of stochastic elements 340 – to Wiener process 347 Cox–Ingersoll–Ross process 69 Cox–Ross–Rubinstein model 11 cylinder sets 323 decomposition – Doob–Meyer 327 – Galtchouk–Kunita–Watanabe 330 “Delta” 40 delta-hedge 14 derivative 6 – contingent claim 6 – contract 6 – security 6 diffusion coefficient 358 discounted – contingent claim 6 – price process 3 – value process 13 Doléans-Dade exponential 336 Doob decomposition 325 Doob–Meyer decomposition 327 drift 358 dual predictable projection 326 equation – Langevin 58 – stochastic differential 358 equivalent – martingale measure 180 – probability measures 321 Euler scheme 361 filtration 2, 322 – natural 323 financial market – arbitrage-free 22 – complete 6 – multi-period 5 – with continuous time 19 – with discrete time 2 finite-dimensional distributions 323 fractional Brownian motion 92, 332 Galtchouk–Kunita–Watanabe decomposition 330

372 | Index

Gaussian process 331 geometric – Brownian motion 23, 28 – Ornstein–Uhlenbeck process 60 Girsanov theorem 336 ℍ-admissible strategy 282 hedge 6 hedgeable – contingent claim 6 hedging strategy 6, 25 Hurst index 332 inequality – Burkholder–Davis–Gundy 326 – Lenglart–Rebolledo 325 integral – Itô 333 – Skorokhod 339 Itô – formula 335 – integral 333 Komlós–Major–Tusnády approximation 266 Kunita–Watanabe decomposition 330 Langevin equation 58 Lenglart–Rebolledo inequality 325 Lindeberg theorem 348 linearly independent components 297 local – de Moivre–Laplace theorem 44 – martingale 327 – submartingale 327 – supermartingale 327 localizing sequence 327 locally bounded process 181 Malliavin – calculus 337 – derivative 338 Markov process 323 martingale 323 – local 327 – measure 5, 325 – equivalent 180 – minimal 284, 325, 331 – multi-dimensional 324 – square-integrable 328

martingales – strongly orthogonal 282, 325, 328 mean self-financing strategy 280 mean squared risk 282 minimal martingale measure 284, 325, 331 modification 322 Molchan – kernel 333 – representation 332 natural filtration 323 Novikov condition 336 option 6 Ornstein–Uhlenbeck process 58, 177 – fractional 204 – geometric 60 payoff – attainable 6 – replicable 6 portfolio 4 predictable 327 – process 3, 325, 327 – σ-algebra 327, 330 probability space – filtered 323 process 321 – adapted 323 – càdlàg 322 – continuous 322 – Cox–Ingersoll–Ross 69 – Gaussian 331 – locally bounded 181 – Markov 323 – Ornstein–Uhlenbeck 58, 177 – fractional 204 – predictable 3, 325, 327 – stochastic 321 – Wiener 331 processes – stochastically equivalent 322 – strongly orthogonal 325 Prokhorov theorem 342 property – NA+ 180 – NAg 180 pseudomoment 167, 350

Index | 373

quadratic – characteristics 326, 328 – mutual 328 – variation 326, 329 Radon–Nikodym – derivative 321 – theorem 321 random variables – smooth 337 rate of convergence – in central limit theorem 348 – to normal law – in terms of pseudomoments 350 replicable payoff 6 replicating strategy 13, 25 scenario 2 scheme of series 8, 347 self-financing strategy 5, 13, 21, 179, 280 self-similar 332 semimartingale 50, 327 – special 50 Skorokhod – integral 339 – metrics 322 – representation theorem 347 – space 322 – topology 322 smooth random variables 337 square characteristic 329 square-integrable martingale 328 stochastic – basis 2, 323 – derivative 337 – differential 335 – differential equation 358 – strong solution 359 – weak solution 360 – element 340 – exponential 336 – process 321 stochastically equivalent processes 322 stopping time 324, 327 strategy 4, 20, 178, 280 – admissible 179 – ℍ-admissible 282

– hedging 6, 25 – mean self-financing 280 – replicating 13, 25 – self-financing 5, 13, 21, 179, 280 strongly orthogonal – martingales 282, 325, 328 – processes 325 submartingale 323 – local 327 supermartingale 323 – local 327 theorem – central limit 347 – rate of convergence 348 – de Moivre–Laplace 44 – Girsanov 336 – Lindeberg 348 – of optimal choice 324 – Prokhorov 342 – Yamada–Watanabe 359 tightness 341 trading strategy 178 trajectory 322 uniform topology 322 uniformly integrable 327 value process 13 – discounted 13 variation – quadratic 326, 329 weak – convergence 30, 340 – in Skorokhod space 342 – of finite-dimensional distributions 341 – of measures 340 – of random variables 340 – of stochastic elements 340 – to Wiener process 347 – relative compactness 341 – solution 360 Wiener process 331 Yamada–Watanabe theorem 359