The Fitted Finite Volume and Power Penalty Methods for Option Pricing [1st ed.] 9789811595578, 9789811595585

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SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY  MATHEMATICAL METHODS

Song Wang

The Fitted Finite Volume and Power Penalty Methods for Option Pricing

SpringerBriefs in Applied Sciences and Technology Mathematical Methods

Series Editors Anna Marciniak-Czochra, Institute of Applied Mathematics, IWR, University of Heidelberg, Heidelberg, Germany Thomas Reichelt, Emmy-Noether research group, Universität Heidelberg, Heidelberg, Germany

More information about this subseries at http://www.springer.com/subseries/11219

Song Wang

The Fitted Finite Volume and Power Penalty Methods for Option Pricing

123

Song Wang School of Electrical Engineering Computing and Mathematical Sciences Curtin University Perth, WA, Australia Shenzhen Audencia Business School Shenzhen University Shenzhen, China

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISSN 2365-0826 ISSN 2365-0834 (electronic) SpringerBriefs in Mathematical Methods ISBN 978-981-15-9557-8 ISBN 978-981-15-9558-5 (eBook) https://doi.org/10.1007/978-981-15-9558-5 Mathematics Subject Classification: 65N08, 65K15, 91-08, 65M12, 91G20 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

A financial derivative is a contract between two parties whose value is dependent on (or derived from) an underlying asset or assets. Derivative securities consist of three major parts: Forwards and Future (obligation to buy or sell), Options (right to buy or sell) and Swaps (simultaneous selling and purchasing). In particular, the first two form the basis of derivative securities. Options are often used for hedging risks in the underlying assets or stocks. An option can be traded on a secondary financial market. Thus, how to price options accurately has attracted much attention in the past few decades from both financial engineers and researchers. There are mainly two types of options: European option and American option. The former can be exercised only on maturity, while the latter is exercisable anytime prior to or on maturity. Mathematically, the values of European and American options are governed, respectively, by a partial differential equation (PDE), known as Black-Scholes equation, and a differential Linear Complementarity Problem (LCP). Thus, pricing options involve solutions of PDEs and LCPs. In recent years, we have developed efficient and accurate discretization and optimization methods for numerically solving the aforementioned PDEs and LCPs in both 1- and 2-dimensions. These methods include a fitted finite volume method for the PDEs and a power penalty approach to the LCPs. This book is to put together some of the methods, algorithms, and mathematical analyses developed by us to provide a reference for both practitioners and researchers on the latest advances in numerical methods for pricing financial options. The book also provides materials which can be used in an advanced course on numerical methods in financial engineering for postgraduate research students. Perth, Australia/Shenzhen, China July 2020

Song Wang

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Contents

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1 1 2 2 4 6 9 9 12 13 30 32

2 American Options on One Asset . . . . . . . . . . . . . . . . . . . . 2.1 The Differential LCP and Its Solvability . . . . . . . . . . . 2.1.1 The Differential LCP . . . . . . . . . . . . . . . . . . . . 2.1.2 The Variational Inequality . . . . . . . . . . . . . . . . 2.2 The Penalty Method and Its Convergence Analysis . . . 2.2.1 The Power Penalty Equation and Its Solvability 2.2.2 Convergence Analysis . . . . . . . . . . . . . . . . . . . 2.3 Numerical Solution of the Penalty Equation . . . . . . . . . 2.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Solution of the Nonlinear System . . . . . . . . . . . 2.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 European Options on One Asset . . . . . . . . . . . . . . . . . 1.1 Stock Price Dynamics and Itô Lemma . . . . . . . . . . 1.2 The Black–Scholes Equation and Its Solvability . . . 1.2.1 The Black–Scholes Equation . . . . . . . . . . . 1.2.2 The Strong Problem . . . . . . . . . . . . . . . . . . 1.2.3 The Variational Problem and Its Solvability 1.3 The Fitted Finite Volume Method (FVM) . . . . . . . 1.3.1 The Formulation of the FVM . . . . . . . . . . . 1.3.2 Time Discretization . . . . . . . . . . . . . . . . . . 1.3.3 Stability and Convergence . . . . . . . . . . . . . 1.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Options on One Asset with Stochastic Volatility . . . . . . . . . . . . . 3.1 The 2-Dimensional PDE Model for Pricing European Options with Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The Pricing Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 The Variational Problem and Its Solvability . . . . . . . . 3.2 The Fitted FVM for (3.1.9)–(3.1.11) . . . . . . . . . . . . . . . . . . . 3.3 Convergence of the FVM . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Reformulation of the FVM . . . . . . . . . . . . . . . . . . . . . 3.3.2 Stability and Convergence . . . . . . . . . . . . . . . . . . . . . 3.4 Power Penalty Method for Pricing American Options with Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Linear Complementarity Problem . . . . . . . . . . . . . 3.4.2 The Penalty Method and Convergence . . . . . . . . . . . . 3.5 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Options on One Asset Revisited . . . . . . . . . . . . . . . . . . 4.1 The Unsymmetric Finite Volume Method . . . . . . . . 4.2 Determination of Superconvergent Points . . . . . . . . . 4.3 Superconvergent Points When b Is Independent of S 4.4 Local Error Estimates at the Superconvergent Points 4.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

European Options on One Asset

Abstract In this chapter we first give a brief account of stochastic differential equations governing risky asset/stock dynamics and Itô lemma to be used for deducing the mathematical model of pricing European options on one asset. We then derive the Black–Scholes (BS) equation using the ideal of Δ-hedging and Itˆo’s lemma. The BS equation is formulated as a variational problem, which is shown to be uniquely solvable. A fitted Finite Volume Method (FVM) is proposed for the discretization of the equation. We prove that the FVM is unconditionally stable and its solution converges to that of the BS equation. Numerical results are presented to demonstrate the usefulness and accuracy of this FVM. Keywords European option valuation · Black–Scholes equation · Fitted finite volume method · Stability and convergence.

1.1 Stock Price Dynamics and Itô Lemma In stochastic mathematics, the Wiener process, denote as Wt , is a stochastic process satisfying the following properties. √ 1. W0 = 0 and ΔWt := Wt+Δt − Wt = ε Δt in a small positive time-increment Δt at time t, where ε has a standardized normal distribution N (0, 1). 2. Increments ΔWt and ΔWs in any two different short time intervals are independent. 3. The path {Wt : t ≥ 0} is continuous in t. The Wiener’s process is the mathematical representation of the 1-dimensional Brownian motion. A stochastic process St is said to follow a Geometric Brownian Motion if St satisfies the following stochastic differential equation: d St = μSt dt + σ St dWt ,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 S. Wang, The Fitted Finite Volume and Power Penalty Methods for Option Pricing, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-15-9558-5_1

(1.1.1)

1

2

1 European Options on One Asset

where μ represents the percentage drift of the process and σ is the percentage volatility (or standard deviation) of the uncertainty in the process. Both μ and σ are constants. In mathematical finance, the price of a risky asset is usually assumed to follow a Geometric Brownian Motion and thus satisfies (1.1.1) (e.g., [8]), which is now called Merton–Black–Scholes model for risky asset price dynamics. An option on a risky asset is a function of time t and the underlying asset price S satisfying (1.1.1). Itô established in [11] a mathematical presentation of the differential of a time-dependent function of a stochastic process, which is given in the following theorem. Theorem 1.1.1 (Itô Lemma) Consider a function f (t, X ), where t is time and X (t) satisfies the following stochastic differential equation: d X = a(t, X )dt + b(t, X )dW,

(1.1.2)

with W a Wiener process. If f is a twice-differentiable scalar function, then we have  df =

∂f 1 ∂2 f ∂f + a(t, x) + b2 (t, x) 2 ∂t ∂x 2 ∂x

 dt + b(t, x)

∂f dW. ∂x

(1.1.3)

Equation (1.1.1) is a special case of (1.1.2). If the value of an option. denoted as V (S, t), on a stock whose price S satisfies (1.1.1), then from (1.1.3) we have  dV =

∂f ∂V 1 ∂2 f + μS + σ 2 S2 2 ∂t ∂S 2 ∂S

 dt + σ S

∂f dW. ∂S

(1.1.4)

1.2 The Black–Scholes Equation and Its Solvability In this section we establish the Black–Scholes PDE model for pricing European options, We then reformulate it as a variational problem and show the variational problem is uniquely solvable.

1.2.1 The Black–Scholes Equation An option is a contract which gives to its owner the right to buy (call option) or sell (put option) a fixed quantity of a specified asset(s) at a fixed price (exercise or strike price) on (European option) or before (American option) a given date (expiry date). An option can usually be traded in a financial market and the market prices of the rights to buy and to sell are called call prices and put prices, respectively. It was shown by Black and Scholes [5] that the price of a European option satisfies a second-order parabolic partial differential equation (PDE), which is now known as

1.2 The Black–Scholes Equation and Its Solvability

3

the Black–Scholes equation. The Black–Scholes equation can be derived in various ways. In what follows, we derive it using the idea of delta-hedging (see [13]). Assume that the market conditions are ideal for a European option and its underlying stock, i.e., (i) the short-term interest rate is constant and known, (ii) the stock price follows the geometric Brownian motion (1.1.1); (iii) the stock pays no dividends or distribution; (iv) there are no transaction costs in buying and selling the stock or option. Under these conditions, let us consider the hedging problem: for every European call option issued by you on a stock, how many shares of the stock you need to hold in order to neutralize the risk of the option issued. Mathematically, let us consider a riskless portfolio Π consisting of one European option shorted and δ shares of the underlying stock. At time t, the value of the portfolio is Π = δS(t) − V (S(t), t), where S denotes the price of the stock and V the value of the option. The differential of V satisfies (1.1.4). In any infinitesimal time interval [t, t + dt), the change in Π is the combination of the changes in S and V , i.e., dΠ = δd S − d V . Therefore, combining this relation with (1.1.1) and (1.1.4) we have      ∂V 1 2 2 ∂2 V ∂V ∂V + μS + σ S dt + σ S δ − dW. dΠ = μSδ − ∂t 2 ∂ S2 ∂S ∂S Since we expect that the portfolio is riskless, the coefficient of dW should vanish, implying that δ = ∂∂VS . Thus, the above equality becomes 

1 ∂V ∂2 V + σ 2 S2 2 dΠ = − ∂t 2 ∂S

 dt.

(1.2.1)

We expect that this portfolio has the riskless return rate r , i.e., dΠ = r Π dt = r (S ∂∂VS − V )dt. Comparing this equality with (1.2.1) gives  −

1 ∂V ∂2 V + σ 2 S2 2 ∂t 2 ∂S



  ∂V =r S −V . ∂S

Thus, from this equation we have 1 ∂V ∂2 V ∂V + σ 2 S2 2 + r S − r V = 0. ∂t 2 ∂S ∂S

(1.2.2)

This is Black–Scholes equation which determines the value of the option. Note that ∂V is called the Delta of the option, and is denoted as Δ. The portfolio Π is said ∂S to be Δ-neutral, i.e., we long (buy) ∂∂VS shares for every option sold, or equivalently long one share and short (sell) ( ∂∂VS )−1 options, as in [5].

4

1 European Options on One Asset

1.2.2 The Strong Problem As derived in Sect. 1.2.1, the value V of a European option issued on an asset with price S satisfying (1.1.1) satisfies (1.2.2). We now consider the following generalized Black–Scholes equation (e.g., [22]): L V := −

1 ∂2 V ∂V ∂V − σ 2 (t)S 2 2 − (r (t)S − D(S, t)) + r (t)V = 0, ∂t 2 ∂S ∂S

(1.2.3)

for (S, t) ∈ I × [0, T ), with the boundary and terminal (or payoff) conditions V (0, t) = g1 (t), V (Smax , t) = g2 (t), V (S, T ) = g3 (S), t ∈ [0, T ), S ∈ I¯, (1.2.4) where I = (0, Smax ) ⊂ R with Smax a positive constant greater than the strike price K of the option, and T > 0 the expiry date. Note that (1.2.2) is defined on (0, ∞) × [0, T ). However, in (1.2.3)–(1.2.4), we truncate it using Smax , in order to be able to solve this problem numerically. We assume that g1 , g2 and g3 satisfy the following compatibility conditions g3 (0) = g1 (T ) and g3 (Smax ) = g2 (T ).

(1.2.5)

For simplicity, we assume D(S, t) = d(S, t)S, where d(S, t) represents the dividend rate and is continuously differentiable. When, d(S, t) = d(t), the problem is said to be path-independent. Otherwise, it is path-independent. There are various choices of final/payoff conditions depending on models. For example, for a call option, the most common final condition is the following payoff function V (x, T ) = max(0, S − K ), S ∈ I¯

(1.2.6)

where K < Smax denotes the strike/exercise price of the option. A second choice is the cash-or-nothing payoff given by V (S, T ) = BH (S − K ), S ∈ I¯,

(1.2.7)

where B > 0 is a constant and H denotes the Heaviside function. Obviously, this final condition is a step function which is zero if S < K and Smax if S ≥ K . Another choice is the bullish vertical spread payoff defined by V (S, T ) = max(0, S − K 1 ) − max(0, S − K 2 ), x ∈ I¯,

(1.2.8)

where K 1 and K 2 are two strike prices satisfying K 1 < K 2 . This represents a portfolio of buying one call option with the exercise price K 1 and writing another call

1.2 The Black–Scholes Equation and Its Solvability

5

option with the same expiry date but a larger exercise price (i.e., K 2 ). For a detailed discussion on this, we refer to [13, 22]. Boundary conditions g1 (t) and g2 (t) are usually difficult to determine exactly when both d and r are non-constant and Smax is finite. The simplest way to determine them for a call option is to choose V (0, t) = 0 and V (Smax , t) = V (Smax , T )er (t−T ) . Boundary conditions for a European call are determined in [9, Eq. (14)] when both d and r are constant, which can be extended to non-constant d and r as follows V (0, t) = 0,

(1.2.9)

      T T V (Smax , t) = Smax exp − d(Smax , τ )dτ − K exp − r (τ )dτ . (1.2.10) t

t

Payoff and boundary conditions for put options can be defined analogously. Before further discussion, we first transform (1.2.3) with the non-homogeneous Dirichlet boundary conditions in (1.2.4) into one with the homogeneous boundary conditions. This can be achieved by adding the term f (S, t) := −eβt L V0 to both sides of (1.2.3) and introducing a new variable u = eβt (V (S, t) − V0 (S, t)), where β > 0 is a constant to be defined, V0 (S, t) = g1 (t) +

g2 (t) − g1 (t) S Smax

(1.2.11)

and L is the Black–Scholes differential operator defined in (1.2.3). Under this transformation, (1.2.3) becomes Lu + βu = f, where L is the operator defined in (1.2.3). This equation can further be written in the following form L u := −

  ∂ ∂u ∂u − a(t)S 2 + b(S, t)Su + c(S, t)u = f (S, t), ∂t ∂S ∂S

(1.2.12)

where 1 2 σ (t), b(S, t) = r (t) − d(S, t) − σ 2 (t), (1.2.13) 2 ∂d ∂ D(S, t) c(S, t) = r (t) + b(S, t) − S + β = 2r (t) − σ 2 (t) − + β. (1.2.14) ∂S ∂S

a(t) =

From (1.2.4) and the definition of u we see that the boundary and final conditions for (1.2.12) now become u(0, t) = 0 = u(Smax , t), t ∈ [0, T ), u(S, T ) = eβT (g3 (S) − V0 (S, T )), x ∈ I¯. (1.2.15) It is also easy to see from (1.2.11) and (1.2.5) that the boundary and final conditions in (1.2.15) satisfy the compatibility conditions since g3 (0) − V0 (0, T ) = 0 = g3 (X ) − V0 (X, T ).

6

1 European Options on One Asset

1.2.3 The Variational Problem and Its Solvability We shall reformulate (1.2.12)–(1.2.15) as a variational problem. To achieve this, it is necessary to introduce some function spaces and norms on them. ¯ be the set of For an open interval Ω ∈ R, we let C m (Ω) (respectively, C m (Ω)) functions, in which a function and its derivatives up to order m are continuous on ¯ for a non-negative integer m. For 1 ≤ p < ∞, we let L p (Ω) = Ω (respectively Ω) 

1/ p < ∞} denote the space of all p-power integrable {v : Ω |v| p dΩ

functions on Ω. For any p, q ≥ 1 satisfying 1/ p + 1/q = 1, we let (v, w) := Ω vwdΩ be the duality between L p (Ω) and L q (Ω), which becomes an inner product when p = q. The inner product on L 2 (Ω) is also denoted by (·, ·). We use  ·  L p (Ω) to denote the norm on L p (Ω). For m = 1, 2, . . ., we let H m, p (Ω) denote the usual Sobolev space with the norm  · m, p,Ω . When p = 2, we simply denote H m,2 (Ω) and  · m,2,Ω as H m (Ω) and  · m,Ω , respectively. When Ω = I , we omit the subscript Ω in the above notation. We put H0m (I ) = {v ∈ H m (I ) : v(0) = v(Smax ) = 0}. To handle the degeneracy in the Black–Scholes equation, we introduce the fol

1/2 S . The space of all weighted lowing weighted L 2 -norm v0,w := 0 max S 2 v2 d S square-integrable functions is defined as L 2w (I ) := {v : v0,w < ∞}. We also define

S a weighted inner product on L 2w (I ) by (u, v)w := 0 max S 2 uvd S. Using a standard argument (cf., for example, [6, Chaps. 1 & 2]), it is easy to show that the pair (L 2w (I ), (·, ·)w ) is a Hilbert space. For brevity, we omit this discussion. Using L 2 (I ) and L 2w (I ), we define the following weighted Sobolev space 1 (I ) := {v : v ∈ L 2 (I ), v ∈ L 2w (I ) and v(Smax ) = 0}. H0,w 1 (I ) defined by Let  · 1,w be a functional on H0,w

 1/2 . v1,w = (v20 + v 20,w )1/2 = (S 2 v , v ) + (v, v)

(1.2.16)

Then, it is easy to check that  · 1,w is a weighted H 1 -norm (energy norm) on 1 (I ). Using the inner products on L 2 (I ) and L 2w (I ), we define a weighted inner H0,w 1 1 product on H0,w (I ) by (·, ·) H := (·, ·) + (·, ·)w . For the pair (H0,w (I ), (·, ·) H ), we have the following lemma. 1 (I ), (·, ·) H ) is a Hilbert space. Lemma 1.2.1 The pair (H0,w

The result is obvious since both pairs (L 2 (I ), (·, ·)) and (L 2w (I ), (·, ·)w ) are Hilbert spaces. For brevity, we omit a formal proof of this lemma, but refer the reader to a similar proof in [6, Chap. 2]. For a detailed discussion of weighted Sobolev spaces, we refer to [14]. 1 Remark 1.2.1 It is easy to check by examples that H0,w (I ) contains the conventional 1 Sobolev space H0 (I ) as a proper subspace. Also, for a v ∈ H0,w (I ), v ∈ L 2w (I ), i.e.,

Smax 2 2 S (v ) d S < ∞. Intuitively, this implies that (Sv )2 ∼ S q for some q > −1 0

1.2 The Black–Scholes Equation and Its Solvability

7

when S is close to 0. From this we see that near 0, Sv ∼ S q/2 , and so Sv ∼ S 1+q/2 for some q > −1. Therefore, if v, w ∈ H0,w (I ), S 2 v w ∼ S 1−q near S = 0 some q > −1. This further implies that lim S→0+ S 2 v w = 0. This intuition will be used in the formulation of the variational problem corresponding to (1.2.12)–(1.2.15). We now show the variational problem corresponding to (1.2.12)–(1.2.15) has a 1 unique solution in H0,w (I ) for t ∈ [0, T ) almost everywhere (a.e.). We will often 1 write u(·, t) as u(t) when we regard it as an element of H0,w (I ). From time to time, we will also suppress the independent time variable t (or τ ) when doing so causes no confusion. Before further discussion, make the the following assumptions: Assumption 1.2.1 are continuous and there exist positive constants 1. The functions σ (t) and ∂ D(S,t) ∂S   σ , σ , r and d such that σ ≤ σ (t) ≤ σ , 0 ≤ r (t) ≤ r and  ∂∂DS  ≤ d.

2. The constant β is chosen such that β − sup(x,t)∈I ×(0,T ) σ 2 (t) + ∂ D(S,t) ≥ β0 , ∂S where β0 is a positive constant. 1 Now, for any v ∈ H0,w (I ), multiplying both sides of (1.2.12) by v, integrating the resulting equation on I and using integration by parts we have

    ∂u ∂v 2 ∂u − , v + aS + (cu, v) = ( f, v) + bSu, ∂t ∂S ∂S for any t ∈ [(0, T ), since S 2 ∂∂uS v → 0 as S → 0 or Smax as discussed in Remark 1.2.1. In the above we used the homogeneous boundary conditions (1.2.15) in the strong form of the problem. This motivates us to pose the following variational problem: 1 1 Problem 1.2.1 Find u(t) ∈ H0,w (I ) for t ∈ (0, T ) a.e. such that for all v ∈ H0,w (I )



 ∂u(t) − , v + A(u(t), v; t) = ( f (t), v) ∂t

(1.2.17)

1 and (1.2.15) is satisfied, where A(·, ·; t) is the following bilinear form on H0,w (I ):

  ∂w ∂v 1 + bSv, + (cv, w), v, w ∈ H0,w A(v, w; t) := aS 2 (I ). ∂S ∂S

(1.2.18)

The following theorem shows that Problem 1.2.1 is uniquely solvable under Assumption 1.2.1. Theorem 1.2.1 Let Assumption 1.2.1 be satisfied. Then, there exists a unique solution to Problem 1.2.1.

8

1 European Options on One Asset

Proof To prove this theorem, it is sufficient to show that A defined in (1.2.18) is 1 (I ), coercive and continuous. Integrating by parts, we have, for any v ∈ H0,w 

Smax

bSvv d S = −



0

Smax 0

   Smax ∂b 2 bSvv d S. b+S v dS − ∂S 0

From this we have      Smax   1 Smax 1 Smax ∂b 2 ∂b 2 bSvv d S = − b+S b−S v dS = − v d S. 2 0 ∂S 2 0 ∂S 0 In the above we used (1.2.13). Substituting the above into the bilinear form on the right-hand side of (1.2.17) and using (1.2.13) and (1.2.14) we have    S ∂d b v, v r+ − 2 2 ∂S    1 ∂D 3r − σ 2 − v, v = (aS 2 v , v ) + 2 ∂S   1 ≥ min{σ 2 , 3r + β0 } (aS 2 v , v ) + (v, v) ≥ Cv21,w 2

A(v, v; t) = (aS 2 v , v ) +

(1.2.19)

by Assumption 1.2.1, where C denotes a positive constant, independent of v and 1  · 1,w is defined in (1.2.16). Therefore, A(·, ·; t) is coercive on H0,w (I ). 1 (I ), We now prove that A(·, ·; t) is Lipschitz continuous. For any v, w ∈ H0,w using Cauchy–Schwarz inequality we get from (1.2.18)  A(v, w; t) ≤ M v 0,w w 0,w + v0 w 0,w + v0 w0 ≤ M(v 0,w + v0 )(w 0,w + w0 ) ≤ Mv1,w w1,w , where M denotes a generic positive constant, independent of v and w, but depending on the constants in Assumption 1.2.1. Therefore, A is Lipschitz continuous with 1 (I ) is a Hilbert space by Lemma respect to the two variables. Furthermore, H0,w 1.2.1. Using an existing theoretical results (cf., for example, [12, Lemma 1 and Theorem 1.33]), we see that Problem 1.2.1 has a unique solution.  Remark 1.2.2 Assumption 1.2.1(2) is not essential for proving the unique solvability of Problem 1.2.1. In fact, the condition (1.2.19) can be replaced by Gårding inequality [17, Theorem 9.17] A(v, v; t) ≥ C1 v21,w − C2 v20 for positive constants C1 and C2 . Thus, when β = 0 and β0 < 0, Problem1.2.1 is still uniquely solvable, which is reasonable as the solvability of the problem does not depends on the scaling factor eβt . Remark 1.2.3 A special regularity assumption on the continuity of u or, equally, V at the boundary S = 0 needs to be made. This is because we cannot conclude that 1 (Ω), u is continuous at S = 0. This continuity has to be established for any u ∈ H0,w

1.2 The Black–Scholes Equation and Its Solvability

9

separately, see, e.g., [1], where the case with d = 0, r = const and σ = σ (S, t) is considered. We assume that the solution possesses the following property: ∃g1 : (0, T ) → R : lim V (S, t) = g1 (t) ∀t ∈ (0, T ). S→0

(1.2.20)

1.3 The Fitted Finite Volume Method (FVM) 1.3.1 The Formulation of the FVM There are several existing methods developed for solving (1.2.3) (or (1.2.12)) such as those in [2, 7, 10, 18, 20, 22]. However, most of these methods are unable to handle the degeneracy of L as S → 0+ . Due to this degeneracy, the solution to (1.2.12) cannot take a ‘trace’ as S → 0+ . A fitted FVM is developed in [21] to handle the degeneracy at the discrete level, based on the fitting techniques used in [15, 16]. The idea of fitting can be traced back to [3]. Let the interval I = (0, Smax ) be divided into N sub-intervals Ii := (Si , Si+1 ), i = 0, 1, . . . , N − 1, with 0 = S0 < S1 < · · · < S N = Smax . For each i = 0, 1, . . . , N − 1, we put h i = Si+1 − Si and h = max0≤i≤N −1 h i . We also let Si−1/2 = (Si−1 + Si )/2 and Si+1/2 = (Si + Si+1 )/2 for each i = 1, 2, . . . , N − 1. These mid-points form a second partition of (0, Smax ) if we define S−1/2 = S0 and S N +1/2 = S N . Integrating both sides of (1.2.12) over (Si−1/2 , Si+1/2 ) we have 

Si+1/2

−

Si−1/2

   Si+1/2  Si+1/2  Si+1/2 ∂u ∂u d S − S aS + bu + cud S = f dS ∂t ∂S Si−1/2 Si−1/2 Si−1/2

for i = 1, 2, . . . , N − 1. Applying the mid-point quadrature rule to the first, third and last terms gives −

  ∂u i li − Si+1/2 ρ(u)| Si+1/2 − Si−1/2 ρ(u)| Si−1/2 + ci u i li = f i li ∂t

(1.3.1)

for i = 1, 2, . . . , N − 1, where li = Si+1/2 − Si−1/2 , ci = c(Si , t), f i = f (Si , t), u i is an approximation to u(Si , t) and ρ(u) is a flux defined by ρ(u) := aS

∂u + bu. ∂S

(1.3.2)

Clearly, we now need to derive approximations to ρ(u) at Si+1/2 , i = 0, 1, . . . , N − 1. This discussion is divided into two cases with i ≥ 1 and i = 0, respectively. Case I. Approximation of ρ at Si+1/2 for i ≥ 1. Let us consider the following two-point boundary value problem:

10

1 European Options on One Asset

(aSv + bi+1/2 v) = 0, S ∈ Ii , v(Si ) = u i , v(Si+1 ) = u i+1 ,

(1.3.3)

where bi+1/2 = b(Si+1/2 , t). Integrating the equation in (1.3.3) yields ρi (v) := aSv + bi+1/2 v = C1 ,

(1.3.4)

where C1 denotes an additive constant. The integrating factor of this 1st-order linear equation is μ = S bi+1/2 /a and the analytic solution is v=S

−bi+1/2 /a

 S

bi+1/2 /a

C1 d S + C2 aS

 =

C1 + C2 S −bi+1/2 /a , bi+1/2

(1.3.5)

where C2 is also an additive constant. In the above, we assume that bi+1/2 = 0. This restriction will be remedied later. Applying the boundary conditions in (1.3.3) to (1.3.5) we obtain ui =

C1 bi+1/2

+ C2 Si−αi , and u i+1 =

C1 bi+1/2

−αi + C2 Si+1 ,

(1.3.6)

where αi = bi+1/2 /a. Solving this linear system gives ρi (u) = C1 = bi+1/2

αi Si+1 u i+1 − Siαi u i . αi Si+1 − Siαi

(1.3.7)

This gives a representation for the flux on the right-hand side of (1.3.4). Note that (1.3.7) also holds when αi → 0. This is because lim

αi →0

αi Si+1 − Siαi S αi − Siαi 1 1 = lim i+1 = (ln Si+1 − ln Si ) > 0, bi+1/2 a αi →0 αi a

(1.3.8)

since Si < Si+1 and a > 0. Obviously, ρi (u) in (1.3.7) provides an approximation to the flux ρ(u) at Si+1/2 . Case II. Approximation of ρ at S1/2 . The analysis in Case I does not apply to the approximation of the flux on (0, S1 ), because (1.3.3) is degenerate. This can be seen from the expression (1.3.5). When α0 > 0, we have to choose C2 = 0 as, otherwise, v blows up as S → 0. However, the resulting solution v = C1 /b1/2 can never satisfy both of the conditions in (1.3.3) unless u 0 = u 1 . To solve this difficulty, let us re-consider (1.3.3) with an extra degree of freedom in the following form (aSv + b1/2 v) = C2 , in (0, S1 )

lim v(S) = u 0 , v(S1 ) = u 1 ,

S→0+

(1.3.9)

where C2 is an unknown constant to be determined and u 0 = g1 (t) by (1.2.20). Integrating (1.3.9) once we have aSv + b1/2 v = C2 S + C3 . Using the condition

1.3 The Fitted Finite Volume Method (FVM)

11

lim S→0+ v(0) = u 0 we have C3 = b1/2 u 0 , and so the above equation becomes ρ0 (u) := aSv + b1/2 v = b1/2 u 0 + C2 S.

(1.3.10)

Solving this problem analytically gives  v=

u0 + u0 +

C2 S + a+b1/2 C2 S ln x a

C4 S −α0 α0 = −1, + C4 S α0 = −1,

(1.3.11)

where C4 is an additive constant (depending on t). To determine the constants C2 and C4 , we first consider the case that α0 = −1. When α0 ≥ 0, v(0) = u 0 implies that C4 = 0. If α0 < 0, C4 is arbitrary, so we also choose C4 = 0. Using v(S1 ) = u 1 in (1.3.9) we obtain C2 = S11 (a + b1/2 )(u 1 − u 0 ). When α0 = −1, from (1.3.11) we see that v(0) = u 0 is satisfied for any C2 and C4 . Therefore, solutions to C2 and C4 are not unique. We choose C2 = 0, and v(S1 ) = u 1 in (1.3.9) gives C4 = (u 1 − u 0 )/S1 . Therefore, from (1.3.10) we have that ρ0 (u) = (aSv + b1/2 v) S1/2 =

1 [(a + b1/2 )u 1 − (a − b1/2 )u 0 ] 2

(1.3.12)

for both α0 = −1 and α0 = −1. Furthermore, (1.3.11) reduces to v = u 0 + (u 1 − u 0 )S/S1

S ∈ [0, S1 ].

(1.3.13)

Now, using (1.3.7) and (1.3.12) obtained in Case I and Case II above respectively, we define a global piecewise constant approximation to ρ(u) by ρh (u) satisfying ρh (u) = ρi (u)

if S ∈ Ii . i = 0, 1, . . . , N − 1.

(1.3.14)

Substituting (1.3.7) into (1.3.14) and then the result and (1.3.12) into (1.3.1) we obtain ∂u i li + ei,i−1 u i−1 + ei,i u i + ei,i+1 u i+1 = f i li , (1.3.15) − ∂t where

and

b1+1/2 S1+1/2 S2α1 S1 (a − b1/2 ), e1,2 = − , 4 S2α1 − S1α1 b1+1/2 S1+1/2 S1α1 S1 = (a + b1+1/2 ) + + c1l1 , 4 S2α1 − S1α1

e1,0 = −

(1.3.16)

e1,1

(1.3.17)

12

1 European Options on One Asset α

ei,i−1

αi i−1 bi−1/2 Si−1/2 Si−1 bi+1/2 Si+1/2 Si+1 =− , ei,i+1 = − , αi−1 αi−1 αi αi Si+1 − Si Si − Si−1

(1.3.18)

α

ei,i =

bi−1/2 Si−1/2 Si i−1 bi+1/2 Si+1/2 Siαi + + ci li , α αi−1 αi Si+1 − Siαi Si i−1 − Si−1

(1.3.19)

for i = 2, 3, . . . , N − 1. These form an (N − 1) × (N − 1) linear system for u := (u 1 (t), . . . , u N (t)) with u 0 (t) and u N (t) in (1.3.15) equal to the given homogeneous boundary conditions in (1.2.15).

1.3.2 Time Discretization We now discuss the time-discretization of the linear ODE system (1.3.15). Let E i , i = 1, 2, . . . , N − 1 be 1 × (N − 1) row vectors defined by E 1 = (e11 (t), e12 (t), 0, . . . , 0), E N −1 = (0, . . . , 0, e N −1,N (t), e N −1,N −1 (t)), E i = (0, . . . , 0, ei,i−1 (t), ei,i (t), ei,i+1 (t), 0, . . . , 0), i = 2, 3, . . . , N − 2, where ei,i−1 , ei,i and ei,i+1 are defined in (1.3.16)–(1.3.19) and those which are not defined are zeros. Obviously, using E i , (1.3.15) can be rewritten as −

∂u i (t) li + E i (t)u(t) = f i (t)li ∂t

(1.3.20)

for i = 1, 2, . . . , N − 1. This is a first order linear ODE system. To discretize this system, we let ti (i = 0, 1, . . . , K ) be a set of partition points on [0, T ] satisfying T = t0 > t1 > · · · > t K = 0. Then, we apply the two-level implicit time-stepping method with a splitting parameter θ ∈ [1/2, 1] to (1.3.20) to yield u ik+1 − u ik li + θ E ik+1 uk+1 + (1 − θ )E ik uk = (θ f ik+1 + (1 − θ ) f ik )li −Δtk

(1.3.21)

for k = 0, 1, . . . , K − 1, where Δtk = tk+1 − tk < 0, E ik = E i (tk ), f ik = f (Si , tk ) and uk denotes the approximation of u at t = tk . Let E k = (E 1k , E 2k , . . . , E Nk −1 ) . Then, the above linear system can be re-written as (θ E k+1 + G k )uk+1 = f k + [G k − (1 − θ )E k ]uk , k = 0, 1, . . . , K − 1, (1.3.22) where G k = diag(l1 /(−Δtk ), . . . , l N −1 /(−Δtk )) and f k = θ ( f 1k+1l1 , . . . , f Nk+1 −1 l N −1 ) + (1 − θ )( f 1k l1 , . . . , f Nk −1 l N −1 ) . When θ = 1/2, the time-stepping scheme becomes that of the Crank-Nicolson and when θ = 1, it is the backward Euler (or fully implicit) scheme. Both of the two cases are unconditionally stable, and they are of second and first order accuracy, respectively.

1.3 The Fitted Finite Volume Method (FVM)

13

We now show that, when |Δtk | is sufficiently small, the system matrix of (1.3.22) is an M-matrix. This is contained in the following theorem. Theorem 1.3.1 For any given k = 1, 2, . . . , K − 1, if |Δtk | is sufficiently small, the the system matrix of (1.3.22) is an M-matrix. Proof Let us first investigate the off-diagonal entries of E k+1 in (1.3.22). From (1.3.16)–(1.3.19) we see that ei, j ≤ 0 for all i, j = 1, 2, . . . , N − 1, j = i. This is b aαi because S αi i+1/2 αi = αi α > 0 for all i = 1, 2, . . . , N − 1 and all bi+1/2  = 0. From Si+1 −Si i i+1 −Si (1.3.8) we see that this also holds when bi+1/2 → 0. This proves that all of the offdiagonal element of the system matrix of (1.3.22) are non-positive. Furthermore, from (1.3.16)–(1.3.19) and the definitions of E ik+1 , i = 1, 2, . . . , N − 1, it is easy to check that the diagonal entries of (θ E k+1 + G k ) are given by ⎛ ⎞   N −1  l1 S1 k+1 1 k+1 k+1 ⎠ k+1 k+1 (a + θ e1,1 =θ⎝ |e1, | + θ + b ) + θ c + l1 , 1/2 1 j −Δtk 4 |Δtk | j=1 ⎛ ⎞   N −1  lj 1 k+1 k+1 ⎠ lj + θ ei,i =θ⎝ |ei,k+1 | + θ c + j j −Δtk |Δtk | j=1 for i = 2, 3, . . . , N − 1. Thus, when |Δtk | is sufficiently small, θ E k+1 + G k is (strictly) diagonally dominant. Therefore, it is an M-matrix [19, p. 85].  Remark 1.3.1 We remark e1,0 defined in (1.3.16) may not be negative. However, this will not make any difficulty to the method because e1,0 does not appear in the system matrix of (1.3.22). Remark 1.3.2 Theorem 1.3.1 shows that the fully discretized system (1.3.22) satisfies the discrete maximum principle. This guarantees that numerical solutions from this method are non-negative as the option prices should be. Finally, we comment that local approximations to ∂u/∂ S and ∂ 2 u/∂ S 2 , can be obtained easily from (1.3.13) and (1.3.5). These two quantities, known respectively as the Δ and  of an option, are useful in practice. In particular, the former is used by financial engineers for constructing portfolios that hedge against risk (or portfolios that are delta neutral). This is also known as delta hedging.

1.3.3 Stability and Convergence We shall re-formulate the FVM as finite element method and present a stability and convergence analysis for the scheme, based on those in [4, 21].

14

1 European Options on One Asset

1.3.3.1

The Finite Element Formulation of the FVM

For any i = 1, 2, . . . , N − 1, let ψi be defined by  ψi =

1, S ∈ Ii := (Si−1/2 , Si+1/2 ), 0, otherwise,

(1.3.23)

We choose the test space to be Vh = span{ψi }1N −1 . To define the trial space, we choose the hat function φi associated with Si in the following way. On Ii we choose φi so that it satisfies (1.3.3) with φi (Si ) = 1 and φi (Si+1 ) = 0. Naturally, the solution to this two-point boundary value problem is given in (1.3.5) where C1 and C2 are determined by (1.3.6) with u i = 1 and u i+1 = 0. Similarly, we can define φi (x) on Ii−1 so that φi (Si−1 ) = 0 and φi (Si ) = 1. Combining these two solutions, choose ⎧ αi−1 −1  αi−1  Si S ⎪ 1 − Si−1 , S ∈ Ii−1 , ⎪ ⎨ 1 − Si−1 αi   αi −1  φi (S) = Si S 1 − Si+1 , S ∈ I¯i , ⎪ ⎪ 1 − Si+1 ⎩ 0, otherwise.

(1.3.24)

One [0, S1 ], φ1 (S) is given by the linear function in (1.3.13) with u 0 = 0 and u 1 = 1. For this set of basis functions we have the following theorem. Theorem 1.3.2 For each i = 1, . . . , N − 1, the function φi is monotonically increasing and decreasing on Ii−1 and Ii respectively. Furthermore, φi and φi+1 satisfy φi (S) + φi+1 (S) = 1 on Ii for i = 1, 2, . . . , N − 1. Proof Differentiating φi on Ii we have φi (S) =

αi −αi αi Sαi S Si+1 1− S i

on (Si , Si+1 ). Since

i+1

αi Si+1 − Si > 0, we have 1−(Si /S α > 0 for all αi . Therefore, φ (x) < 0, and so φ is i+1 ) i monotonically decreasing in Ii . Similarly, it is easy to show that φi is monotonically

increasing in Ii−1 . From (1.3.24) we have, for S ∈ Ii , φi + φi+1 =

1− 1−



S Si+1 Si Si+1

αi

αi

αi 1 − SSi S αi − S αi Siαi − S αi αi = i+1 + αi αi + αi αi = 1. Si+1 − Si Si − Si+1 1 − SSi+1 i 

Thus, we have proved this theorem.

Two examples of these hat functions with constant α are plotted in Fig. 1.1. The finite element trial space is chosen to be Uh = span{φi }1N −1 . For an arbitrary function v ∈ C( I¯), we define the mass lumping operator L h : C( I¯) → Vh by L h (v) :=

N  i=0

v(Si )i (S),

(1.3.25)

1.3 The Fitted Finite Volume Method (FVM)

15

Fig. 1.1 Examples of the hat functions for different values of α

where i is defined in (1.3.23). Using Uh , Vh and L h , we define the following Petrov–Galerkin problem. Problem 1.3.1 Find u h (t) ∈ Uh such that for all vh ∈ Vh − (L h (u), ˙ vh ) + Ah (u h , vh ; t) = (L h ( f ), vh ),

(1.3.26)

where Ah (·, ·; t) is the bilinear form on Uh × Vh defined by

Ah (u h , vh ; t) := −

N −1   j=1

x(a(t)x

∂u h (t) ˆ + b(t)u h (t)) ∂x

 S j+1/2

vh + (L h (c(t)u h (t)), vh )

S j−1/2

(1.3.27) ˆ = b j+1/2 (t) when S ∈ Ii for all feasible and bˆ is piecewise constant satisfying b(t) i. From the constructions of the FVM, φi and ψ j , it is easy to verify that if u h (t) =  N −1 j=1 u j (t)φ j and vh = ψi , (1.3.26) becomes the semi-discretized system (1.3.15) (or (1.3.20)). We will leave this as an exercise for the reader. Note that, when restricted on Uh , the lumping operator P is surjective from Uh to Vh . Using this P, we rewrite Problem 1.3.1 as the following equivalent Galerkin finite element formulation. Problem 1.3.2 Find u h (t) ∈ Uh such that for all vh ∈ Uh − (L h (u), ˙ L h (vh )) + Bh (u h , vh ; t) = (L h ( f ), L h (vh )), where Bh (u h , vh ; t) = A(u h , L h (vh ); t) with A(·, ·; t) defined in (1.3.27). We first define functionals  · 0,h and  · 1,h on Uh by

(1.3.28)

16

1 European Options on One Asset

vh 20,h :=

N −1 

N −1 

v2j l j , vh 21,h :=

j=1

b j+1/2 S j+1/2

j=1

α

α

α

α

j S j+1 + Sj j j S j+1 − Sj j

(v j+1 − v j )2

(1.3.29)  for any vh = Nj=0 v j φ j ∈ Uh , with v N = 0. It is easy to show that  · 1,h is a norm αj α + S j j )/b j+1/2 > 0 for any α j (= b j+1/2 /a). (The limiting case on Uh , because (S j+1 that α j → 0 is given in (1.3.8)). Obviously, it is a weighted discrete energy norm on Uh . Using (1.3.29), we define the following weighted discrete H 1 -norm on Uh vh 2h = vh 20,h + vh 21,h with the convention that v0 = v N = 0. Thus, we have the following theorem. Theorem 1.3.3 Let Assumption 1.2.1 be fulfilled. If h is sufficiently small, then, for all vh ∈ Uh , we have (1.3.30) Bh (vh , vh ; t) ≥ Cvh 2h where C denotes a positive constant, independent of h and vh .  −1 Proof We omit the time variable t. Let vh = Nj=1 vi φi ∈ Uh . We have Bh (vh , vh ) = −

N −1  

 S j+1/2

x(ax

j=1

=−

N −1  

S j+ 1 (ax 2

j=1

∂vh ˆ h) + bv ∂x

P(vh ) + (P(cvh ), P(vh ))

S j−1/2

∂vh ˆ h ) S 1 − S 1 (ax ∂vh + bv ˆ h )S 1 + bv j− 2 j+ 2 j− 2 ∂x ∂x

 vj +

N −1 

c j v 2j l j .

j=1

Re-arranging the first sum and using (1.3.14) (with ρi given by (1.3.7)) and (1.3.12) (with u 0 = 0) we have Bh (vh , vh ) =

S1/2 (a + b 21 )

+

2 N −1 

v12

+

N −1 

α

b j+ 21 S j+ 21

α

j S j+1 v j+1 − S j j v j

j=1

α

α

j S j+1 − Sj j

(v j+1 − v j )

N −1

c j v2j l j =: S1/2

j=1

 (a + b1/2 ) 2 v1 + I + c j v2j l j , 2 j=1

(1.3.31)

since S1 = 2 S1/2 and v0 = 0. For the term I , we have I =

N −1 

α

b j+1/2 S j+ 21

α

α

N −1 

α

b j+1/2 S j+ 21

j=1

=

N −1  j=1

α

j S j+1 − Sj j

j=1

=

α

j S j+1

2 α j (v j+1 − v j ) +

j S j+1 − Sj

α

j S j+1

α

j S j+1 − Sj j

N −1  j=1

α

b j+ 21 S j+ 21

α

j j S j+1 (v j+1 − v j ) + (S j+1 − S j j )v j

(v j+1 − v j )2

(v j+1 − v j )

b j+ 21 S j+ 21 v j (v j+1 − v j )

1.3 The Fitted Finite Volume Method (FVM)

+ =

17

  1 1 b j+ 21 S j+ 21 − (v j+1 − v j )2 + (v2j+1 − v2j ) 2 2 j=1

N −1 

αj α N −1 S j+1 + Sj j 1 1 2 2 2 b j+ 21 S j+ 21 α j α (v j+1 − v j ) + [b1+ 21 S1+1/2 (v2 − v1 ) 2 j=1 2 S j+1 − S j j

+b2+1/2 S2+ 21 (v32 − v22 ) + · · · + b(N −1)+1/2 S(N −1)+ 21 (v2N − v2N −1 )] = =

N −1

1 1  1 vh 21,h + S j− 21 b j− 21 − S j+ 21 b j+ 21 v2j − S 21 b 21 v12 2 2 j=1 2

 N −1  b j+ 21 − b j− 21 1 1 1 S j− 21 vh 21,h − + b j+ 21 v2j l j − S 12 b 12 v12 , 2 2 j=1 lj 2

since v N = 0. Note that r and σ in (1.2.13) are functions of t only. Therefore, substituting the above into (1.3.31) and using (1.3.29), (1.2.13) and (1.2.14) we have (a + b 21 )

1 v12 + vh 21,h 2 2 N −1   S j− 21 b j+ 21 − b j− 21 b j+ 21  2 1 cj − v j l j − S 12 b 12 v12 + − 2 lj 2 2 j=1

Bh (vh , vh ) = S1/2

a 1 = S1/2 v12 + vh 21,h 2 2 N −1   S j− 21 b j+ 21 − b j− 21 b j+ 21  2 r + b j − S j d (S j ) + β − vjlj + − 2 lj 2 j=1 N −1   d j − d j+ 21 bj Sj a 1 r+ + − d (S j ) + β = S1/2 v12 + vh 21,h + 2 2 2 2 2 j=1   d j+ 21 − d j− 21 1 v2j l j S j d (S j ) − S j− 21 − 2 lj

≥

N −1 1 1 vh 21,h + (3r − σ 2 − D (S j ) + β)v2j l j 2 2 j=1

+

N −1   d j − d j+ 21 j=1

2

−

1 2



S j d (S j ) − S j− 21

d j+ 21 − d j− 21  lj

v2j l j .

Note d j − d j+1/2 and S j d (S j ) − S j−1/2 (d j+1/2 − d j−1/2 )/l j are of order O(h). When h is sufficiently small, the absolute values of these terms are smaller than, say, β0 /2. Thus, (1.3.30) follows from the above estimate and Assumption 1.2.1  because (3r − D (S j ) − σ 2 + β)/2 ≥ 3r + β0 . This completes the proof.

18

1 European Options on One Asset

The lower bound of Bh (vh , vh ) is just a discrete analogue of that in Theorem 1.2.1. We remark that (1.3.30) implies that Bh (·, ·, t) is coercive with respect to  · h . This result is crucial for the convergence analysis.

1.3.3.2

Full Discretization and Stability

Using the mesh for (0, T ) defined in Sect. 1.3.2, we pose the following problem: Problem 1.3.3 Find a sequence u 1h , . . . , u hK ∈ Uh such that for k ∈ {0, . . . , K − 1} ⎧  k ⎪ L h (u k+1 ⎪ h ) − L h (u h ) ⎪ , L (v ) +Bh (θ u k+1 + (1 − θ)u kh , vh ; tk+θ ) ⎪ h h ⎨ h −Δt k

⎪ ⎪ ⎪ ⎪ ⎩

=(θ L h ) f k+1 ) + (1 − θ)L h ( f k ), vh )

∀vh ∈ Uh ,

u 0h = g3I ,

(1.3.32) where tk+θ := θ tk+1 + (1 − θ )tk = tk + θ Δtk , f k := f (tk ) and g3I ∈ Uh is the interpolant of the terminal condition g3 (S) in Uh .  −1 k It is easy to verify that if we choose u kh = Nj=1 u j φ(x) and vh = φi for i = 1, 2, . . . , N − 1, in (1.3.32), (1.3.32) reduces to (1.3.21). We will leave this as an exercise for the reader. The stability of the scheme is established in the following theorem: Theorem 1.3.4 If θ ∈ [1/2, 1] and g3 ≥ 0 in I × (0, T ), then any solution to (1.3.32) satisfies 2T sup  f (s)20,h u kh 20,h ≤ g3I 20,h + C0 s∈(0,T ) for a positive constant C0 , independent of i and u ih for all i = 1, 2, . . . , K . Proof In (1.3.32), we take the particular test function vh = u θh := θ h k+1 + (1 − h θ )u kh . Since θ u k+1 h

+ (1 −

θ )u kh

   1  k+1  1  k+1 u h − u kh + u h + u kh , = θ− 2 2

we have, for θ ∈ [1/2, 1], 

k L h (u k+1 h ) − uh )

−Δtk =

 , L h (u θh )

+ B(u θh , u θh ; tk+θ )

1 2θ − 1 (L h (u k+1 − u kh ), L h (u k+1 + u kh )) + (L h (u k+1 − u kh ), L h (u k+1 − u kh )) h h h h −2Δtk −2Δtk  1  k+1 2 u h 0,h − u kh 20,h + C0 u θh 2h , +B(u θh , u θh ; tk+θ ) ge −2Δtk

1.3 The Fitted Finite Volume Method (FVM)

19

where C is the constant used in (1.3.30) and  · 0,h is the norm defined in (1.3.29). Therefore we get from (1.3.32)    1  k+1 2 u h 0,h − u kh 20,h + C0 u θh 2h ≤ θ  f k+1 0,h + (1 − θ ) f k 0,h u θh h . −2Δtk  In the above we used the fact that u  h 0,h ≤ u h h . Applying the ε-inequality y2 εz 2 yz ≤ 2ε + 2 for any y, z and ε > 0 to the RHS of the above inequality with ε = 2C0 we have

  1  k+1 2 1  k+1 2 0,h +  f k 20,h , u h 0,h − u kh 20,h ≤ f −Δtk C0 or, 2 k 2 u k+1 h 0,h ≤ u h 0,h +

 −Δtk  k+1 2 f 0,h +  f k 20,h C0

for k = 0, 1, . . . , K − 1. Fro this recursive relationship we obtain u hK 20,h ≤ u 0h 20,h + ≤ u 0h 20,h +

K −1   1  (−Δtk )  f k+1 20,h +  f k 20,h C0 k=0

2T sup  f (s)20,h . C0 s∈(0,T ) 

1.3.3.3

Error Estimate

We now establish an upper bounds for the difference between the approximate and the exact solutions in  · h . We start this discussion with the assumption that the spatial mesh is quasi-uniform as given below. Assumption 1.3.1 There exists a constant q0 > 0 such that q0−1 h i+1 ≤ h i ≤ q0 h i+1 , i = 0, . . . , N − 1. Using this assumption we have the following lemma. Lemma 1.3.1 The discrete flux density ρh defined by (1.3.12), (1.3.14) and (1.3.7) can be written as ⎧ vh1 − vh0 ⎪ ⎨ a(1 + α0 )S1/2 + b1/2 vh0 , i = 0, h0 ρh (vh )| Ii = v − v hi ⎪ ⎩ a(1 + γi ) Si+1 hi+1 + bi+1/2 vhi , i = 1, . . . , N − 1 , hi

20

1 European Options on One Asset h S

αi −1

i+1 where γi := αi S αii −S αi − 1. Furthermore, under Assumption 1.3.1, there exists a i+1 i constant Cγ > 0 depending only on maxi αi and q0 such that

|γi xi+1 | ≤ Cγ h i , i = 1, . . . , N − 1 .

(1.3.33)

Proof Let us first consider the case i = 0. By (1.3.12), ρh (vh )| I0 =

vh1 − vh0 1 1 ah 0 + b1/2 [vh1 + vh0 ] . 2 h0 2

vh1 − vh0 h 0 , replacing vh1 in the last term of the above equah0 h0 tion by this expression yields ρh (vh )| I0 = (a + b1/2 ) h20 vh1h−v + b1/2 vh0 . Finally it 0 remains to observe that a + b1/2 = a(1 + α0 ) and h 0 /2 = S1/2 . αi vhi in the numerator of When i = 1, . . . , N − 1, adding and subtracting Si+1 (1.3.7), we get Since vh1 = vh0 +

 ρh (vh )| Ii = bi+1/2 But

αi −1 h i Si+1 vhi+1 − vhi + vhi . αi αi Si+1 Si+1 − Si hi

αi −1 αi −1 h i Si+1 h i Si+1 bi+1/2 αi = aαi αi = a(1 + γi ) Si+1 − Siαi Si+1 − Siαi

with γi defined in the Lemma. Using a Taylor expansion we have αi = Siαi + αi ξ αi −1 h i , where Si < ξ < Si+1 , Si+1

where ξ < η

αi −1

α −1

i Si+1 −ξ αi −1 . ξ αi −1 αi −1 using the Taylor expansion Si+1 = ξ αi −1 + (αi − 1)ηαi −2 (Si+1 αi −2 , and thus, < Si+1 , we have γi = (αi − 1) (xi+1ξ −ξ ) ηξ

from which we have γi = Similarly,



(1.3.34)

Si+1 ξ

|γi xi+1 | ≤ |αi − 1|

Si+1 ξ

−1=

− ξ ),

 αi −2 η h i with Si < ξ < η < Si+1 . ξ

To estimate the factor (η/ξ )αi −2 , we have to consider the cases αi < 2 and αi ≥ 2 separately. In the former case, we use that ξ < η and so  αi −2  2−αi  2−αi ξ η η = < = 1. ξ η η

1.3 The Fitted Finite Volume Method (FVM)

21

αi −2 αi −2 In the latter case, we use Si < ξ and η < Si+1 to obtain ηξ ≤ SSi+1 . i Since Si−1 ≥ 0 for all i = 1, . . . , N − 1, we have the following estimate: Si + h i hi hi hi Si+1 = =1+ =1+ ≤1+ ≤ 1 + q0 Si Si Si Si−1 + h i−1 h i−1 by Assumption 1.3.1. αi −2 Therefore ηξ ≤ (1 + q0 )αi −2 . Similarly, Si+1 < SSi+1 ≤ 1 + q0 . So we finally ξ i obtain the estimate  αi < 2 , |αi − 1|(1 + q0 )h i , |γi xi+1 | ≤ |αi − 1|(1 + q0 )αi −1 h i , αi ≥ 2 . 

This is (1.3.33), and thus we have proved this lemma. Now we are ready to prove the following consistency result.

1 Lemma 1.3.2 Let w ∈ H0,w (Ω) be such that ρ (w, ·, t) ∈ L 2 (Ω) for all t ∈ (0, T ). Under Assumption (1.3.1), there exists a constant C > 0 depending only on q0 , Cγ and a and b, such that the following estimate holds:

 |ρh (w I , Si+1/2 , t) − ρ(w, Si+1/2 , t)| ≤ C

Si+1



 |ρ (w, S, t)| + |w (S, t)| + |w(S, t)| d S

Si

for i = 0, . . . , N − 1, where w I (·, t) denotes the Uh -interpolant of w(·, t). Proof We let C > 0 denote a generic positive constant, depending only on q0 , Cγ and on certain norms of the coefficients a and b. By Lemma 1.3.1 and (1.3.2), for i = 1, . . . , N − 1, we can write ρh (w I , Si+1/2 , ·) − ρ(w, Si+1/2 , ·) wi+1 − wi =a(1 + γi ) Si+1 − aSi+1/2 w (Si+1/2 ) + bi+1/2 [wi − w(Si+1/2 )] hi   wi+1 − wi − w (Si+1/2 ) + bi+1/2 [wi − w(Si+1/2 )] =aSi+1/2 hi wi+1 − wi + a[Si+1 − Si+1/2 + γi Si+1 ] =: ϑ1i + ϑ2i + ϑ3i . hi To estimate the first term, we use the following Taylor expansion with an integral remainder:  y w(y) = w(Si+1/2 ) + w (Si+1/2 )(y − Si+1/2 ) + w (x)(y − S)d S. Si+1/2

Replacing y by y = Si+1 and Si in the above expression, we have respectively

22

1 European Options on One Asset

wi+1



hi = w(Si+1/2 ) + w (Si+1/2 ) + 2

wi = w(Si+1/2 ) − w (Si+1/2 )



hi − 2

Si+1

w (S)(Si+1 − S)d S

Si+1/2 Si+1/2

w (S)(Si − S)d S.

Si

Therefore, from these equalities we have wi+1 − wi 1 − w (Si+1/2 ) = hi hi



Si+1/2

w (S)(Si − S)d S +

Si

1 hi



Si+1

w (S)(Si+1 − S)d S.

Si+1/2

(1.3.35) Note    h  Si+1  Si+1   i w (S)(Si+1 − S)d S  ≤ |w |d S,    Si+1/2 2 Si+1/2

   

Si+1/2 Si

   h i Si+1/2 w (Si − S)d S  ≤ |w |d S. 2 Si

Combining these estimates with (1.3.35) we obtain     1 xi+1  wi+1 − wi   ≤ − w (x ) |w |d x, i+1/2   hi 2 xi and so |ϑ1i | ≤

a Si+1/2 2



Si+1

|w |d S =

Si

S

a 2



Si+1 Si

 Si+1/2 a q0 Si+1 |Sw |d S ≤ |Sw |d S, 1+ S 2 2 Si

S

where we used i+1/2 ≤ i+1/2 ≤ 1 + 2hSi i ≤ 1 + q20 by Assumption (1.3.1). Since, by S Si the definition of ρ, axw = ρ − (a + b)w − b w, we obtain   c Si+1  1 1+ |ρ | + |a + b||w | + |b ||w| d S 2 2 Si  Si+1   |ρ | + |w | + |w| d S. ≤C

|ϑ1i | ≤

Si

S To estimate ϑ2i , we apply the formula w(xi+1/2 ) − wi = Si i+1/2 w d S, and hence

S

S |ϑ2i | ≤ |bi+1/2 | Si i+1 |w |d S. Similarly, since wi+1 − wi = Si i+1 w d S, we obtain from (1.3.33) in Lemma 1.3.1 that  |ϑ3i | ≤ a

hi + Cγ h i 2



  Si+1  |wi+1 − wi | 1 ≤ a Cγ + |w |d S. hi 2 Si

Putting the above three estimates together, we have the following estimate:  |ρh (w I , Si+1/2 , ·) − ρ(w, Si+1/2 , ·)| ≤ C

Si+1 Si

  |ρ | + |w | + |w| d S.

1.3 The Fitted Finite Volume Method (FVM)

23

When i = 0, we have ρh (w I , S1/2 , ·) − ρ(w, S1/2 , ·) w1 − w0 = a(1 + α0 ) S1/2 − aS1/2 w (S1/2 ) + b1/2 [w0 − w(S1/2 )] h0   w1 − w0 w1 − w0 = aS1/2 − w (S1/2 ) + b1/2 [w0 − w(S1/2 )] + aα0 S1/2 . ! "# $ h0 h0 ! "# $ "# $ ! ϑ20 ϑ30

ϑ10

To estimate ϑ10 , we first proceed as in the general case and get w1 − w0 1 − w (S1/2 ) = h0 h0



S1/2 S0

1 w (S)(S0 − S)d S + h0



S1

w (S)(S1 − S)d S.

S1/2

Therefore    S1/2  S1  w1 − w0  1 1   − w (S1/2 ) ≤ |Sw (S)|d S + |w (S)||S1 − S|d S.  h h 0 S0 h 0 S1/2 0 For S ∈ [S1/2 , S1 ], it is easy to see that  |S1 − S| = S1 − S = S and we obtain

   S1 S1 −1 ≤ S − 1 = S, S S1/2

   S1   w1 − w0 1   ≤ − w (S ) |Sw |d S. 1/2   h h 0 S0 0

It follows that  S1     S1/2 S1 1 S1 |ρ | + |w | + |w| d S. |ϑ10 | ≤ a |Sw |d S = |aSw (S)|d S ≤ C h 0 S0 2 S0 S0

The term ϑ20 can be estimated as in the general case, i.e. |ϑ20 | ≤ |b1/2 | Finally, we have |ϑ30 | = a|α0 |S1/2

|b1/2 | |w1 − w0 | = h0 2



S1

S1 S0

|w |d S.

|w |d S.

S0

In summary, we get  |ρh (w I , S1/2 , ·) − ρ(w, S1/2 , ·)| ≤ C

x1 x0

  |ρ | + |w | + |w| d S.



24

1 European Options on One Asset

We are ready to prove the main convergence result for the FVM in the previous subsections. For simplicity and clarity, we only consider the case that θ = 1, i.e., the fully implicit or backward Euler’s scheme. We also assume that the solution u to Problem 1.2.1 is sufficiently smooth to avoid intensive discussion on the regularity requirements. We will estimate R k := u I (tk ) − u kh in a discrete energy norm to be defined, where u I (·, t) is the Uh -interpolation of u(·, t) defined above. In what follows, we use C > 0 to denote a generic positive constant, independent of both h i and Δtk for all feasible i and k. For any v ∈ C( I¯), multiplying (1.2.12) by L h (v) and integrating the 2nd term by parts, we have (1.3.36) (−u, ˙ L h (v)) + Bˆ h (u, v); t) = ( f, L h (v)), where L h is the operator defined in (1.3.25) and Bˆ h (u, v; t) = −

N −1 

S

[Sρ(u, x, t)] Si+1/2 v(Si ) + (cu, L h (v)). i−1/2

i=1

Adding and subtracting appropriate terms and using (1.3.36) with the test functions v = vh ∈ Uh and (1.3.32), we easily derive the following equation with respect to R k+1 :   L h (R k+1 ) − L h (R k ) , L h (vh ) + Bh (R k+1 , vh ; tk+1 ) −Δtk   L h (u I (tk+1 )) − L h (u I (tk )) , vh + Bh (u I (tk+1 ), vh ; tk+1 ) = −Δtk   k L h (u k+1 h ) − L h (u h ) − , vh − Bh (u k+1 h , vh ; tk+1 ) −Δtk   L h (u I (tk+1 )) − L h (u(tk )) ˙ k+1 ), L h (vh )) + , vh = −(u(t ˙ k+1 ), L h (vh )) + (u(t −Δtk ˆ ˆ + Bh (u(tk+1 ), vh ; tm+1 ) − Bh (u(tk+1 ), vh ; tm+1 ) + Bh (u I (tk+1 ), vh ; tk+1 )   k L h (u k+1 ) − L (u ) h h h − , L h (vh ) − Bh (u k+1 h , vh ; tk+1 ) −Δtk    L h (u I (tk+1 )) − L h (u I (tk )) ˙ k+1 ), L h (vh )) , vh + (u(t = −Δtk "# $ ! Y1k



 + Bh (u I (tk+1 ), vh ; tk+1 ) − Bˆ h (u(tk+1 ), vh ; tk+1 ) "# $ ! 

+ (f !

Y2k

k+1

 L h (vh )) − (L h ( f k+1 ), L h (vh )) =: Y1k + Y2k + Y3k . "# $ Y3k

(1.3.37)

1.3 The Fitted Finite Volume Method (FVM)

25

Now we estimate the terms |Y jk |, j = 1, 2, 3 separately. (i) Estimation of |Y1k |: ))−L h (u(tk )) + u(t ˙ k+1 )k. Then, by Cauchy–Schwarz inequality, Let wk := L h (u(tk+1−Δt k k we have |Y1 | ≤ wk 0 vh 0,h . Thus, we need to derive a bound for wk 0 . A simple algebraic manipulation yields wk =

 1  (L h (u(tk+1 ) − u(tk+1 )) − (L h (u(tk )) − u(tk )) −Δtk 1 + (u(tk+1 ) − u(tk )) + u(t ˙ k+1 ). −Δtk

(1.3.38)

To estimate the first term on the RHS of (1.3.38), we use a Taylor expansion with integral remainder for it to obtain 

 1  1 (L h (u(tk+1 )) − u(tk+1 ) − (L h (u(tk )) − u(tk ) = −Δtk −Δtk

tk+1 tk

d [(L h (u(s)) − u(s)]ds, ds

and so % %  tk % 1  % 1 % % L h (u(s)) ˙ − u(s) ˙ 0 ds. % −Δt (L h (u(tk+1 )) − u(tk+1 ) − (L h (u(tk )) − u(tk ) % ≤ −Δt k k tk+1 0

For the the second term on the RHS of (1.3.38), we have, using a Taylor expansion with an integral remainder, 

tk

˙ k+1 )(−Δtk ) + u(tk ) = u(tk+1 ) + u(t

(tk − s)u(s)ds, ¨

tk+1

from which, we get 1 1 (u(tk+1 ) − u(tk )) + u(t ˙ k+1 ) = −Δtk −Δtk



tk+1

(tk − s)u(s)ds. ¨

tk

Taking the norm on both sides, we have the following estimate for the second term on the RHS of (1.3.38): % %  % 1 % % % ≤ (u(t ) − u(t )) + u(t ˙ ) k+1 k k+1 % −Δt % k

0

tk

u(s) ¨ 0 ds.

tk+1

Combining the estimates for the two terms on the RHS of (1.3.38) and using the triangular inequality, we obtain the following estimate for |Y1k |: |Y1m | ≤ wm 0 vh 0,h ≤ Q m 1 (Δtk , h)vh 0,h , where

(1.3.39)

26

1 European Options on One Asset

Q k1 (Δtk , h) :=

1 −Δtk





tk

L h (u(s)) ˙ − u(s) ˙ 0 ds +

tk+1

tk

u(s) ¨ 0 ds.

(1.3.40)

tk+1

(ii) Estimation of |Y2k |: From the definitions of Bh and Bˆ h in (1.3.28) and (1.3.36) respectively we have Y2k = Bh (u I (tk+1 ), vh ; tk+1 ) − Bˆ h (u(tk+1 ), vh ; tk+1 ) =−

N −1  

S[ρh (u I , S, tk+1 ) − ρ(u, S, tk+1 )]

 Si+1/2 Si−1/2

vhi + (L h (cu I ) − cu, L h (vh )),

i=1

where vhi = vh (Si ). Re-arranging the first sum and using the boundary conditions vh0 = vh N = 0, we get |Y2k |

 N −1      ≤ Si+1/2 [ρh (u I , Si+1/2 , tk+1 ) − ρ(u, Si+1/2 , tk+1 )](vhi − vhi+1 )   i=0

+|(L h (cu I ) − cu, L h (vh ))| =: δ1 + δ2 . For δ1 , we have δ1 ≤ |ρh (u I , S1/2 , tk+1 ) − ρ(u, S1/2 , tk+1 )|S1/2 |vh0 − vh1 | +

N −1 

|ρh (u I , Si+1/2 , tk+1 ) − ρ(u, Si+1/2 , s)|Si+1/2 |vhi − vhi+1 |.

(1.3.41)

i=1

Since S1/2 = h 0 /2 and vh0 = 0, we can write |ρh (u I , S1/2 , tk+1 ) − ρ(u, S1/2 , tk+1 )|S1/2 |vh0 − vh1 |  S1  &   |ρ | + |w | + |w| d S |vh1 | ≤ Ch 0 ≤ Ch 0 S0

S1



2 |ρ | + |w | + |w| d S

S0

&1/2 '

2 . h 0 vh1

(1.3.42)

The estimate of the remaining sum in (1.3.41) is more completed. We start with the consideration of the terms Si+1/2 |vhi − vhi+1 | for i = 1, . . . .N − 1. By a simple algebraic manipulation, it holds 1/2 αi Si+1 − Siαi × αi Si+1 + Siαi 1/2  S αi + Siαi × Si+1/2 bi+1/2 i+1 |vhi − vhi+1 |. αi Si+1 − Siαi 

Si+1/2 |vhi − vhi+1 | =

Si+1/2 bi+1/2

1/2 

(1.3.43)

αi − Siαi = αi ξ αi −1 h i , ξ ∈ Now, using the Taylor expansion (1.3.34), we have Si+1 (Si , Si+1 ). Hence

1.3 The Fitted Finite Volume Method (FVM)

27

αi Si+1 − Siαi ξ αi αi h i . αi αi = αi Si+1 + Si ξ Si+1 + Siαi

If αi < 0, then ξ αi < Siαi , and so

ξ αi αi α Si+1 +Si i

≤

α

Si i αi α Si+1 +Si i

(1.3.44)

≤ 1. α

i Si+1 αi α Si+1 +Si i

αi

αi Analogously, if αi ≥ 0, then ξ αi < Si+1 . Thus, S αi ξ +S αi ≤ i+1 i Combining these estimates with (1.3.44), we have

≤ 1.

 αi   Si+1 − Siαi  |αi |h i Si+1/2 |αi |h i q0 |αi |h i  α  ≤ 1+ .  S i + S αi  ≤ ξ = ξ S 2 Si+1/2 i+1/2 i+1 i So, the 1st product on the RHS of (1.3.43) can be estimated as follows:  αi 

− Siαi  Si+1/2  Si+1  ≤ 1 + c h i ≤ Ch i , α α i |bi+1/2 |  Si+1 + Si i  2 a where C > 0 denotes a generic positive constant, independent of h i ’s and Δtk ’s. Using this estimate, we have from (1.3.43) N −1 

|ρh (u I , Si+1/2 , tk+1 ) − ρ(u, Si+1/2 , tk+1 )|Si+1/2 |vhi − vhi+1 |

i=1

≤C

N −1 ( 

 hi

N −1 



N −1  Si+1 

 N −1   



& Si+1/2 bi+1/2

2  |ρ | + |w | + |w| d S

&1/2 

αi Si+1 + Siαi

 2 |ρ | + |w | + |w| d S

SN

xi+1

2  |ρ | + |w | + |w| d S

&1/2  Si+1/2 bi+1/2 )1/2  N −1 

Si

2  |ρ | + |w | + |w| d S

|vhi − vhi+1 |

αi Si+1 + Siαi αi Si+1

−

αi Si+1

Si+1/2 bi+1/2

−

1/2

Siαi

αi Si+1 + Siαi

i=1

&1/2

1/2

αi Si+1 − Siαi

Si+1/2 bi+1/2

Si

i=1

= Ch

 |ρ | + |w | + |w| d S

Si

i=1

≤ Ch

Si+1

hi

i=1

≤ Ch



Si

i=1

≤C

Si+1

|vhi − vhi+1 |

1/2

Siαi

αi Si+1 + Siαi αi Si+1 − Siαi

|vhi − vhi+1 | )1/2 |vhi − vhi+1 |2

vh 1,h .

S1

Together with the estimate (1.3.42) we have (recalling Assumption (1.3.1)) 

 2 |ρ | + |w | + |w| d S

S1

δ1 ≤ Ch 0

&1/2 '

S0



SN

+ Ch  ≤ Ch

 2 |ρ | + |w | + |w| d S

S1 SN

 2 |ρ | + |w | + |w| d S

S0

≤ Ch (|ρ|1 + w1 ) vh h ,

2 h 0 vh1

&1/2

&1/2

vh 1,h * 2 +1/2 l1 vh1 + vh 21,h

28

1 European Options on One Asset

where |w|1 = w 0 denotes the 1st-order semi-norm of w. For the second term δ2 , we have by a standard argument δ2 ≤ Ch (|c(s)|1 + |w|1 ) vh 0,h . Combining the bounds on δ1 and δ2 yields |Y2k | ≤ Ch (|c(s)|1 + |ρ|1 + w1 ) vh h . (iii) Estimation of |Y3k |: For Y3k , we simply have |Y3k |

= |( f

k+1

− Lh( f

k+1

), L h (vh )) =

N −1  i=1

≤ Ch| f |1

N −1 



Si+1/2

vhi Si−1/2



 f (S, tk+1 ) − f ik+1 d S

vhi li ≤ Ch| f |1 vh 0,h .

(1.3.45)

i=1

Combining the estimates (1.3.39)–(1.3.45), we get from (1.3.37) 

 L h (R k+1 ) − L h (R k ) , L h (vh ) + Bh (R k+1 , vh ; tk+1 ) −Δtk  k   ≤ C Q 1 + h| f |1 vh 0,h + h (|c|1 + |ρ|1 + u1 ) v j h ≤ Q k (Δtk , h)vh h , (1.3.46)

because v0,h ≤ vh h , where Q k1 is defined in (1.3.40) and  Q k (Δtk , h) := C Q k1 (Δtk , h) + (|c|1 + |ρ|1 + u1 + | f |1 )h .

(1.3.47)

We now choose in (1.3.46) the particular test function: vh = R k+1 . Applying the same argument as in the proof of Theorem 1.3.4 (with θ = 1), we get the following estimate:   L h (R k+1 ) − L h (R k ) , L h (R k+1 ) + Bh (R k+1 , R k+1 ; tk+1 ) −Δtk  1 k+1 2 ≥ R 0,h − R k 20,h + C0 R k+1 2h . −2Δtk Combining this with (1.3.46) leads to  1 k+1 2 R 0,h − R k 20,h + C0 R k+1 2h ≤ Q k (Δtk , h)vh h . −2Δtk Applying the ε-inequality given above to the RHS of the above with ε = C0 , we get  1 k+1 2 1 C0 k+1 2 R h , R 0,h − R k 20,h + C0 R k+1 2h ≤ [Q k (Δtk , h)]2 + −2Δtk 2C0 2 and thus it follows that

1.3 The Fitted Finite Volume Method (FVM)

29

 C0 k+1 2 1 k+1 2 1 R h ≤ R 0,h − R k 20,h + [Q k (Δtk , h)]2 . −2Δtk 2 2C0 Multiplying this inequality by −2Δtk (> 0), we obtain R k+1 20,h − R k 20,h − C0 Δtk R k+1 2h ≤

−Δtk k [Q (Δtk , h)]2 . C0

Summing up both sides of the above from k = 0 to k = K − 1 and using the definition of Q k in (1.3.47) we have R K 20,h + C0

K −1 

(−Δtk )R k+1 2h ≤

k=0

K −1 1  (−Δtk )[Q k (Δtk , h)]2 C0 k=0

 K −1   ≤C (−Δtk )[Q k1 (Δtk , h)]2 + (|c|1 + |ρ|1 + u1 + | f |1 )2 h 2 T . (1.3.48) k=0

It remains to estimate the sum on the RHS of (1.3.48). From (1.3.40) we get [Q k1 (Δtk , h)]2   tk &2  tk &2 1 ≤2 L ( u(s)) ˙ − u(s) ˙ ds +  u(s) ¨ ds h 0 0 (Δtk )2 tk+1 tk+1    tk  tk 1 2 2 (L h − I )u(s) ˙ ds − Δt  u(s) ¨ ds , ≤2 k 0 0 −Δtk tk+1 tk+1 and, therefore, K −1 

(−Δtk )[Q k1 (Δtk , h)]2 ≤ 2

k=0

K −1  tk  tk+1

k=0



T

≤2 0

where Δt :=

max

k=0,...,K −1

 2 2 L h (u(s)) ˙ − u(s) ˙ 0 ds + (Δtk )

2 2 L h (u(s)) ˙ − u(s) ˙ 0 ds + (Δt)



T 0

tk tk+1

 2 u(s) ¨ 0 ds

 2 u(s) ¨ ds , 0

|Δtk |. By a standard argument, L h (u(s)) ˙ − u(s) ˙ 0 ≤

h|u(s)| ˙ 1 , and so we arrive at K −1 

  (−Δtk )[Q k1 (Δtk , h)]2 ≤ 2 h 2 u ˙ 2L 2 (0,T ;H 1 (I )) + (Δt)2 u ¨ 2L 2 (0,T ;L 2 (I )) .

k=0

(1.3.49) Replacing the sum in (1.3.48) by the upper bound in (1.3.49), we finally get the estimate

30

1 European Options on One Asset

R K 20,h + C0

K −1 

(−Δtk )R k+1 2h ≤ C(h 2 + Δt 2 ).

(1.3.50)

k=0

To summarize, we have proven, in the above, the following theorem: Theorem 1.3.5 Let u and {u kh ∈ Uh : k = 1, 2, . . . , K } be respectively the solutions to Problems 1.3.2 and 1.3.3 with θ = 1 and assume (1.3.1) holds. If u is such that is sufficiently smooth, then there exists a positive constant C > 0, independent of h and Δt, such that u I (t0 ) −

u hK 0,h

+

 K −1 

1/2 |Δtk |u I (tk ) −

u kh 2h

≤ C(h + Δt),

(1.3.51)

k=0

where u I (tk ) denotes the Uh -interpolant of u(·, tk ) for k = 0, 1, . . . , K . Proof The estimate (1.3.51) follows immediately from (1.3.50).



Remark 1.3.3 Since  · 0,h in (1.3.51) depends only on the nodal values of a given function, Theorem 1.3.5 still holds if we replace u I by u, as both of them have the same nodal values. We also comment, in Theorem 1.3.5 we used a ‘vague’ assumption that ‘u is sufficiently smooth’ to avoid introducing more spaces and norms.

1.4 Numerical Experiments We now present some numerical examples to demonstrate the usefulness of the finite volume method developed in the previous sections. For all the examples given below, we choose Smax = 100 and T = 1. The uniform mesh with 51 × 51 mesh nodes is used for solving all the tests below. Example 1.1 European call option with K = 50 and market parameters σ = 0.4, r = 0.03, d = 0.02. The payoff and boundary conditions are given in (1.2.6) and (1.2.9)–(1.2.10) respectively. The computed option value and its derivative with respect to S (i.e., Δ) are plotted in Fig. 1.2. Example 1.2 European call option with K = 50. The market parameters are chosen to be σ = 0.4 + 0.2 sin(10t), r = 0.06 + 0.02 sin(10t) and d = 0.0005S, and the payoff and boundary conditions are given in (1.2.6)–(1.2.10). The computed option value V and the corresponding Δ are plotted in Fig. 1.3. Example 1.3 European call option with the market parameters σ = 0.4, r = 0.04, d = 0.02S and strike price K = 50. This is a Cash-Or-Nothing option and its payoff condition is given in (1.2.7) with B = 1. The boundary conditions are V (0, t) = 0 and V (Smax , t) := exp(−r (T − t)) for t ∈ [0, T ).

1.4 Numerical Experiments

31

Fig. 1.2 Computed value V and Δ for Example 1.1

Fig. 1.3 Computed value V and Δ for Example 1.2

Fig. 1.4 Computed value V and Δ for Example 1.3

The computed option value V and Δ are plotted in Fig. 1.4. As can be seen from Fig. 1.4, the derivative of V does not exist at S = 50.

32

1 European Options on One Asset

Fig. 1.5 Computed value V and Δ for Example 1.4

Example 1.4 Our last test is the European call option with the bullish vertical spread payoff (1.2.8) where K 1 = 40 and K 2 = 55. The market parameters chosen for this test are σ = 0.4, r = 0.04 and d = 0.02S. The boundary conditions are chosen to be V (0, t) = 0 and V (Smax , t) := K 2 − K 1 for t ∈ [0, T ). The computed option value and its derivative with respect to S and Δ are plotted in Fig. 1.5.

References 1. Achdou Y (2005) An inverse problem for a parabolic variational inequality arising in volatility calibration with American options. SIAM J Control Optim 43:1583–1615 2. Achdou Y, Pironneau O (2005) Computational methods for option pricing. SIAM, New York 3. de Allen DN, G, Southwell RV, (1955) Relaxation methods applied to determine the motion, in two dimensions, of a viscous fluid past a fixed cylinder. Quart J Mech Appl Math 8:129–145 4. Angermann L, Wang S (2007) Convergence of a fitted finite volume method for the penalized Black-Scholes equations governing European and American option pricing. Numer Math 106:1–40 5. Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81:637– 659 6. Brenner SC, Scott LR (1994) The mathematical theory of finite element methods. Springer, New Yok 7. Courtadon G (1982) A more accurate finite difference approximation for the valuation of options. J Financ Econ Q Anal 17:697–703 8. Cox JC, Ross S, Rubinstein M (1979) Option pricing: a simplified approach. J Financ Econ 7:229–264 9. Dewynne JN, Howison SD, Rupf I, Wilmott P (1993) Some mathematical results in the pricing of American options. Euro J Appl Math 4:381–398 10. Duffy D (2006) Finite difference methods in financial engineering - a partial differential equation approach. Wiley, New Work 11. Itô K (1951) On stochastic differential equations. Mem Amer Math Soc 4:1–51 12. Haslinger J, Miettinen M (1999) Finite Element Method for Hemivariational Inequalities. Kluwer Academic Publisher, Dordrecht

References

33

13. Hull J, White A (1996) Hull-White on derivatives. Risk Publications, London 14. Kufner A (1985) Weighted Sobolev spaces. Wiley, New York 15. Miller JJH, Wang S (1994) A new non-conforming Petrov-Galerkin method with triangular elements for a singularly perturbed advection-diffusion problem. IMA J Numer Anal 14:257– 27 16. Miller JJH, Wang S (1994) An exponentially fitted finite element volume method for the numerical solution of 2D unsteady incompressible flow problems. J Comput Phys 115:56–64 17. Renardy M, Rogers RC (2004) An introduction to partial differential equations. Texts in applied mathematics, 2nd edn., vol. 13. Springer, New York 18. Rogers LCG, Tallay D (1997) Numerical methods in finance. Cambridge University Press, Cambridge 19. Varga RS (1962) Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs 20. Vazquez C (1998) An upwind numerical approach for an American and European option pricing model. Appl Math & Comput 97:273–286 21. Wang S (2004) A novel fitted finite Volume method for the Black-Scholes equation governing option pricing. IMA J Numer Anal 24:699–720 22. Wilmott P, Dewynne J, Howison J (1993) Option pricing: mathematical models and computation. Oxford Financial Press, Oxford

Chapter 2

American Options on One Asset

Abstract In this chapter, we first give a brief account of the derivation of the differential Linear Complementarity Problem (LCP) governing American put option valuation. We then write this LCP as a variational inequality, which is shown to be uniquely solvable. A partial differential equation with a nonlinear penalty term is proposed to approximate the LCP. We prove that the penalty equation is uniquely solvable and its solution converges to the weak solution to the LCP exponentially. The finite volume method in Chap. 1 is used for the penalty equation. A Newton’s algorithm is proposed to solve the discretized nonlinear system. Numerical experimental results are presented to demonstrate this method produces financially meaningful numerical solutions to the American put option pricing problem. Keywords American option valuation · Linear complementarity problem · Variational inequality · Power penalty method · Convergence.

2.1 The Differential LCP and Its Solvability An American option is one which can be exercised any time prior to or on the maturity/expiry date T . Therefore, a question that arises is what is the optimal time to exercise an American option? A solution to an American option pricing problem contains two components—the the option value and optimal exercise curve which divides the solution domain in space and time into two sub-domains, so that in one sub-domain, the option should be exercised and in the other, it should be held. An American call option has the same value as its European counterpart. An intuitive explanation for this is that, at time t < T , the option should be held if its holder thinks the underlying stock price will increase. On the other hand, if the option holder thinks the stock price will decrease, he/she can short the stock without exercising the option, and the risk of this shorting is covered by the call option. Therefore, the option should be exercised only at expiry date T , as for its European counterpart. In what follows, we will consider American put options only. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 S. Wang, The Fitted Finite Volume and Power Penalty Methods for Option Pricing, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-15-9558-5_2

35

36

2 American Options on One Asset

2.1.1 The Differential LCP Consider an American put option (put) with strike price K , maturity T and payoff V ∗ (S) := max(K − S, 0), under the market assumptions in Sect. 1.2. Let us consider from the option holder’s point of view the hedging problem: for every American put on a stock you buy, from how many shares of the stock you can neutralize the risk. Equivalently, let us consider a riskless portfolio Π consisting of one put option and δ shares of the underlying stock. At t, the value of the portfolio is Π = δS + V (S, t), where S denotes the price of the stock and V the value of the option. Using the same argument in Sect. 1.2.1, we have that S and V should satisfy (1.1.4) and      1 2 2 ∂2 V ∂V ∂V ∂V + σ S dt + σ S δ + dW. + μS dΠ = μSδ + ∂t 2 ∂ S2 ∂S ∂S Note that ∂∂VS ≤ 0 for a put. Since we expect Π is riskless, the coefficient of dW should vanish, implying δ = − ∂∂VS . Thus, the above equality becomes  dΠ =

∂V 1 ∂2 V + σ 2 S2 2 ∂t 2 ∂S

 dt.

(2.1.1)

We expect that, if the option is exercised at the optimal time, this portfolio has the riskless return rate r , that is dΠ = r Π dt = r (−S ∂∂VS + V )dt. However, if the option is exercised not optimally, Π should rate from  less than r . Thus,  have a return ∂V 1 2 2 ∂2 V ∂V (2.1.1) and the above analysis we have ∂t + 2 σ S ∂ S 2 ≤ r −S ∂ S + V , or 1 ∂2 V ∂V ∂V + σ 2 S2 2 + r S − r V ≤ 0. ∂t 2 ∂S ∂S

(2.1.2)

This inequality becomes an equation if the option is exercised at the optimal time. Furthermore, at any t, if the spot price S < K , the holder of the American put can buy a share of the stock from the share market and exercise the put so that the holder makes a profit of K − S. On the other hand, V (S, t) ≥ 0. Thus, combining these two cases, we have V (S, t) ≥ V ∗ (S) := max(0, K − S), where V ∗ (S) is called the intrinsic value of the American put option. From the above analysis, we see that there is an optimal exercise curve Sopt (t) such that when S(t) > Sopt (t) the option should be held and (2.1.2) becomes an equation, as Π has the return rate r . When S(t) < Sopt (t), it should be exercised, as otherwise the return rate of Π is smaller than r and the option value falls below V ∗ (S). Combining these with (2.1.2), we see V (S, t) satisfies

2.1 The Differential LCP and Its Solvability

37

 1 2 2 ∂2 V ∂V ∂V + σ S − r V ≥ 0, , V − V ∗ ≥ 0, − + rS ∂t 2 ∂ S2 ∂S   ∂V ∂V 1 2 2 ∂2 V + rS − + σ S − r V (V − V ∗ ) = 0 ∂t 2 ∂ S2 ∂S 

(2.1.3) (2.1.4)

for (S, t) ∈ (0, ∞) × [0, T ) with V (S, T ) = V ∗ (S). This is an LCP.

2.1.2 The Variational Inequality The LCP (2.1.3)–(2.1.4) is defined on an infinite domain. In computation, we usually truncate this infinite domain into I × [0, T ), where I = (0, Smax ) for an Smax  K . Let us consider the following general LCP governing a American put option valuation: L V (S, t) ≥ 0, V (S, t) ≥ V ∗ (S),

L V (S, t) · (V (S, t) − V ∗ (S)) = 0 (2.1.5)

for (S, t) ∈ I × [0, T ) with the payoff condition V (S, T ) = V ∗ (S), where L is the differential operator defined in (1.2.3). The above LCP is equivalent to the HamiltonJacobi-Bellman equation min{L V, V − V ∗ } = 0 for (S, t) ∈ I × [0, T ). We now define boundary conditions for V (S, t). When Smax is sufficiently large, we have V (Smax , t) = 0. When S → 0+ , it is obvious that the option should be exercised and the payoff is V (0, t) = V ∗ (0) = K . Clearly, (2.1.5) and these boundary and payoff conditions determine the value of an American put with variable (but not stochastic) volatility, interest rate and dividend, which contain (2.1.3)–(2.1.4) as a special case.   S K , a special case of (1.2.11). Clearly, V0 (S) satisfies the Let V0 (S) = 1 − Smax same boundary conditions as V (S, t) does. Introduce a new variable u(S, t) = −eβt (V (S, t) − V0 (S)),

(2.1.6)

where β > 0 is a constant to be determined later. Using this transformation and the operator L defined in (1.2.12), we rewrite (2.1.5) as the following LCP: L u ≤ f, u − u ∗ ≤ 0, (L u − f ) (u − u ∗ ) = 0, (S, t) ∈ I × [0, T ), (2.1.7) where f (t) = eβt L V0 (S) and u ∗ (x) = eβt (V0 (S) − V ∗ (S)). It is easily seen ⎧  ⎨ eβt 1 −  u ∗ (S, t) = eβt (V0 (S) − V ∗ (S)) = ⎩ eβt 1 −



K Smax  S Smax

S, 0 ≤ S ≤ K , K , K < S ≤ Smax .

From (2.1.6) we see that the boundary and payoff conditions for (2.1.7) are

(2.1.8)

38

2 American Options on One Asset

u(0, t) = 0 = u(Smax , t) t ∈ [0, T ) and u(S, T ) = u ∗ (S, T ). 1 1 (I ) : v ≤ u ∗ }, where H0,w (I ) is the Sobolev space defined Let K = {v ∈ H0,w 1 in Sect. 1.2.3. It is easy to verify that K is a convex and closed subset of H0,w (I ). Using K , we define the following problem.

Problem 2.1.1 Find u(t) ∈ K such that, for all v ∈ K ,   ∂u(t) − , v − u(t) + A(u(t), v − u(t); t) ≥ ( f (t), v − u(t)) ∂t

(2.1.9)

 a.e. in [0, T ), where A(u, v; t) = aS 2 ∂∂uS + bSu, ∂∂vS + (cu, v) as in (1.2.18). For this variational inequality problem, we have the following theorem. Theorem 2.1.1 Problem 2.1.1 is the variational form corresponding to the linear complementarity problem (2.1.7). Proof Note that w − u ∗ (t) ≤ 0 a.e. on I × [0, T ) for any w ∈ K . Multiplying both sides of L u ≤ f in (2.1.7) by w − u ∗ for an arbitrary w ∈ K and integrating the second term by parts as in the proof of Theorem 1.2.1, we obtain   ∂u − , w − u ∗ + A(u, w − u ∗ ; t) ≥ ( f, w − u ∗ ), t ∈ [0, T ) a.e.. ∂t

(2.1.10)

1 Since K is a convex subset of H0,w (I ), we may write w as w = θ v + (1 − θ )u(t), where v ∈ K and θ ∈ [0, 1]. Therefore, (2.1.10) becomes

    ∂u(t) ∂u(t) , u(t) − u ∗ (t) + − , θ (v − u(t)) + A(u(t), u(t) − u ∗ (t); t) − ∂t ∂t +A(u(t), θ (v − u(t)); t) ≥ ( f (t), u(t) − u ∗ (t)) + ( f (t), θ (v − u(t))). (2.1.11) Integrating (L u − f ) (u − u ∗ ) = 0 in (2.1.7) by parts, we have  −

 ∂u(t) , u(t) − u ∗ (t) + A(u(t), u(t) − u ∗ (t); t) = ( f (t), u(t) − u ∗ (t)). ∂t

The difference between (2.1.11) and the above equality is   ∂u(t) − , θ (v − u(t)) + A(u(t), θ (v − u(t)); t) ≥ ( f (t), θ (v − u(t))). ∂t Since θ ≥ 0, eliminating it from the above inequality we have (2.1.9).



For Problem 2.1.1 we have the following theorem. Theorem 2.1.2 Let Assumption (1.2.1) be fulfilled. Then, Problem 2.1.1 has a unique solution in the convex set K defined above.

2.1 The Differential LCP and Its Solvability

39

Proof When Assumption (1.2.1) is satisfied by σ, r, D and β in (2.1.6), it has been 1 (I ), there exist positive shown in the proof of Theorem 1.2.1 that, for any v, w ∈ H0,w constants C and M, independent of v and w, such that A(v, v; t) ≥ C v 21,w ,

A(v, w; t) ≤ M v 1,w w 1,w ,

(2.1.12)

i.e., A(·, ·; t) is coercive and Litschitz continuous with respect to any of its two variables. Therefore, the unique solbability of Problem 2.1.1 is just a consequence of (2.1.12) and [8, Theorem 1.33], in which the unique solvability for a general variational inequality problem is established. 

2.2 The Penalty Method and Its Convergence Analysis Numerical solution of variational inequalities of the form (2.1.9) (or (2.1.3)–(2.1.4)) has been discussed extensively in the open literature (e.g., [6, 8]). Numerical methods for pricing American options have been proposed by numerous authors (e.g., [1, 4, 7, 10–12]). Since (2.1.3)–(2.1.4) is closely related to a constrained optimization problem, optimization techniques are expected to be used for it. In recent years, linear penalty methods have been used successfully for LCPs including American option pricing problems (e.g., [3, 5, 6]). A power penalty method for (2.1.3)–(2.1.4) has been proposed and analyzed in [13].

2.2.1 The Power Penalty Equation and Its Solvability Let us consider the following simple constrained optimization or obstacle problem:  find u ∈ H01 (I ) such that u = arg inf v∈H01 (I ),v≤0 I 21 |∇v|2 − f v d S. Its solution u satisfies [9]: (∇u, ∇(v − u)) ≥ f (v − u), ∀v ∈ {w ∈ H01 (I ) : w ≤ 0}. The above constrained optimization problem can be approximated by   inf

v∈H01 (I )

I

 1 λ 1+1/κ 2 d S, |∇v| − f v + [v] 2 1 + 1/κ +

where [z]+ = max{z, 0} for any function z, and κ > 0 and λ > 1 are positive constants. From Calculus of Variation we see that the optimality condition (Euler– 1/κ Lagrange equation) for this optimization problem is −∇ 2 v + λ[v]+ = f. Motivated by the above example, we propose to approximate (2.1.9)/(2.1.7): by the following nonlinear equation system: L u λ (S, t) + λ[u λ (S, t) − u ∗ (S, t)]+ = f (S, t), (S, t) ∈ I × [0, T ), (2.2.1) u λ (0, t) = 0 = u λ (Smax , t) and u λ (S, T ) = u ∗ (S, T ), (2.2.2) 1/κ

40

2 American Options on One Asset

where λ > 1 and κ > 0 are parameters. In (2.2.1), λ[u λ (S, t) − u ∗ (S, t)]+ is the penalty term which penalizes the positive part of u λ − u ∗ . Using the argument for deducing (1.2.17), we have the variational problem corresponding to (2.2.1)–(2.2.2) as follows. 1/κ

1 1 (I ) such that, for all v ∈ H0,w (I ), Problem 2.2.1 Find u λ (t) ∈ H0,w

    ∂u λ 1/κ − , v + A(u λ , v; t) + λ [u λ − u ∗ ]+ , v = ( f, v), a.e. in (0, T ). ∂t (2.2.3) For any Hilbert space H (I ), we let L p (0, T ; H (I )) denote the space defined by L p (0, T ; H (I )) = {v(·, t) : v(·, t) ∈ H (I ) a.e. in (0, T ); v(·, t) H ∈ L p ((0, T ))}, where 1 ≤ p ≤ ∞ and · H denotes the natural norm on H (I ). The norm on this space is denoted by · L p (0,T ;H ) , i.e., 

T

v L p (0,T ;H (I )) = 0

1/ p p v(·, t) H dt

.

(2.2.4)

Clearly, L p (0, T ; L p (I )) = L p (I × (0, T )) = L p (Ω). Using this space, we have the following theorem. 1 (I ). Theorem 2.2.1 Problem 2.2.1 has a unique solution in H0,w

Proof First, from (1.2.11), we see f (S, t) = eβt L V0 is sufficiently smooth in (S, t). We now prove this theorem by showing that the variational form of the nonlinear operator on the LHS of (2.2.1) is strongly monotone and continuous. For any 1 (I ) with v1 (T ) = v2 (T ) = u ∗ (S, T ), let e = v1 − v2 . Then, using v1 (t), v2 (t) ∈ H0,w integration by parts, we have   1/κ 1/k (L e, e) + λ [v1 − u ∗ ]+ − [v2 − u ∗ ]+ , e   ∂e 1/κ 1/κ = − , e + A(e, e; t) + λ([v1 − u ∗ ]+ − [v2 − u ∗ ]+ , e). ∂t 1/κ

(2.2.5)

1/κ

By definition, [v]+ = (max{v, 0})1/κ . Clearly, [v]+ is monotone in v, and so   1/κ 1/κ λ [v1 − u ∗ ]+ − [v2 − u ∗ ]+ , e ≥ 0. Integrating both sides of (2.2.5) from 0 to T and using the above inequality and (2.1.12), we have

2.2 The Penalty Method and Its Convergence Analysis



41

   1/κ 1/κ (L e(τ ), e(τ )) + λ [v1 (τ ) − u ∗ (τ )]+ − [v2 (τ ) − u ∗ (τ )]+ , e(τ ) dτ 0   T  T  ∂e(τ ) 1/κ − A(e(τ ), e(τ )) + λ([v1 (τ ) − u ∗ (τ )]+ = , e(τ ) dτ + ∂τ 0 0   T  T  ∂(τ ) 1/κ ∗ − , e(τ ) dτ + C − [v2 (τ ) − u (τ )]+ , e(τ )) dτ ≥ e(τ ) 21,w dτ. ∂t 0 0 (2.2.6) T

For any t ∈ (0, T ), using integrating by parts we have  t

T

    T ∂e(τ ) ∂e(τ ) − − , e(τ ) dτ = (e(t), e(t)) − , e(τ ) dτ, ∂τ ∂τ t

because e(T ) = 0. From this, it follows that  t

T

  ∂e(τ ) 1 − , e(τ ) dτ = (e(t), e(t)) ≥ 0. ∂τ 2

(2.2.7)

Therefore, from (2.2.6), (2.2.7) and (2.2.4), we get 

T 0

   1/κ 1/κ (L e, e) + λ [v1 − u ∗ ]+ − [v2 − u ∗ ]+ , e dτ ≥ C e 2L 2 (0,T ;H 1

0,w (I ))

with e = v1 − v2 . This implies that the operator on the right-hand side of (2.2.1) is strongly monotone. 1 (I )), it is easy to show using the 2nd Moreover, for any v, w ∈ L 2 (0, T ; H0,w inequality in (2.1.12) and a standard argument that  0

T

[A(v, w; t)] dt ≤ C v L 2 (0,T ;H 1 0,w(I )) w L 2 (0,T ;H 1 0,w(I )) .

Also, it is easily seen that ([v − u ∗ ]+ , w) is also continuous in both v and w. Therefore, using the result in [8, p. 37], we have that Problem 2.2.1 is uniquely solvable.  1/κ

2.2.2 Convergence Analysis Since Problems 2.1.1 and 2.2.1 are not equivalent, it is necessary to prove that the solution to Problem 2.2.1 converges to that of Problem 2.1.1 as λ → ∞ in a proper norm. We shall prove this in two different stages. In Stage 1, we establish an error bound for [u λ − u ∗ ]+ . In Stage 2, we use this bound to derive a bound for u λ − u . We begin this analysis with the following lemma.

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2 American Options on One Asset

Lemma 2.2.1 Let u λ be the solution to Problem 2.2.1. If u λ ∈ L p (Ω), then there exists a positive constant C, independent of u λ and λ, such that C , λκ C ≤ κ/2 , λ

[u λ − u ∗ ]+ L p (Ω) ≤ 1 [u λ − u ∗ ]+ L ∞ (0,T ;L 2 (I )) + [u λ − u ∗ ]+ L 2 (0,T ;H0,w (I ))

(2.2.8) (2.2.9)

where p = 1 + 1/κ. Proof Let C be a generic positive constant, independent of u λ and λ. In what follows, we use this single constant C in our bound estimations. To simplify the notation, we 1 (I ) a.e. in (0, T ) . introduce φ(·, t) = [u λ (·, t) − u ∗ (·)]+ ∈ H0,w Since v in (2.2.3) is arbitrary, replacing it with φ gives   ∂u λ − , φ + A(u λ , φ; t) + λ(φ 1/κ , φ) = ( f, φ) a.e. in (0, T ). ∂t ∗

Subtracting −( ∂u , φ) + A(u ∗ , φ; t) from both sides of the above equation gives ∂t  −

 ∗   ∂u ∂(u λ − u ∗ ) , φ + A(u λ − u ∗ , φ; t) + λ(φ 1/κ , φ) = ( f, φ) + , φ − A(u ∗ , φ; t). ∂t ∂t

Integrating this from t to T and using (2.1.8) and Hölder inequality yield 

T

t

   T  T ∂(u λ − u ∗ ) − , φ dτ + A(u λ − u ∗ , φ; τ )dτ + λ (φ 1/κ , φ) ∂τ t t  T  T ∗   T ∂u , φ dτ − ( f, φ)dτ + A(u ∗ , φ; τ )dτ = ∂t t t t  T 1/q  T 1/ p q p ≤ f (τ ) L q (I ) dτ φ(τ ) L p (I ) dτ t

 +β

t

T

e

βτ





∗



V0 − V , φ(τ ) dτ −

t

T

A(u ∗ (τ ), φ(τ ); τ )dτ,

t

where q = 1 + κ so that 1/ p + 1/q = 1. ∗ From the definition of φ, we see ∂(u λ∂t−u ) · φ = ∂φ · φ. Thus, from the above ∂t estimate, (2.2.7) and the 1st inequality in (2.1.12) we get  T 1/ p  T  T 1 p p 2 (φ, φ) + C φ H 1 (I ) dτ + λ φ L p (I ) dτ ≤ C φ L p (I ) dτ 0,w 2 t t t  T  T  +β eβτ V0 − V ∗ , φ dτ − A(u ∗ , φ; τ )τ. (2.2.10) t

t

2.2 The Penalty Method and Its Convergence Analysis

43

Let us consider the last two integrals in (2.2.10). From (2.1.8), we see that |V0 (S) − V ∗ (S)| ≤ (1 − K /Smax )K for S ∈ [0, Smax ]. Using this bound and Hölder inequality we have the following estimate: 

T

 eβτ V0 − V ∗ , φ dτ ≤ C

 T

t

t

Smax



T

φd Sdτ ≤ C t

0

1/ p p

φ L p (I ) dτ

. (2.2.11)

From (1.2.18), we see that the integrand of the last term in (2.2.10) is   ∗ 2 ∂u ∗ ∂φ + bSu , + (cu ∗ , φ). − A(u , φ; τ ) = − aS ∂S ∂S ∗

(2.2.12)

Using (2.1.8), we have that  ∂u ∗ eβt (1 − K /Smax ), S ∈ (0, K ) = ∂S S ∈ (K , Smax ). −eβt K /Smax ,

(2.2.13)

Therefore, integrating by parts and using u ∗ (0, t) = 0 = u ∗ (Smax , t), we obtain 

∂u ∗ (S, τ ) ∂φ(S, τ ) · dS a(τ )S 2 ∂S ∂S 0   K  Smax K ∂φ K ∂φ = −eβτ 1 − aS 2 d S + eβτ aS 2 d S S ∂S Smax K ∂S 0   K K = −eβτ a K 2 φ(K , τ ) + eβτ 1 − 2aSφd S S 0  Smax  Smax βτ K 2aSφd S ≤ C φ(S, τ )d S +e Smax K 0 −

Smax

for τ ∈ (0, T ), because a(τ ) = 21 σ 2 (τ ) > 0 and φ ≥ 0. Similarly, since u ∗ ∈ H01 (I ) and b, c ∈ L ∞ (I ), using integration by parts and (2.2.13) we have 

∂φ − bSu , ∂S ∗



  Smax ∂u ∗ + (cu , φ) = b(u + S ), φ + (cu ∗ , φ) ≤ C φd S. ∂S 0 ∗



∗

Integrating from t to T and using the above two estimates, we have from (2.2.12) 

T

− t

A(u ∗ , φ; τ )dτ ≤ C



T t



Smax 0

 φd Sdτ ≤ C t

T

1/ p p

φ L p (I ) dτ

Combining the above inequality and (2.2.11) with (2.2.10), we obtain

.

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2 American Options on One Asset

1 (φ, φ) + 2

 t

T

 φ 2H 1

0,w

dτ + λ (I )

T t



T

p

φ L p (I ) dτ ≤ C

t

1/ p p

φ L p (I ) dτ (2.2.14)

for all t ∈ (0, T ), which implies  λ t

T



T

p

φ L p (I ) dτ ≤ C

t

1/ p p

φ L p (I ) dτ

 , or

T

t

1−1/ p p

φ L p (I ) dτ

≤

C . λ

Taking the ( p − 1)th root on both sides of the above estimate gives 

T t

1/ p p φ(τ ) L p (I ) dτ

≤

C λ1/( p−1)

=

C , λκ

(2.2.15)

since p = 1 + 1/κ. Thus, we have proved (2.2.8). From (2.2.14) and (2.2.15), we have 1 (φ(t), φ(t)) + 2



T t

 φ(τ ) 2A dτ ≤ C

from which it follows that (φ(t), φ(t))1/2 + (0, T ). Thus, we have proved (2.2.9).

T t



T t

1/ p p

φ(τ ) L p (I ) dτ φ(τ ) 2A dτ

1/2

≤ ≤

C λκ/2

C , λκ for all t ∈ 

Using the results obtained in Lemma 2.2.1 we are able to establish the rate of convergence for u λ in the following theorem. ∈ L κ+1 (Ω) and the assumptions in Lemma 2.2.1 be fulfilled. Theorem 2.2.2 Let ∂u ∂t Then, the solutions u and u λ to Problems 2.1.1 and 2.2.1 respectively. satisfy 1 u − u λ L ∞ (0,T ;L 2 (I )) + u − u λ L 2 (0,T ;H0,w (I )) ≤

C , λκ/2

(2.2.16)

where C is a positive constant, independent of u, u λ and λ. Proof We continue to use φ(·, t) introduced in the proof of Lemma 2.2.1. The term u − u λ can be decomposed as u − u λ = u − u ∗ − (u λ − u ∗ ) = u − u ∗ + [u λ − u ∗ ]− − [u λ − u ∗ ]+ =: rλ − φ, (2.2.17) where [u λ − u ∗ ]− = − min{u λ − u ∗ , 0} and rλ = u − u ∗ + [u λ − u ∗ ]− . From the definition of φ, we see (φ α , [u λ − u ∗ ]− ) = [u − u ∗ ]α+ [u λ − u ∗ ]− = 0 for any α > 0. To establish an upper bound for u − u λ , we need only to find an upper bound for rλ , as that for φ is given in Lemma 2.2.1. Setting v = u − rλ in (2.1.9) and v = rλ in (2.2.3) respectively, we have

2.2 The Penalty Method and Its Convergence Analysis

45

  ∂u − , −rλ + A(u, −rλ ; t) ≥ ( f, −rλ ), ∂t   ∂u λ − , rλ + A(u λ , rλ ; t) + λ(φ 1/κ , rλ ) = ( f, rλ ). ∂t Adding up the above inequality and equality gives   ∂(u λ − u) , rλ + A(u λ − u, rλ ; t) + λ(φ 1/κ , rλ ) ≥ 0. − ∂t

(2.2.18)

Using the definition of rλ , we have (φ 1/κ , rλ ) = (φ 1/κ , u − u ∗ + [u λ − u ∗ ]− ) = (φ 1/κ , u − u ∗ ) ≤ 0,

(2.2.19)

since φ ≥ 0, u − u ∗ ≤ 0 and φ[u λ − u ∗ ]− = 0. From (2.2.18) and (2.2.19), we get  λ) , rλ + A(u − u λ , rλ ; t) ≤ 0, and using this inequality and the decompo− ∂(u−u ∂t sition of u − u λ in (2.2.17), we have 

   ∂rλ ∂φ − , rλ + A(rλ , rλ ; t) ≤ − , rλ + A(φ, rλ ; t). ∂t ∂t

Integrating both sides of the above estimate from t to T and using (2.2.7) and Cauchy– Schwarz inequality, we obtain  T 1 (rλ (t), rλ (t)) + A(rλ (τ ), rλ (τ ); τ )dτ 2 t   T  T ∂φ(τ ) − , rλ (τ ) dτ + ≤ A(φ(τ ), rλ (τ ); τ )dτ ∂τ t t    T  T ∂rλ (τ ) dτ + φ(τ ), A(φ(τ ), rλ (τ ); τ )dτ ≤ (φ(t), rλ (t)) + ∂τ t t 1 1 ≤ φ L ∞ (0,T ;L 2 (I )) rλ L ∞ (0,T ;L 2 (I )) + C φ L 2 (0,T ;H0,w (I )) r λ L 2 (0,T ;H0,w (I ))   T ∂rλ (τ ) φ(τ ), dτ, t ∈ (0, T ). (2.2.20) + ∂t t From (2.1.8) and the definition of rλ , we have   ∂rλ (τ ) ∂u(τ ) ∂u ∗ (τ ) ∂u(τ ) φ(τ ) = φ(τ ) − = φ(τ ) − φ(τ )βeβτ (V0 − V ∗ ). ∂τ ∂τ ∂τ ∂τ Thus, using (2.2.8) we obtain

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2 American Options on One Asset

   T  T  T  ∂rλ (τ ) ∂u(τ ) dτ = dτ − β φ(τ ), φ(τ ), eβτ φ(τ ), V0 − V ∗ dτ ∂τ ∂τ t t t     ∂u  C  + V0 − V ∗ L q (Ω) ≤ κ , ≤ C φ L p (Ω)   ∂t  q λ L (Ω)

since ∂u ∈ L κ+1 (Ω), where p = 1 + 1/κ and q = κ + 1. Substituting the above ∂t upper bound into (2.2.20) and using (2.2.9), we obtain 1 (rλ (t), rλ (t)) + 2



T

A(rλ (τ ), rλ (τ ); τ )dτ

t

C 1 1 ≤ φ L ∞ (0,T ;L 2 (I )) rλ L ∞ (0,T ;L 2 (I )) + C φ L 2 (0,T ;H0,w (I )) r λ L 2 (0,T ;H0,w (I )) + κ λ    ∞ 2 1 ∞ 2 1 ≤ C φ L (0,T ;L (I )) + φ L 2 (0,T ;H0,w (I )) rλ L (0,T ;L (I )) + rλ L 2 (0,T ;H0,w (I ))     −k 1 + Cλ−κ ≤ C λ−k/2 rλ L ∞ (0,T ;L 2 (I )) + rλ L 2 (0,T ;H0,w + λ . (I )) On the other hand, from (2.1.12), (2.2.7), (2.2.4) and the above estimate, we have  2 C 1 rλ L ∞ (0,T ;L 2 (I )) + rλ L 2 (0,T ;H0,w ≤ rλ 2L ∞ (0,T ;L 2 (I )) + C rλ 2L 2 (0,T ;H 1 (I )) (I )) 0,w 2  T C ≤ (rλ (t), rλ (t)) + C A(rλ (τ ), rλ (τ ); τ )dτ 2 t     −κ 1 . (2.2.21) ≤ C λ−κ/2 rλ L ∞ (0,T ;L 2 (I )) + rλ L 2 (0,T ;H0,w (I )) + λ This is of the form y 2 ≤ Cρ 1/2 y + Cρ which can be rewritten as (y − 21 Cρ 1/2 )2 ≤   2 C + C4 ρ. From this inequality, we have y ≤ Cρ 1/2 . (Recall that C > 0 is a generic constant.) Thus, applying this analysis to (2.2.21) yields rλ L ∞ (0,T ;L 2 (I )) + rλ L 2 (0,T ;H01 (I )) ≤

C . λκ/2

Using the triangle inequality, the above estimate and (2.2.9), we have from (2.2.17), u − u λ L ∞ (0,T ;L 2 (I )) + u − u λ L 2 (0,T ;H 1 (I )) ≤ rλ L ∞ (0,T ;L 2 (I )) + rλ L 2 (0,T ;H 1 (I )) 0,w 0,w   C + φ L ∞ (0,T ;L 2 (I )) + φ L 2 (0,T ;H 1 (I )) ≤ κ/2 . 0,w λ

This is (2.2.16), and thus the theorem is proved.



Remark 2.2.1 The assumption ∂u ∈ L k+1 in Theorem 2.2.2 is not restrictive. Any ∂t function which has a bounded derivative w.r.t. t satisfies this A function

condition. κ+1 | d S < ∞. v with a singular t-derivative, ∂v/∂t, is also in L k+1 (Ω) if I | ∂v ∂t

2.3 Numerical Solution of the Penalty Equation

47

2.3 Numerical Solution of the Penalty Equation We now consider the numerical solution of (2.2.1) which is a nonlinear parabolic PDE. We discretize it using the FVM in Sect. 1.3 and use a Newton’s method for the resulting nonlinear system.

2.3.1 Discretization To simplify notation, we suppress the subscript λ of u λ , but bear in mind u is a solution to Problem 2.2.1. In the following discussion, we use the notation defined in Sect. 1.3. Let the interval I = (0, Smax ) be divided into N sub-intervals Ii := (Si , Si+1 ), i = 0, 1, . . . , N − 1 with 0 = S0 < S1 < · · · < S N = Smax . For each i = 0, 1, . . . , N − 1, we put h i = Si+1 − Si and h = max0≤i≤N −1 h i . We also let Si−1/2 = (Si−1 + Si )/2 and Si+1/2 = (Si + Si+1 )/2 for each i = 1, 2, . . . , N − 1. These mid-points form a second partition of (0, Smax ) if we define S−1/2 = S0 and S N +1/2 = S N . For any i = 1, 2, . . . , N − 1, integrating (2.2.1) over (Si−1/2 , Si+1/2 ) we have −

 S 1  S 1  S 1  S 1 Si+ 1 i+ 2 ∂u i+ 2 i+ 2 i+ 2 1/κ d x − [Sρ(u)] S 2 + cud S + λ [u − u ∗ ]+ d S = f d S, i− 21 Si− 1 ∂t Si− 1 Si− 1 Si− 1 2

2

2

2

where ρ(u) is the flux defined in (1.3.2). Applying a 1-point quadrature rule to the first, third, fourth and last terms, we obtain from the above     ∂u i 1/κ li − Si+ 21 ρ(u)| Si+ 1 − Si− 21 ρ(u)| Si− 1 + ci u i + λ[u i − u i∗ ]+ li = f i li , 2 2 ∂t (2.3.1) where li = Si+1/2 − Si−1/2 , ci = c(Si , t), f i = f (Si , t), u i∗ = u ∗ (Si , t), and u i is the nodal approximation to u(xi , t) to be determined. In Sect. 1.3, approximations to ρ(u) at Si+ 21 , i > 0, and S 21 are constructed and given in (1.3.7) and (1.3.10) respectively. Using these, we have from (2.3.1) −

∂u i (t) li + ei,i−1 u i−1 (t) + ei,i u i (t) + ei,i+1 u i+1 (t) + di (u i (t), t) = f i (t)li , ∂t (2.3.2) where ei,i−1 , ei,i and ei,i+1 are defined in (1.3.16)–(1.3.19) and −

di (u i (t), t) = λli [u i (t) − u i∗ (t)]+

1/κ

(2.3.3)

for i = 1, 2, . . . , N − 1. These form an N − 1 nonlinear ODE system for u(t) := (u 1 (t), . . . , u N (t)) with the homogeneous boundary condition u 0 (t) = 0 = u N (t).

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2 American Options on One Asset

We now consider the time discretization of the linear ODE system (2.3.2). Let E i := (0, . . . , 0, ei,i−1 (t), ei,i (t), ei,i+1 (t), 0, . . . , 0), where ei,i−1 , ei,i and ei,i+1 are defined in (1.3.16)–(1.3.19) and those which are not defined are zeros. Using E i , we rewrite (2.3.2) as the following matrix form: −

∂u i (t) li + E i (t)u(t) + di (u i (t), t) = f i (t)li , i = 1, 2, . . . , N − 1. ∂t

(2.3.4)

To discretize the above ODE system, we choose a partition for [0, T ] with mesh nodes ti (i = 0, 1, . . . , M) satisfying T = t0 > t1 > · · · > t M = 0. Apply the twolevel implicit time-stepping scheme with a splitting parameter θ ∈ [1/2, 1] to (2.3.4), we obtain, for m = 0, 1, . . . , M − 1,     u im+1 − u im li + θ E im+1 um+1 + dim+1 (u im+1 ) + (1 − θ ) E im um + dim (u im ) −Δtm = (θ f im+1 + (1 − θ ) f im )li , where Δtm = tm+1 − tm < 0, E im = E i (tm ), f im = f (xi , tm ), dim (u im ) = di (u im , tm ) m m and um = (u m 1 , u 2 , . . . , u N −1 ) denotes the approximation of u(t) at t = tm . Let m E be the (N − 1) × (N − 1) matrix given by E m = (E 1m , E 2m , . . . , E Nm −1 ) and m m m m D m (um ) = (d1m (u m 1 ), d2 (u 2 ), . . . , d N −1 (u N −1 )) . Then, the above system can be re-written as (θ E m+1 + G m )um+1 + θ D m+1 (um+1 ) = f¯m + [G m − (1 − θ)E m ]um − (1 − θ)D m (um )

(2.3.5) for m = 0, 1, . . . , M − 1, where G m = diag(l1 /(−Δtm ), . . . , l N −1 /(−Δtm )) is an (N − 1) × (N − 1) diagonal matrix and m m f¯m = θ ( f 1m+1l1 , . . . , f Nm+1 −1 l N −1 ) + (1 − θ )( f 1 l 1 , . . . , f N −1 l N −1 ) .

The boundary and terminal/payoff conditions for this system are m 0 ∗ ∗ ∗ um 0 = 0 = u N , m = 0, 1, . . . , M, u = (u 1 (T ), u 2 (T ), . . . , u N −1 (T )) .

Remark 2.3.1 When θ = 1/2, the time-stepping scheme becomes that of the Crank–Nicholson, and when θ = 1, it is the backward Euler (or full implicit) scheme. Both are unconditionally stable. A stability and error analysis for this finite volume method with D = 0 in (2.3.5) is given in Sect. 1.3.3. For clarity, we will skip this discussion and refer the author to [2].

2.3 Numerical Solution of the Penalty Equation

49

2.3.2 Solution of the Nonlinear System Equation (2.3.5) is nonlinear in um+1 . We now apply a Newton method to it. When κ > 1, from (2.3.3) we see that di (u im ) → ∞ as u im − u i∗ → 0+ . To overcome this difficulty, we smooth out di (u im ) in the neighbourhood of [u im − u i∗ ]+ = 0 by redefining di as  m 1 m m (u i − u i∗,m )1/κ , u im − u i∗,m ≥ ε, (2.3.6) di (u i ) = W ([u im − u i∗,m ]+ ), u im − u i∗,m < ε λli for k > 0, where 1  ε > 0 is a transition parameter and W (z) is a function which smooths out the original di (z) around z = 0. We choose W (z) = c1 + c2 z + · · · + cn z n−1 + cn+1 z n for n ≥ 3 and impose that W (z) is such that di (·) is smooth. This requires that W (z) satisfies W (0) = W (0) = 0, W (ε) = ε1/κ , and W (ε) = k1 ε1/κ−1 . Thus, the function defined in (2.3.6) is globally smooth. Using these four conditions and setting c3 = · · · = cn−1 = 0, we can easily find that     1 1 , cn+1 = ε1/κ−n −n+1 . c1 = c2 = 0, cn = ε1/κ−n+1 n − κ κ Substituting these into (2.3.6) gives ⎧ m ∗,m u im − u i∗,m ≥ ε, ⎨ (u i − u i  )1/κ , 1 m m ∗,m n−1 1 m d (u ) = ε1/κ−n+1  n − κ [u i − u i ]+ ⎩ 1/κ−n 1 λli i i +ε − n + 1 [u im − u i∗,m ]n+ , u im − u i∗,m < ε. κ

(2.3.7)

We comment that the intuition of this choice of dim is as follows. When u im − ≥ ε, a given tolerance, (u im − u i∗,m )1/κ offers a convergence rate of order κ/2 by the results from the previous section. When [u im − u i∗,m ] < ε, we choose dim (u im ) = W ([u im − u i∗,m ]+ ) to slow down the convergence. For the function W (z) used to smooth the penalty term, we have the following theorem. u i∗,m

Theorem 2.3.1 The function W (z) is strictly increasing on [0, ε] when κ ≥ 1/n and n ≥ 3. Proof When z = 0, W (0) = 0. We now show that W (z) > 0 for z ∈ (0, ε). Differentiating W (z) gives     1 n−2 1 z − n + 1 z n−1 + nε1/κ−n W (z) = (n − 1)ε1/κ−n+1 n − κ κ       1 1 z +n −n+1 = z n−2 ε1/κ−n+1 (n − 1) n − κ κ ε = z n−2 ε1/κ−n+1 G(z),   where G(z) := (n − 1) n − κ1 + n κ1 − n + 1 εz .

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2 American Options on One Asset

 We now show G(z) > 0 on (0, ε] when κ ≥ 1/n. In fact, G(0) = (n − 1) n − κ1 ≥ 0 and G(ε) = 1/κ > 0. Since G(z) is linear, we have G(z) > 0 on (0, ε]. Therefore, we have W (z) > 0 on (0, ε), and so W (z) is strictly increasing on [0, ε].  Using this theorem, we have the following corollary. Corollary 2.3.1 The nonlinear function dim (u ik ) defined in (2.3.7) is smooth and increasing on (−∞, ∞) when k ≥ 1/n and n ≥ 3. Proof By Theorem 2.3.1, dim (u im ) is increasing in the region (u i∗ , u i∗ + ε). Also, from (2.3.7) we see that d(u im ) = (u im − u i∗,m )1/k if u ik ≥ u i∗ + ε and d(u im ) = 0 if u im ≤ u i∗ . Both of these are increasing. The smoothness of the function is obvious because of the choice of W . This completes the proof.  We use the following Newton’s method to (2.3.5) with dim defined in (2.3.7). Algorithm Newton 1. Choose penalty and smoothing parameters λ > 1, κ ≥ 1 and 0 < ε  1, tolerance 0 < δ  1, and damping parameter γ ∈ (0, 1]. Calculate terminal condition u 0 and let m = 0. 2. Let w0 = u m and l = 1. 3. Solve the following system for wl . [θ E m+1 + G m + θ J D m (wl−1 )]Δl = f¯m + [G m − (1 − θ)E m ]um − (1 − θ)D m (um ) − (θ E m+1 + G m )wl−1 − θ D m (wl−1 ), w =w l

l−1

(2.3.8)

+ γΔ , l

where J Dm (w) denotes the Jacobian of the column vector D m (w). 4. If wl − wl−1 2 ≤ δ, then go to Step 5, where · 2 denotes the Euclidean norm. Otherwise, let l := l + 1 and go to Step 3. 5. Let u m+1 = wl . If m = M − 1, then stop and the problem is solved. Otherwise, let m := m + 1 and go to Step 2. For the system matrix of (2.3.8), we have the following theorem. Theorem 2.3.2 The system matrix θ E m+1 + G m + θ J Dm (wl−1 ) of (2.3.8) is an Mmatrix when |Δt| := maxm |Δtm | is sufficiently small. Proof We have already shown in Theorem 1.3.1 that θ E m+1 + G m is an M-matrix. Thus we only need to show that the diagonal entries of J Dm are all non-negative. From the definition of D m in (2.3.5), we see that, for any w, its Jacobian is J Dm (w) = diag((d1m ) (w1 ), (d2m ) (w2 ), . . . , (d Nm−1 ) (w N −1 )). By Corollary 2.3.1, we have di (wi ) ≥ 0 for i = 1, 2, . . . , N − 1, and thus the diagonal entries of J Dm are non-negative. Combining this with Theorem 1.3.1 yields  θ E m+1 + G m + θ J Dm (wl−1 ) is an M-matrix.

2.4 Numerical Experiments

51

2.4 Numerical Experiments We now use the methods developed in the previous sections to solve some model problems. As in Sect. 1.4, for all the tests given below, we choose Smax = 100 and T = 1. The uniform mesh with 51 × 51 mesh nodes is used for solving the test unless mentioned otherwise. Example 2.1 The American put option with K = 50. The market parameters are σ = 0.4, r = 0.03, d = 0.02. To solve this problem, we choose λ = 104 and κ = 2 in (2.2.1). The computed option 2 value V , its Greeks Δ and Γ := ∂∂ SV2 , and V − V ∗ are plotted in Fig. 2.1. From Δ and Γ in Fig. 2.1 we see that an interior curve can clearly be seen, while the Δ of its European counterpart depicted in Fig. 1.2 does not contain such a curve. To further demonstrate the optimal exercise curve, we have also solved this problem on the 301 × 301 uniform mesh and use this numerical solution to estimate the optimal exercise curve, which is depicted in the V plot in Fig. 2.1. We now show the influence of κ on the quality of solutions. We choose λ = 5 and solve the problem for κ = 1, 3, 5. The computed V and Δ for each κ are plotted in

Fig. 2.1 Computed value V , optimal exercise curve, Δ, Γ and V − V ∗ for Example 2.1

52

2 American Options on One Asset

Fig. 2.2 Computed value V and Δ of Test 2 for various values of κ when λ = 5

Fig. 2.2, from which we see that when κ = 1, there is a large error near S = 0. When κ = 3, the approximation errors are still noticeable, particularly in Δ. However, when κ = 5, the computed V and Δ are very close to those in Fig. 2.1. Thus, Fig. 2.2 shows the quality of the solution improves as κ increases for a fixed λ.

References

53

References 1. Achdou Y, Pironneau O (2005) Computational methods for option pricing. SIAM, New York 2. Angermann L, Wang S (2007) Convergence of a fitted finite volume method for the penalized Black-Scholes equations governing European and American option pricing. Numer Math 106:1–40 3. Bensoussan A, Lions JL (1982) Applications of variational inequalities in stochastic control. North-Holland, Amsterdam 4. Benth FE, Karlsen KH, Reikvam K (2004) A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence. Interfaces Free Bound 6:379–404 5. Forsyth PA, Vetzal KR (2002) Quadratic convergence for valuing American options using a penalty method. SIAM J Sci Comput 23:2095–2122 6. Glowinski R (1984) Numerical methods for nonlinear variational problems. Springer, New York 7. Han H, Wu X (2003) A fast numerical method for the Black-Scholes equation of American options. SIAM J Numer Anal 41:2081–2095 8. Haslinger J, Miettinen M (1999) Finite Element Method for Hemivariational Inequalities. Kluwer Academic Publisher, Dordrecht 9. Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic, New York 10. Nielsen BF, Skavhaug O, Tveito A (2001) Penalty and front-fixing methods for the numerical solution of American option problems. J Comp Fin 5:69–97 11. Wang G, Wang S (2006) On stability and convergence of a finite difference approximation to a parabolic variational inequality arising from American option valuation. Stoch Anal Appl 24:1185–1204 12. Wang G, Wang S (2010) Convergence of a finite element approximation to a degenerate parabolic variational inequality with non-smooth data arising from American option valuation. Optim Methods Softw 25:699–723 13. Wang S, Yang XQ, Teo KL (2006) Power penalty method for a linear complementarity problem arising from American option valuation. J Optim Theory Appl 129:227–254

Chapter 3

Options on One Asset with Stochastic Volatility

Abstract In this chapter we develop numerical methods for pricing European and American options whose underlying asset price and volatility follow two separate geometric Brownian motions. These methods include a fitted Finite Volume Method (FVM) for the discretization of the resulting 2D Black–Scholes equation and a power penalty approach to the differential Linear Complementarity Problem involving the 2D differential operator of Black–Scholes type. A mathematical analysis will be presented for the convergence of the FVM and power penalty approach. These methods can also be used for pricing options on two assets such as a basket option. Keywords Option pricing under two-factor models · Stochastic volatility · 2-dimensional Black–Scholes equation · Linear complementarity problem · Finite volume methods, penalty method

3.1 The 2-Dimensional PDE Model for Pricing European Options with Stochastic Volatility There are different stochastic volatility models in option pricing, such as Heston’s model [6] and the Constant Elasticity of Variance (CEV) Model [1]. In what follows, we shall consider the latter which is widely used by researchers (e.g., [5]). We will first derive the PDE governing the pricing problem. We will then formulate the ‘strong’ problem as an variational one and show that the latter is uniquely solvable.

3.1.1 The Pricing Problem Assume a stock price S and its instantaneous volatility σ =: stochastic equations. d S = μSdt +

√

ν satisfy the following

√ ν SdW1 and dν = μν vdt + σν νdW2 ,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 S. Wang, The Fitted Finite Volume and Power Penalty Methods for Option Pricing, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-15-9558-5_3

(3.1.1) 55

56

3 Options on One Asset with Stochastic Volatility

where μ is the drift coefficient for S, μv and σv are the constant drift coefficient and volatility of v, and W1 and W2 are two Wiener processes with correlation ρ. It is shown in [2] that a security F with a price depending on assets Si , i = 1, 2, . . . , N satisfies the following PDE (also see [5]): N N  ∂F 1  ∂2 F ∂F + ρi j σi σ j −rF = Si [−μi + βi (μ∗ − r )], ∂t 2 i, j=1 ∂ Si ∂ S j ∂ S i i=1

(3.1.2)

where σi is the instantaneous standard deviation of Si , ρi j is the instantaneous correlation between Si and S j , μi is the expected return rate of Si , {βi } the vector of instantaneous sensitivity (or betas) of the expected asset returns (d S/S) to the expected market return, and μ∗ is the instantaneous expected return on the market portfolio. When Si is traded, it should satisfy the continuous time (N + 1)-factor Capital Asset Pricing Model (CAPM). Therefore, its return rate should satisfy μi = r + βi (μ∗ − r ) [4, Page 795]. Thus, the ith term of the sum on the RHS of (3.1.2) becomes −r Si ∂∂ SFi . In our Problem (3.1.1), there are two assets S and ν. Thus, if V is a European option with strike price K and maturity T whose price depends on S and ν, using (3.1.2) and the above analysis we see that V satisfies the following equation. ∂V 1 ∂2 V ∂2 V ∂2 V  + ν S 2 2 +2ρσν Sν 3/2 + σν2 ν 2 − r V ∂t 2 ∂S ∂ S∂ν ∂ν ∂V ∂V = −r S − [μν − βν (μ∗ − r )]ν , ∂S ∂ν

(3.1.3)

where βν denotes the sensitivity (beta) of the expected return of ν. As in [5], we assume βν (μ∗ − r ) = 0, i.e., the volatility is uncorrelated with aggregated consumption, or the volatility has zero systematic risk. Thus, (3.1.3) becomes the following 2-dimensional Black–Scholes equation: −

 2 2 ∂2 V 1 ∂V ∂V ∂V 3/2 ∂ V 2 ∂ V − ν S2 + σ − μν ν + rV = 0 + 2ρσ Sν ν − rS ν ν ∂t 2 ∂ S2 ∂ S∂ν ∂ν 2 ∂S ∂ν

(3.1.4)

for S > 0, ν > 0 and t ∈ [0, T ). It is easy to show [7] that (3.1.4) can be written as − where A =

∂V − ∇ · (A∇V + bV ) + cV = 0, ∂t

(3.1.5)

  a11 a12 , b = (b1 , b2 ) and c is scalar with a21 a22

1 2 1 1 S ν, a22 = σν2 ν 2 , a12 = a21 = ρσν Sν 3/2 , 2 2 2 3 1 b1 = r S − ρσν Sν − Sν, b2 = μν ν − ρσν ν 3/2 − σν2 ν 4 2

a11 =

(3.1.6) (3.1.7)

3.1 The 2-Dimensional PDE Model for Pricing European Options …

3 c = 2r − ρσν ν 1/2 − ν + μν − σν2 . 2

57

(3.1.8)

To solve (3.1.4) numerically, we need to truncate its solution domain and define boundary and payoff conditions for it. Let Smax , νmin and νmax be positive integers satisfying Smax > K and 0 < νmin < νmax . Let κ(V ) := A∇V + bV . Following [7], we rewrite (3.1.5) as the following problem: −

∂V − ∇ · κ(V ) + cV = 0, ∂t V (0, ν, t) = g1 (t), V (Smax , ν, t) = g2 (t), V (S, ν, T ) = g3 (S),

(3.1.10)

V (S, νmin , t) = G 1 (S, t), V (S, νmax , t) = G 2 (S, t)

(3.1.11)

(3.1.9)

for (S, ν, t) ∈ Ω :=∈ (0, Smax ) × (νmin , νmax ) × [0, T ), where gi , i = 1, 2, 3 in (3.1.10) are the same as defined in (1.2.6)–(1.2.10). The boundary conditions G 1 and G 2 in (3.1.11) can not normally be found explicitly, but we can calculate them numerically by solving (1.2.3)–(1.2.4) in Sect. 1.2.2 at σ 2 = νmin and νmax .

3.1.2 The Variational Problem and Its Solvability The discussions below follow those in [8]. Without loss of generality, we assume that g1 , g2 , G 1 and G 2 in (3.1.10)–(3.1.11) are all zero. The case with non-zero boundary conditions can be transformed into one with homogeneous boundary conditions by subtracting a known smooth function V0 satisfying the non-homogeneous boundary conditions (e.g., the solution of −∇ 2 V0 = 0 satisfying the boundary conditions in (3.1.10)–(3.1.11)). This transformation creates a non-zero RHS function. We introduce the transformation u = eβt (V − V0 ) for any constant β > 0. Under this transformation, (3.1.9)–(3.1.11) can be written as the following problem: ∂u − ∇ · κ(u) + cu ˆ = f, (3.1.12) ∂t u(S, ν, t) = 0, (S, ν, t) ∈ ∂Ω × [0, T ), u(S, ν, T ) = gˆ 3 (S, ν), (S, ν) ∈ Ω, (3.1.13) L2D u := −

where cˆ = β + c and ∂Ω denotes the boundary of Ω. Before further discussion, we first introduce some notation to be used in the analysis. Let L 2 (Ω) be the space of all square-integral  functions on Ω. We√define an inner product and norm on L 2 (Ω) by (u, v) := Ω u  vdΩ and v0 = (v, v) respectively for u, v ∈ L 2 (Ω). a weighted inner product on (L 2 (Ω))2 defined by (u, v)w :=  We2 now introduce 2   2 2 Ω (S u 1 v1 + ν u 2 v2 )dΩ for any u = (u 1 , u 2 ) , v = (v1 , v2 ) ∈ (L (Ω)) . Let

58

3 Options on One Asset with Stochastic Volatility

√ v0,w := (v, v)w be the norm generated by (·, ·)w . We define a space of all weighted square-integrable functions in (L 2 (Ω))2 by Lw2 (Ω) := {v ∈ (L 2 (Ω))2 : ∇v0,w < ∞}. Using a standard argument it is easy to show that the pair (Lw2 (Ω), (·, ·)w ) is a Hilbert space (e.g., [9]). Based on this space, we define a weighted Sobolev space Hw1 (Ω) = {v ∈ L 2 (Ω) : ∇v ∈ Lw2 (Ω)} with the energy norm defined by v1,w = (|v|21,w + v20 )1/2 , where |v|1,w = ∇v0,w . is a semi-norm. Let ∂Ω D = {(S, ν) ∈ ∂Ω : S = 0} denote the boundary segments of Ω excluding the part where S = 0. 1 (Ω) = {v ∈ Hw1 (Ω) : v|∂Ω D = 0}. Let H0,w 1 (Ω) and applying integration by parts yield Multiplying (3.1.12) by v ∈ H0,w  −

  ∂u ,v − vκ(u) · nds + (κ(u), ∇v) + (cu, ˆ v) = ( f, v), ∂t ∂Ω

where n denotes the unit vector outward-normal to ∂Ω. Since v|∂Ω D = 0, from the definition of κ(u) and (3.1.6)–(3.1.7) we have 

  ∂u ∂u a11 + a12 + b1 u vds = 0 vκ(u) · nds = − ∂S ∂ν ∂Ω ∂Ω\∂Ω D 

(3.1.14)

since a11 , a12 and b1 are all equal to zero at S = 0. Therefore, if we define the following bi-linear form on Hw1 (Ω) B(u, v; t) = (A∇u + bu, ∇v) + (cu, ˆ v),

(3.1.15)

we then propose the following variational problem. 1 (Ω), satisfying the final condition in (3.1.13), such Problem 3.1.1 Find u(t) ∈ H0,w 1 that for all v ∈ H0,w (Ω),

  ∂u(t) − , v + B(u(t), v; t) = ( f, v) a.e. in (0, T ). ∂t

(3.1.16)

From the above analysis we see that Problem 3.1.1 is the variational problem corresponding to (3.1.12)–(3.1.13). Note that we do not need the boundary condition that V (0, ν, t) = 0 due to the degeneracy of L2D at S = 0. In this case, a solution to Problem 3.1.1 cannot take a ‘trace’ (boundary condition) at S = 0. However, in computation, we choose a particular solution with a given value at S = 0. The unique solvability of Problem 3.1.1 is given in the following theorem. Theorem 3.1.1 Problem 3.1.1 has a unique solution. 1 (Ω), using integraProof We show B is coercive and continuous. For any φ ∈ H0,w tion by parts and (3.1.14), we have

 (bφ, ∇φ) =

∂Ω

 φ 2 b · nds −

Ω

 φ∇ · (bφ) dΩ = −

Ω

 φb · ∇φdΩ −

Ω

φ 2 ∇ · bdΩ.

3.1 The 2-Dimensional PDE Model for Pricing European Options …

59

Thus, (bφ, ∇φ) = − 21 (φ∇ · b, φ), and so from (3.1.15) and (3.1.6)–(3.1.8), we have    1 B(φ, φ; t) = (A∇φ, ∇φ) + cˆ − ∇ · b φ, φ 2       1 1 1 2 2 2 2 2 2 ν S φ S + 2ρν σν ν Sφ S φν + σν ν φν dΩ + cˆ − ∇ · b φ, φ = 2 Ω 2    1 1 (1 − ρ)ν S 2 φ S2 + ρ(ν 2 Sφ S + σν νφν )2 + (1 − ρ)σν2 ν 2 vφν2 dΩ = 2 Ω     3 1 3 1 1 φ, φ β+ r− ρσν ν 2 + y − μν + σ + ν 2 + ((cˆ − ∇ · b)φ, φ) ≥ 2 2 2 2  2 2

(3.1.17) S φ S + ν 2 φν2 dΩ ≥ Cφ21,w +C Ω

when β is chosen to be sufficiently large, where C denotes a generic positive constant, independent of v. Therefore, B is coercive. 1 To show B(φ, ψ; t) is continuous, we first note that, for φ, ψ ∈ H0,w (Ω), we have |B(φ, ψ; t)| ≤ |(A∇φ, ∇ψ)| + |(bφ, ∇ψ)| + |(cφ, ˆ ψ)|. (3.1.18) For |(A∇φ, ∇ψ)| in (3.1.18), using the triangular inequality, we have    1  2 2 2 |(A∇φ, ∇ψ)| ≤  (ν S φ S ψ S + σν ν φν ψν )dΩ  2 Ω

  I1

   1  3 2 +  ρσν Sν (φ S ψν + φν ψ S )dΩ  2 Ω

 

(3.1.19)

I2

We now estimate I1 and I2 . Using Cauchy–Schwarz inequality, we obtain            1 1  I1 ≤  ν S 2 φ S ψ S dΩ  +  σν2 ν 2 φν ψν dΩ  = (ν 2 Sφ S , ν 2 Sψ S ) + |(σν νφν , σν νψν )| Ω

1

Ω

1

1

1

1

1

1

1

≤ (ν 2 Sφ S , ν 2 Sφ S ) 2 · (ν 2 Sψ S , ν 2 Sψ S ) 2 + (σν νφν , σν νφν ) 2 · (σν νψν , σν νψν ) 2   1   1   1  1 2 2 2 2 = ν S 2 φ S2 dΩ · ν S 2 ψ S2 dΩ + σν2 ν 2 φν2 dΩ · σν2 ν 2 ψν2 dΩ  ≤

Ω

Ω

Ω

Ω

Ω

 1  1 2 2 (ν S 2 φ S2 + σν2 ν 2 φν2 )dΩ · (ν S 2 ψ S2 + σν2 ν 2 ψ y2 )dΩ ≤ M|φ|1,w |ψ|1,w , Ω

where M denotes a generic positive constant, independent of φ and ψ.

60

3 Options on One Asset with Stochastic Volatility

Similarly, we have         3 3 I2 ≤  ρσν Sν 2 φ S ψν dΩ  +  ρσν Sν 2 φν ψ S dΩ  Ω Ω     1 1     = (ρν 2 Sφ S , σν νψν ) + (ρσν νφν , ν 2 Sψ S )   1   1  ≤M

Ω

 ≤M

Ω

2

ν S 2 φ S2 dΩ

·

Ω

ν 2 ψν2 dΩ

2

+

Ω

ν 2 φν2 dΩ

 1  1  2 2 2 2 · ν S ψ S dΩ Ω

 1  1 2 2 (ν S 2 φ S2 + ν 2 φν2 )dΩ · (ν S 2 ψ S2 + ν 2 ψν2 )dΩ ≤ M|φ|1,w |ψ|1,w . Ω

Thus, using the above estimates for I1 and I2 , we obtain from (3.1.19) |(A∇φ, ∇ψ)| ≤ M|φ|1,w |ψ|1,w . For |(bv, ∇w)| = |

 Ω

(3.1.20)

bφ · ∇ψdΩ| in (3.1.18), using integration by parts gives

           |(bφ, ∇ψ)| =  φψb · nds − ψ∇ · (bφ)dΩ  = − ψb · ∇φdΩ − φψ∇ · bdΩ  ∂Ω Ω Ω Ω             ≤  ψb · ∇φdΩ  +  φψ∇ · bdΩ  ≤  ψb · ∇φdΩ  + Mv0 w0 . Ω

Ω

Ω

Furthermore, from (3.1.15) and by Cauchy–Schwarz inequality, we obtain            3 1 1 1 ψb · ∇φdΩ  =  ψ r − ρν 2 σν − ν Sν S + μν − ρν 2 σν − σν2 νφν dΩ  4 2 Ω Ω     1  1   2 2 ≤ M  ψ(Sφ S + νφν )dΩ  ≤ M ψ 2 dΩ · (Sφ S + νφν )2 dΩ

   

Ω

Ω

Ω

≤ Mw0 |v|1,w .

Therefore, combining the above two estimates, we have,   |(bφ, ∇ψ)| ≤ M ψ0 |φ|1,w + v0 w0 .

(3.1.21)

It is trivial to show |(cφ, ˆ ψ)| ≤ Mφ0 ψ0 . Thus, Combining (3.1.18), (3.1.20) and (3.1.21) estimate, we have   |B(φ, ψ; t)| ≤ M |φ|1,w |ψ|1,w + ψ0 |φ|1,w + φ0 ψ0 ≤ Mφ1,w w1,w . Therefore, B(φ, ψ; t) is Lipschitz continuous in φ and ψ. Since B(·, ·; t) is coercive and continuous, from [3, Theorem 1.33], we see that Problem 3.1.1 has a unique solution, completing the proof. 

3.2 The Fitted FVM for (3.1.9)–(3.1.11)

61

3.2 The Fitted FVM for (3.1.9)–(3.1.11) We now present the FVM for (3.1.5) proposed in [7]. In what follows, we assume that r and μν are also functions of t and σν is a function of S and t. For given positive integers N S and Nν , we define a mesh for Ω with mesh nodes 0 = S0 < S1 < · · · < S N S = Smax , νmin = ν0 < ν1 < · · · < ν Nν = νmax . (3.2.1) Dual to this mesh, we define a secondary mesh for Ω with nodes (Si+ 21 , ν j+ 21 ) for i = −1, 0, 1, . . . , N S and j = −1, 0, 1, . . . , Nν , where S− 21 = 0, S N S + 21 = Smax , ν− 21 = νmin , ν Nν + 21 = νmax , Si+ 21 = 21 (Si + Si+1 ) for i = 0, 1, . . . , N S − 1 and ν j+ 21 = 21 (ν j + ν j+1 ) for j = 0, 1, . . . , Nν − 1. For each i = 0, 1, . . . , N S and j = 0, 1, . . . , Nν , we put h Si = Si+1/2 − Si−1/2 and h ν j = ν j+1/2 − ν j−1/2 . A typical local stencil of the meshes is depicted in Fig. 3.1 in which Ωi j := (Si− 21 , Si+ 21 ) × (ν j− 21 , ν j+ 21 ). Integrating (3.1.9) over Ωi j and using integration by parts, we have  −

Ωi j

∂V dΩ − ∂t





∂Ωi j

κ(V ) · nds +

Ωi j

cV dΩ = 0

for i = 1, 2, . . . , N S − 1 and j = 1, 2, . . . , Nν − 1, where ∂Ωi j denotes the boundary of Ωi j ans n denotes the unit vector outward-normal to ∂Ωi j . Applying the one-point quadrature rule to the first and third terms, we obtain from the above ∂ Vi, j |Ωi j | − − ∂t

 ∂Ωi j

κ(V ) · nds + ci, j Vi, j |Ωi j | = 0,

(3.2.2)

where | · | denote the measure or absolute value, depending on the context, ci, j = c(Si , ν j , t), and Vi, j denotes the nodal approximation to V (Si , ν j , t). We now consider the approximation of the 2nd term in (3.2.2). Note that ∂Ωi j is the boundary of the rectangle oriented counter-clockwise as depicted Fig. 3.1 by dashed lines. Thus, from the definition of κ we have

Fig. 3.1 A local structure of the meshes

62

3 Options on One Asset with Stochastic Volatility



 ∂Ωi j



 ∂V ∂V + a12 + b1 V dν ∂S ∂ν (Si+ 1 ,ν j− 1 ) (Si− 1 ,ν j− 1 ) 2 2 2 2      (S 1 ,ν 1 ) (Si+ 1 ,ν j− 1 ) i+ 2 j+ 2 ∂V ∂V 2 2 a21 + − + a22 + b2 V d S ∂S ∂ν (Si− 1 ,ν j− 1 ) (Si− 1 ,ν j+ 1 )

κ(V ) · nds =

2

=:

Ji,1 j

−

(Si+ 1 ,ν j+ 1 ) 2

2

2

Ji,2 j

2

+

Ji,3 j

−



−

(Si− 1 ,ν j+ 1 ) 2

2

a11

2

Ji,4 j .

(3.2.3)

To approximate Ji,1 j and Ji,2 j on the RHS of (3.2.3), we first develop approximations to their integrand at the mid-points of I Si−1 and I Si , where I Sk := (Sk , Sk+1 ). Let us consider a constant approximation to this integrand on i Sk , k = i, i − 1, in the following two cases. Case 1. Approximation of Ji1j and Ji2j when i > 1. Using (3.1.6)–(3.1.7), we rewrite the integrand as a11

∂V ∂V + a12 + b1 V = S ∂S ∂ν

 p(ν)S

∂V ∂V + q(ν)V + d(ν) ∂S ∂ν

1

 ,

3

where p = 21 ν, q = r − 43 ρν 2 σν − ν and d = 21 ρσν ν 2 . Following the discussion in Sect. 1.3.1, we approximate the term pS ∂∂VS + q V on I Si and ν = ν j by solving the following two-point boundary value problem  pj S

∂W + qi+ 21 , j W ∂S



= 0, W (Si , ν j ) = Vi, j , W (Si+1 , ν j ) = Vi+1, j , (3.2.4)

where p j = p(ν j ), qi+ 21 , j = q(xi+ 21 , y j ), and Vi, j and Vi+1, j are approximations to the nodal values of V at (Si , ν j ) and (Si+1 , ν j ), respectively. Using the same argument as that for (1.3.7), we have  pS

∂V + qV ∂S



α

(Si+ 1 ,ν j )

≈ qi+ 21 , j

2

α

i, j Si+1 Vi+1, j − Si i, j Vi, j

α

α

i, j Si+1 − Si i, j

=: τi, j (V ),

(3.2.5)

where αi, j = qi+ 21 , j / p j . Therefore, we define the following approximation: Ji,1 j ≈ Si+ 21

  Vi, j+1 − Vi, j τi, j + d j hν j . hν j

(3.2.6)

Similarly, we approximate pS ∂∂VS + q V on I Si−1 using (3.2.5) with i replaced with i − 1, yielding the following approximation to Ji,2 j :  Ji,2 j

≈ Si− 21

τi−1, j

Vi, j+1 − Vi, j + dj hν j

 hν j .

(3.2.7)

3.2 The Fitted FVM for (3.1.9)–(3.1.11)

63

Case 2. Approximation of J11j and J12j . Note that (3.2.6) also holds true for J11j . However, as mentioned in Sect. 1.3.1, (3.2.4) becomes degenerate on I S0 . Thus, we need to approximate pS ∂∂VS + q V on I S0 in a different way. More specifically, following Case II in Sect. 1.3.1, we consider the following problem. 

∂W pj S + q 21 , j W ∂S



= C,

lim W (S, ν j ) = V0, j , W (S1 , ν j ) = V1, j , (3.2.8)

S→+0

where C is an additive constant. Solving (3.2.8) exactly and following the same argument for (1.3.12), we have the following approximation:  pS

∂V + qV ∂S

 (S 1 ,ν j )

≈

 1 ( p j + q 21 , j )V1, j − ( p j − q 21 , j )V0, j =: τ0, j (V ). 2

2

(3.2.9)

Similarly to (3.2.7), we define the following approximation to J0,2 j . J1,2 j

  V1, j+1 − V1, j hν j . ≈ S 21 τ0, j + d j hν j

(3.2.10)

To approximation of Ji3j and Ji4j , we first write their common integrand as ∂V ∂V + a22 + b2 V = ν a21 ∂S ∂ν



∂V ¯ ν) ∂ V + q(S, ¯ ν)V + d(S, p(S)ν ¯ ∂ν ∂S

 ,

where p¯ = 21 σν , q¯ = μν − 21 ρσν ν 1/2 − σν2 and d¯ = ρσν Sν 1/2 . Then, consider the following 2-point boundary value problem on (ν j , ν j+1 ):  p¯ i ν

∂W + q¯i, j+ 21 W ∂ν



= 0, W (Si , ν j ) = Vi, j , W (Si , ν j+1 ) = Vi, j+1 ,

¯ i ) and q¯i, j+ 21 = q(S ¯ i , ν j+ 21 ). Using an argument similar to that in where p¯ i = p(S Case 1 above, we define the following approximation:  pν ¯

∂V + q¯ V ∂ν



α¯

(Si ,ν j+ 1 ) 2

≈ q¯i, j+ 21

α¯

i, j ν j+1 Vi, j+1 − ν j i, j Vi, j

α¯

α¯

i, j ν j+1 − ν j i, j

=: τ¯i, j (V )

(3.2.11)

for i = 1, 2, . . . , N S − 1 and j = 0, 1, . . . , Nν − 1, where α¯ i, j = q¯i, j+ 21 / p¯ i . Thus, Ji1j and Ji2j are approximated respectively by

64

3 Options on One Asset with Stochastic Volatility

Ji,3 j ≈ ν j+ 1 2

    Vi+1, j − Vi, j Vi+1, j − Vi, j τ¯i, j + d¯i, j h Si , Ji,4 j ≈ ν j− 1 τ¯i, j−1 + d¯i, j h Si , 2 h Si h Si

(3.2.12) ¯ i , ν j ). Note Case 2 above does not apply for j = 1, 2, . . . , Nν − 1, where d¯i, j = d(S to the approximation of Ji3j and Ji4j , as ν0 = νmin > 0. Replacing the integrals on the RHS of (3.2.3) with their respective approximations in (3.2.6)–(3.2.12), we have the following system approximating (3.2.2): −

∂ Vi, j |Ωi j | + ei, j;i, j−1 Vi, j−1 + ei, j;i−1, j Vi−1, j + ei, j;i, j Vi, j ∂t + ei, j;i+1, j Vi+1, j + ei, j;i, j+1 Vi, j+1 = 0,

(3.2.13)

for i = 1, 2, . . . , N S − 1, j = 1, 2, . . . , Nν − 1, where α¯

e1, j;1, j−1 = −h S1 ν j− 1 q¯1, j− 1 2

2

α

e1, j;1, j = S 3 h ν j 2

+

q 3 , j S1 1, j

2 α1, j

α

− S1 1, j

S2

1, j−1 ν j−1

1 , e1, j;i−1, j = − S 1 h ν j ( p j − q 1 , j ), (3.2.14) α¯ α¯ 1, j−1 2 2 2 ν j 1, j−1 − ν j−1 ⎛ ⎞ α¯ α¯ q¯1, j+ 1 ν j+ 1 ν j 1, j ν j 1, j−1 2 2 ⎝ ⎠ + +h S1 + ν j− 1 q¯1, j− 1 α¯ α¯ 1, j α¯ α¯ 1, j−1 2 2 ν j+1 − ν j 1, j ν j 1, j−1 − ν j−1

1 S 1 ν y ( p j + q 1 , j ) + h ν j d¯1, j + d j h S1 + c1, j |Ωi j |, 2 2 2 j α q 3 , j S2 1, j 2 α1, j α S2 − S1 1, j

e1, j;2, j = −d¯1, j h ν j − S 3 h ν j 2

(3.2.15)

, e1, j;1, j+1 = −d j h S1 − h S1

α¯ 1, j b¯1, j+ 1 ν j+ 1 ν j+1 α¯

2

2

α¯

1, j S j+1 − ν j 1, j

(3.2.16) for j = 1, 2, . . . , N y − 1, and α

α¯

ei, j;i, j−1 = −h Si ν j− 1 b¯i, j− 1 2

⎛ 2

α¯ b¯i, j+ 1 ν j+ 1 y j i, j α¯

2

α¯

α¯

i, j−1 ν j i, j−1 − ν j−1

α qi+ 1 , j Si i, j 2 αi, j α Si+1 − Si i, j

ei, j;i, j = h ν j ⎝ Si+ 1

+ h Si (

2

i, j−1 ν j−1

2

α¯

i, j ν j+1 − ν j i, j

+ Si− 1 2

, ei, j;i+1, j = −Si− 1 h ν j 2

α qi− 1 , j Si i−1, j 2 α αi−1, j Si i−1, j − Si−1

i−1, j qi− 1 , j Si−1

2 αi−1, j

Si

α

i−1, j − Si−1

,

(3.2.17)

⎞ ⎠ + h ν j d¯i, j + d j h Si

α¯

+ ν j− 1 q¯i, j− 1 2

2

ν j i, j−1 α¯

α¯

i, j−1 ν j i, j−1 − ν j−1

) + ci, j |Ωi j |,

α

ei, j;i+1, j = −d¯i, j h ν j − Si+ 1 h ν j 2

i, j qi+ 1 , j Si+1

α

2

α

i, j Si+1 − Si i, j

(3.2.18) α¯

, ei, j;i, j+1 = −d j h Si − h Si

i, j q¯i, j+ 1 ν j+ 1 ν j+1

α¯

2

2

α¯

i, j ν j+1 − ν j i, j

,

(3.2.19) for i = 2, 3, . . . , N S − 1, j = 1, 2, . . . , Nν − 1.

3.2 The Fitted FVM for (3.1.9)–(3.1.11)

65

Equation (3.2.13) is a linear system in Vi, j , i = 0, 1, . . . , N S and j = 0, 1, . . . , Nν . However, Vi, j equal to the boundary values when i = 0, i = N S , j = 0 or j = Nν . Thus, these known terms are cast to the RHS of (3.2.13). Rearrange the 2D array {Vi, j } as a 1D one in the order V1,1 , V2,1 , . . . , VN S −1,1 , . . . , V1,Nν −1 , …,VN S −1,Nν −1 . This mapping is achieved under the transformation n = i + ( j − 1)(N S − 1) for i = 1, 2, . . . , N S − 1 and j = 1, 2, . . . , Nν − 1. Let V¯n = Vi, j for a feasible (i, j). Then, (3.2.13) is an N × N linear system in V¯ = (V¯1 , . . . , V¯ N ) , which can be written as ∂ V¯n (t) |Ωn | + E¯ n V¯ = f n (3.2.20) − ∂t for n = 1, 2, . . . , N , where E¯ n = (e¯n,1 , . . . , e¯n,n ) and f n is the contribution from the boundary conditions g1 , g2 , G 1 and G 2 in (3.1.10)–(3.1.11). From our analysis we see that E¯ n has up to 5 non-zero entries given by e¯n,n−(N S −1) = ei, j;i, j−1 , e¯n,n−1 = ei, j;i−1, j , e¯n,n = ei, j;i, j , e¯n,n+1 = ei, j;i+1, j , and e¯n,n+(N S −1) = ei, j;i, j+1 with ei, j;l,m defined in (3.2.14)–(3.2.19). Other entries of E¯ n are zeros. The RHS term f n is nonzero only when its corresponding mesh node index (i, j) satisfies i = 1, i = N S − 1, j = 1 or j = Nν − 1. Equation (3.2.20) is semi-discrete. To discretize it in t, we choose a mesh for [0, T ] with nodes tk (k = 0, 1, . . . K ) satisfying T = t0 > t1 > · · · > t K = 0. As in Sect. 1.3.2, the implicit time-stepping scheme with a splitting parameter θ ∈ [ 21 , 1] is used to discretize the time derivative in (3.2.20), yielding V¯nk+1 − V¯nk |Ωn | + θ E¯ nk+1 V¯ k+1 + (1 − θ ) E¯ nk V¯ k = θ f nk+1 + (1 − θ ) f nk , −Δtk for k = 0, 1, 2, . . . , K − 1 and n = 1, 2, . . . , N with V¯ 0 the payoff condition g3 in (3.1.10), where Δtk = tk+1 − tk < 0, E¯ nl = E¯ n (tl ) and f nl = f n (tl ) for l = k and k + 1. The above system can be written as the following matrix equation: 

(θ E¯ k+1 + G k )V¯ k+1 = [G k − (1 − θ ) E¯ k ]V¯ k + θ f nk+1 + (1 − θ ) f nk , V¯n0 = g3 (Si , ν j ), n = i + ( j − 1)(N S − 1), i = 1, . . . , Ns , j = 1, . . . , Nν , (3.2.21) 1 diag(|Ω |, . . . , |Ω |). The cases that for k = 0, 1, · · · , K − 1, where G k = −Δt 1 N k θ = 21 and θ = 1 correspond to Crank–Nicolson and Backward Euler (or fully implicit) time-stepping schemes respectively. We now show that the system matrix of (3.2.21) is an M-matrix in the following theorem. Theorem 3.2.1 Let k ∈ {0, 1, . . . , K − 1}. If |Δtk | is sufficiently small, then the system matrix θ E¯ k+1 + G k of (3.2.21) is an M-matrix.

66

3 Options on One Asset with Stochastic Volatility

Proof We have shown E¯ k+1 is penta-diagonal with non-zero entries e¯n,n−(N S −1) = ei, j;i, j−1 , e¯n,n−1 = ei, j;i−1, j , e¯n,n = ei, j;i, j , e¯n,n+1 = ei, j;i+1, j , and e¯n,n+(N S −1) = ei, j;i, j+1 evaluated at tk in the nth row, with ei, j;l,m defined in (3.2.14)–(3.2.19). We now show that E¯ k+1 is an M-matrix when ci,k+1 j := ci, j (tk+1 ) ≥ 0. From the proof of Theorems 1.3.1 and (1.3.8), we have that S α α−S α > 0 and i+1 i α > 0 for any α > 0 and feasible (i, j). This, from their definitions we see ν α −ν α j+1

j

k+1 k+1 that ei,k+1 j;l,m < 0 if i = l or j = m, and ei, j;i, j > 0 when ci, j ≥ 0. Furthermore, from (3.2.14)–(3.2.19) we also have k+1 k+1 k+1 k+1 k+1 ei,k+1 j;i, j ≥ |ei, j−1;i, j | + |ei−1, j;i, j | + |ei+1, j;i, j | + |ei, j+1;i, j | + ci, j |Ωi j |,

and d¯i,k+1 since d k+1 j j are non-negative. Using the mapping n = i + ( j − 1)(N S − 1),  k+1 ¯ k+1 is ≥ 1≤l≤N ,l =n |e¯n+1 we see that the above inequality is e¯n,n j,n |, Therefore, E k+1 diagonally dominant with respect to its columns when ci, j ≥ 0. Also, when i = 1, i = N S − 1, j = 1 or j = Nν − 1, the corresponding row n = i + ( j − 1)(N S − 1) k+1 > has only 4 or less non-zero off-diagonal entries (and f n = 0). In this case, e¯n,n  n+1 k+1 ¯ is irreducible, as otherwise, (3.2.21) can be 1≤l≤N ,l =n |e¯ j,n |. Obviously, E decomposed into two disjoint sub-problems which can be solved sequentially. Thus, E¯ k+1 is an irreducibly diagonally dominant matrix. Using the result from [10, p.85], we conclude that E¯ k+1 is an M-matrix, when ci,k+1 j ≥ 0. The matrix G k in (3.2.21) is a diagonal matrix with positive diagonal entries. When |Ωi j | ¯ k+1 + G k is an |Δtk | is sufficiently small, we expect ci,k+1 j + −Δtk > 0. Therefore θ E  M-matrix when |Δtk | is sufficiently small, proving this theorem.

3.3 Convergence of the FVM We present a convergence analysis for the above FVM, based on the discussion in [8]. For simplicity, we assume ∂u/∂t = 0 in (3.1.16).

3.3.1 Reformulation of the FVM For the dual meshes constructed in Sect. 1.2, we let ψi, j denote the characteristic function given by ψi, j (S, ν) = 1 when (S, ν) ∈ Ωi j ans 0 otherwise. We choose a test space Th := span {ψi, j , i = 1, 2, . . . , N1 − 1, j = 1, 2, . . . , N2 − 1}. To define the trial space, let

3.3 Convergence of the FVM

67

Fig. 3.2 Hat function φi, j (S, ν) and its support

φi, j

⎧   ! "αi−1, j −1  ! "αi−1, j  Si αi−1, j Si S ⎪ ⎪ 1 − 1 − , (S, ν) ∈ Di, j,1 , ⎪ S Si−1 Si−1 ⎪ ⎪  ! "   ! " −1 ⎪ αi, j αi, j   Si αi, j ⎪ S i ⎪ 1 − SSi+1 1 − Si+1 , (S, ν) ∈ Di, j,3 , ⎪ ⎪ S ⎪    ⎨ −1 ! ! "α¯ i, j−1 "α¯ i, j−1   ν j α¯ i, j−1 νj ν = 1 − ν j−1 1 − ν j−1 , (S, ν) ∈ Di, j,2 , (3.3.1) ν ⎪ ⎪ ⎪     ⎪  ! ! "α¯ i, j −1 "α¯ i, j ⎪ ⎪ νi α¯ i, j νi ν ⎪ ⎪ 1 − 1 − , (S, ν) ∈ Di, j,4 , ⎪ ν ν j+1 ν j+1 ⎪ ⎪ ⎩ 0, otherwise,

for i = 2, 3, . . . , N S − 1 and j = 1, 2, . . . , Nν − 1, where αi, j and α¯ i, j are as defined in Sect. 3.2 and Di, j,l , l = 1, 2, 3, 4, denote the quadrilateral domains depicted in Fig. 3.2a. For example Di, j,1 is the diamond region with vertices (Si−1 , ν j ), (Si− 21 , ν j− 21 ), (Si , ν j ) and (Si− 21 , ν j+ 21 ). Each of the expressions in (3.3.1) is a solution to a local two-point boundary value problem such as (3.2.4). When i = 1, φ1, j = SS1 on D1, j,1 , which is a particular solution to (3.2.8). From (3.3.1) we see that the sup4 Di, j,l which is also a quadrilateral region as shown in Fig. 3.2a and port of φi, j is ∪l=1 an examples of φi, j is shown in Fig. 3.2b, from which it is seen that it is piecewise monotone and discontinuous across the inter-element boundaries. For these basis functions we have the following theorem: Theorem 3.3.1 The function in (3.3.1) has the following properties. 1. φi, j is increasing on Di, j,1 and Di, j,2 and decreasing on Di, j,3 and Di, j,4 . 2. φi, j + φi+1, j = 1 for (S, ν) ∈ Di, j,3 and φi, j + φi, j+1 = 1 for (S, ν) ∈ Di, j,4 . Proof For Item 1, we only show φi, j is decreasing on Di, j,3 in S. Differentiating φi, j with respect to S on Di, j,3 , we have α

S i, j ∂φi, j −αi, j ! "αi, j αii, j +1 , (S, ν) ∈ Di, j,3 . (S, ν) = ∂S S i 1 − SSi+1

68

3 Options on One Asset with Stochastic Volatility

Since Si+1 − Si > 0, we have αi, j /[1 − (Si /Si+1 )αi, j ] > 0 when αi, j = 0. Therefore, ∂φi, j (x, y) < 0 and so φi, j is monotonically decreasing in (xi , xi+1 ). Similarly, we ∂x can show φi, j is monotonically decreasing in Di, j,4 and increasing in Di, j,1 and in Di, j,2 . We now prove Item 2. From (3.3.1) we have α

φi, j + φi+1, j = α

=

α

α

α

i, j S αi, j (Si+1 − Si i, j )

α

α

+ α

α

i, j Si+1 (Si i, j − S αi, j )

α

α

i, j S αi, j (Si i, j − Si+1 )

α

α

α

α

i, j i, j i, j i, j − S αi, j )(Si i, j − Si+1 ) + Si+1 (Si i, j − S αi, j )(Si+1 − Si i, j ) Si i, j (Si+1

α

α

=

α

i, j − S αi, j ) Si i, j (Si+1

α

α

α

α

α

i, j i, j S αi, j (Si+1 − Si i, j )(Si i, j − Si+1 )

α

α

α

α

α

α

α

i, j i, j i, j i, j − Si i, j )(Si+1 Si i, j − Si+1 S αi, j − xi i, j Si+1 + Si i, j S αi, j ) (Si+1

α

α

i, j i, j S αi, j (Si+1 − Si i, j )(Si i, j − Si+1 )

=1

for all (S, ν) ∈ Di, j,3 . Similarly, φi, j + φi, j+1 = 1, for all (S, ν) ∈ Di, j,4 .



We choose the trial space Uh = span{φi, j : i = 1, . . . , N S − 1, j = 1, . . . , Nν − 0 ¯ 1}. PNνdenote the mass lumping operator from C (Ω) to Th defined by P(v) =  NLet S v(S , ν )ψ and Q and Q denote other mass lumping operators from i j i, j 1 2 i=0 j=0  N S  Nν −1 v(Si ,ν j+1 )−v(Si ,ν j ) ∂v 1 ¯ 1 ¯ C (Ω) to Th such that for any v ∈ C (Ω), Q 1 ( ∂ν )= i=0 j=0 ψi, j hν j   ,ν )−v(S ,ν ) v(S N S −1 Nν i+1 j i j and Q 2 ( ∂∂vS ) = i=0 ψi, j where ψi, j is defined in before. j=0 hSj Similarly to Problem 1.3.1, we define the following Petrov–Galerkin problem. Problem 3.3.1 Find u h ∈ Uh such that for all vh ∈ Th Ah (u h , vh ) = (P( f ), vh ),

(3.3.2)

where Ah (·, ·) denotes the bilinear form on Uh × Th defined by ! ∂u "   ! ∂u "  h h ˆ h ), vh ) , vh − Q 2 d¯ , vh + (P(cu Ah (u h , vh ) = − Q 1 d ∂ν ∂S N S −1 N ν −1 # $  Si+ 21 τi, j (u h ) − Si− 21 τi−1, j (u h ) h ν j vh |Ωi j − 

i=1

−

j=1

N S −1 N ν −1 # 

$ ν j+ 21 τ¯i, j (u h ) − ν j− 21 τ¯i, j−1 (u h ) h Si vh |Ωi j ,

i=1

(3.3.3)

j=1

where τi, j (u) is defined in (3.2.5) and (3.2.9) and τ¯i, j is given in (3.2.11). From the construction of the FVM, particularly the arguments leading to the approximation of Ji,l j for l = 1, 2, 3, 4, in Sect. 3.2 we see if vh = 1 when (S, ν) ∈ Ωi j and 0 otherwise, then (3.3.2) becomes (3.2.13). Using the lumping operator P, Problem 3.3.1 can be written as the following equivalent Galerkin finite element problem.

3.3 Convergence of the FVM

69

Problem 3.3.2 Find u h ∈ Uh such that for all vh ∈ Uh , Bh (u h , vh ) := Ah (u h , P(vh )) = (P( f ), vh ).

(3.3.4)

3.3.2 Stability and Convergence For any vh ∈ Uh , we define the following discrete norms and semi-norms. vh 21,h S

:=

N S −1 N ν −1  i=1

vh 21,h ν

:=

vh 20,h :=

j=1

N S −1 N ν −1  i=1

ν 2j+1/2 b¯i, j+1/2 h Si

j=1

N S −1 N ν −1  i=1

2 Si+1/2 bi+1/2, j h ν j

α

α

α

α

α¯

α¯

α¯

α¯

i, j Si+1 + Si i, j i, j Si+1 − Si i, j i, j ν j+1 + ν j i, j i, j ν j+1 − ν j i, j

(vi+1, j − vi, j )2 , (vi, j+1 − vi, j )2 ,

vi,2 j |Ωi j |.

j=1

Using these norms and semi-norms, we define the following weighted discrete H 1 norm vh 2h = vh 21,h S + vh 21,h ν + vh 20,h . The following theorem establishes the coercivity of Bh (·, ·). Theorem 3.3.2 Let h be sufficiently small. For any vh ∈ Uh , we have Bh (vh , vh ) ≥ Cvh 2h , where C denotes the positive constant, independent of h and vh .  N1 −1  N2 −1 Proof Let vh := i=1 j=1 vi, j φi, j ∈ Uh . Replacing vh in (3.3.3) with P(vh ) gives Bh (vh , vh ) = −

N S −1 N ν −1 #  i=1

−

N S −1 N ν −1 #  i=1

+

j=1

N S −1 N ν −1  i=1

j=1

j=1

$ Si+ 1 τi, j (vh ) − Si− 1 τi−1, j (vh ) h ν j vi, j 2

2

N S −1 N ν −1 $  di, j 2 ν j+ 1 τ¯i, j (vh ) − ν j− 1 τ¯i, j−1 h Si vi, j + (v − vi, j+1 vi, j )|Ωi j | 2 2 h ν j i, j i=1

d¯i, j 2 (v − vi+1, j vi, j )|Ωi j | + h Si i, j

N S −1 N ν −1  i=1

j=1

cˆi, j vi,2 j |Ωi j |.

j=1

Rearranging the first and second sums with v0, j = 0, we have

70

3 Options on One Asset with Stochastic Volatility

B(vh , vh ) =

N ν −1 

S1/2 ( p j + q 21 , j )h ν j 2

j=1

−

N S −1 N ν −1  i=1

+ − +

i=1

+ +

i=1

=

j=1

N S −1 N ν −1 

N ν −1 

α

α



α¯

i, j ν j+1 vi, j+1 − ν j i, j vi, j

α¯

α¯

· h Si vi, j

α¯



i, j−1 ν j i, j−1 vi, j − ν j−1 vi, j−1

α¯

α¯

i, j−1 ν j i, j−1 − ν j−1

· h Si vi, j

N S −1 N ν −1  di, j 2 d¯i, j 2 (vi, j − vi, j+1 vi, j )|Ωi j | + (v − vi+1, j vi, j )|Ωi j | hν j h Si i, j i=1 j=1

cˆi, j vi,2 j |Ωi j |

2

2 v1, j +

N S −1

α¯

ν 21 q¯i, 21 h Si

i=1

ν1 i,0

α¯

α¯

ν1 i,0 − ν0 i,0

α α S i, j vi+1, j − Si i, j vi, j Si+ 21 qi+ 21 , j h ν j i+1 αi, j α Si+1 − Si i, j j=1

N S −1 N ν −1 





i, j ν j+1 − ν j i, j

α¯

ν j−1/2 q¯i, j− 21

· h ν j vi, j

α

i−1, j Si i−1, j − Si−1

α¯

q¯i, j+1/2

S 21 ( p j + q 21 , j )h ν j

i=1



α

i−1, j Si i−1, j vi, j − Si−1 vi−1, j

 ν j+ 21

· h ν j vi, j

α

i, j Si+1 − Si i, j

α

qi− 21 , j



j=1

j=1

+

qi+ 21 , j

α

i, j Si+1 vi+1, j − Si i, j vi, j

j=1

N S −1 N ν −1  i=1

Si− 21

j=1

N S −1 N ν −1 

α



j=1

N S −1 N ν −1  i=1

Si+ 21

j=1

N S −1 N ν −1  i=2



2 v1, j

2 vi,1

(vi+1, j − vi, j )





I1

+

N S −1 N ν −1  i=1



α¯

ν j+1/2 q¯i, j+ 21 h Si

j=1

α¯

i, j ν j+1 vi, j+1 − ν j i, j vi, j

α¯

α¯

i, j ν j+1 − ν j i, j



· (vi, j+1 − vi, j ) 

I2

+

N S −1 N ν −1  i=1



j=1

N S −1 N ν −1  di, j 2 d¯i, j 2 (vi, j − vi, j+1 vi, j )|Ωi j | + (v − vi+1, j vi, j )|Ωi j | hν j h Si i, j i=1 j=1

  I3

+

N S −1 N ν −1  i=1

cˆi, j vi,2 j |Ωi j |.

(3.3.5)

j=1

Using the argument for simplifying the term I in (1.3.31), we see the terms I1 and I2 can be rewritten as follows respectively.

3.3 Convergence of the FVM I1 =

  N S −1 N N2 −1 ν −1 qi+ 1 , j − qi− 1 , j 1 1  1  2 2 2 + qi+ 1 , j vi,2 j |Ωi j | − S 1 q 1 , j h ν j v1, Si− 1 vh 21,h S − j, 2 2 2 2 2 2 h Si 2 i=1

I2 =

71

1 1 vh 21,h ν − 2 2

j=1

j=1

N S −1 N ν −1  i=1

 ν j− 1

2

j=1

+ q¯i, j+ 1

2

hν j

2

(3.3.6)



q¯i, j+ 1 − q¯i, j− 1

vi,2 j |Ωi j | −

2

1 2

N S −1 i=1

2 ν 1 q¯i, 1 h Si vi,1 , 2

2

(3.3.7) since v N S , j = 0 and vi,Nν = 0. For the term I3 , we have I3 =

N S −1 N ν −1  i=1

j=1

N 1 −1 N 2 −1   di, j 2 d¯i, j 2 (vi, j − vi, j+1 vi, j )|Ωi j | + (v − vi+1, j vi, j )|Ωi j | hν j h xi i, j i=1 j=1

N S −1 N ν −1 

≥ −C

i=1

vi,2 j |Ωi j |,

(3.3.8)

j=1

where C denotes a generic positive constant, independent of h S and h ν . Using (3.3.6)–(3.3.8), we have from (3.3.5) B(vh , vh ) =

N ν −1

S 1 ( p j + q 1 , j )h ν j 2

2

2

j=1

+

N S −1 N ν −1  

cˆi, j −

i=1

−

j=1

N S −1 i=1

Si− 1 qi+ 1 , j − qi− 1 , j 2

2

2

2

h Si

α¯

ν 1 q¯i, 1 h Si 2

2

qi+ 1 , j

−

2

2

−

ν1 i,0

α¯

α¯

ν1 i,0 − ν0 i,0

2 vi,1 +

1 1 vh 21,h S + vh 21,h ν 2 2

ν j− 1 q¯i, j+ 1 − q¯i, j− 1 2

2

2

2

hν j

−

q¯i, j+ 1  2

2

vi,2 j |Ωi j |

N S −1 Nν −1 1  1  2 2 S 1 q 1 , j h ν j v1, − ν1/2 q¯i, 1 h Si vi,1 + I3 j 2 2 2 2 2 j=1

≥

2 v1, j +

N ν −1

2

2

j=1

−

qi+ 1 , j 2

⎛ ≥C

i=1

S1 pj

2

2 h ν j v1, j +

N S −1 N ν −1  Si− 1 qi+ 1 , j − qi− 1 , j  1 1 2 2 2 β + ci, j − vh 21,h S + vh 21,h ν + 2 2 2 h Si i=1

−

⎝vh 21,h S

ν j− 1 q¯i, j+ 1 − q¯i, j− 1 2

2 + vh 21,h ν

2

2

hν j +

−

N S −1 N ν −1  i=1

q¯i, j+ 1 2

2



j=1

− K vi,2 j |Ωi j | ⎞

vi,2 j |Ωi j |⎠

≥ Cvh 2h ,

j=1

when h S and h ν are sufficiently small and β is sufficiently large, completing the proof.  Theorem 3.3.2 implies that the FVM in Sect. 3.2 is numerically stable. Let τ (u) := pS ∂∂uS + qu and τ¯ (u) := pν ¯ ∂u + qu. ¯ Analogous to (1.3.14) or the expression in ∂ν Lemma 1.3.1, we define the constant approximations to τh and τ¯h respectively by

72

3 Options on One Asset with Stochastic Volatility

 τh (u) =  τ¯h (u) =

τi, j (u) if (S, ν) ∈ Di, j,3 , i = 0, . . . , N S − 1, j = 0, . . . , Nν − 1, 0 otherwise, τ¯i, j (u) if (S, ν) ∈ Di, j,4 , i = 0, . . . , N S − 1, j = 0, . . . , Nν − 1, 0 otherwise,

where τi, j and τ¯i, j are given in (3.2.5), (3.2.9) and (3.2.11), and Di, j,k are defined in Fig. 3.2a with Di, j,3 = Di+1, j,1 and Di, j,4 = Di, j+1,2 . For these flux approximations, we have the following lemma. Lemma 3.3.1 Let w be a sufficiently smooth function, and w I be the Uh -interpolant of w. Then, there exists a constant C > 0, independent of h, such that τ (w) − τh (w I )∞,Di, j,3 ≤ C(τ (w)1,∞,Di, j,3 + q1,∞,Di, j,3 w∞,Di, j,3 )h Si , ¯ 1,∞,Di, j,4 w∞,Di, j,4 )h y j , τ¯ (w) − τ¯h (w I )∞,Di, j,4 ≤ C(τ¯ (w)1,∞,R Di, j,4 + b for i = 0, 1, . . . , N1 − 1 and j = 0, 1, . . . , N2 − 1. The proof of this lemma is essentially the same as that of Lemma 1.3.2 and thus we omit it. Using this lemma we have our following theorem. Theorem 3.3.3 Let u and u h the solutions to Problems 3.1.1 and 3.3.2 respectively. Then, there exists a constant C, independent of h, u h and u, such that ¯ 1,∞ u I − u h h ≤ Ch(τ (u)||1,∞ + τ¯ (u)1,∞ + u1,∞ + q1,∞ + q ¯ 1,∞ + c +d1,∞ + d ˆ 1,∞ + u2,∞ +  f 1 ), (3.3.9) where u I is the Uh -interpolant of u. Proof We assume that C is a generic positive constant, independent of h, u h and u = 0) by P(vh ) and using integration For any vh ∈ Sh , multiplying (3.1.12) (with ∂u ∂t by parts and (3.3.4), we have ⎧

−

N S −1 N ν −1 ⎨  i=1

j=1

⎩

Sτ (u) + d

∂u ∂ν

⎫ (Si ,ν j+ 1 ) ⎬  ∂u 2 2 h ν j + ν(τ¯ (u)) + d¯ h Si P(vh ) ⎭ ∂ S (Si ,ν 1 ) 1 ,ν j )

(Si+ 1 ,ν j ) (Si−

2

j− 2

+ (cu, ˆ P(vh )) = ( f, P(vh )) = ( f − P( f ), P(vh )) + Bh (u h , vh ). Taking Bh (u I , vh ) away from both side of the above equality and using an argument similar to that for (3.3.5), we have from the above equality,

3.3 Convergence of the FVM

73

  N S −1 N ν −1 (Si+ 1 ,ν j )   |Bh (u h − u I , vh )| = (cu ˆ − P(cu ˆ I ), P(vh )) − [S(τ (u) − τh (u I ))](S 21 ,ν j ) h ν j vi, j i− 2  i=1 j=1     ∂u ∂u I h v − d ) d − Q , P(v 1 h ) Si i, j ∂ν ∂ν i=1 j=1       ∂u I ∂u , P(vh ) − ( f − P( f ), P(vh )) − d¯ − Q 2 d¯ ∂S ∂S N S −1 N ν −1   |cu ˆ − cˆi, j u i, j |dΩvi, j ≤ −

N S −1 N ν −1 

i=1



(Si ,ν j+ 1 )

[ν(τ¯ (u) − τ¯h (u I )))](Si ,ν

Ωi j

j=1



2 j− 21



R1

+

N S −1 N ν −1  i=1



 vi, j

j=1

Ωi j

  ( )  ∂u u i, j+1 − u i, j   ∂u u i+1, j − u i, j   ¯ ¯ − di, j − di, j d  + d  dΩ  ∂ν  hν j ∂S h Si

  R2

    N ν −1   S −1 N +  [Si+ 1 (τ (u) − τh (u I ))(S 1 ,ν j ) (vi+1, j − vi, j )]h ν j  i+ 2 2   i=0 j=1

  R3

    N ν −1   S −1 N  + [ν j+ 1 (τ¯ (u) − τ¯h (u I )))(Si ,ν 1 ) (vi, j+1 − vi, j )]h Si  + |( f − P( f ), P(vh ))| . j+ 2 2

    i=1 j=0 R5

  R4

(3.3.10) Since the mass lumping operator P preserves constants, it is easy to show that R1 + R5 ≤ Ch(c ˆ 1,∞ + u1,∞ +  f 1 )vh h . For the first part of R2 , we have      N S −1 N ν −1  ∂u   u i, j+1 − u i, j  ∂u    (d − d dΩ v ) − d d dΩ ≤   i, j i, j i, j   hν j ∂ν  Ωi j  ∂ν Ωi j i=1 j=1 i=1 j=1     N S −1 N ν −1  u i, j+1 − u i, j  ∂u  + vi, j − di, j  dΩ ≤ Ch(dq,∞ + u2,∞ )vh h  ∂S hν j Ωi j 

N S −1 N ν −1 

i=1



vi, j

j=1

Similarly, we have estimate for the 2nd part of R2 . Thus, we conclude ¯ 1,∞ + ||u2,∞ )vh h . R2 ≤ Ch(d1,∞ + ||d|| Using Lemma 3.3.1 and Cauchy–Schwarz inequality we estimate R3 as follows.

74

3 Options on One Asset with Stochastic Volatility

R3 ≤ Ch(τ (u)1,∞ + q1,∞ u∞ )

N S −1 N ν −1  i=0

⎛

≤ Ch(τ (u)1,∞ + q1,∞ u∞ ) ⎝ S1

Si+ 1 h ν j |vi+1, j − vi, j | 2

j=1 N ν −1

h ν j |v1, j |

j=1

+

⎡

N S −1 N ν −1  i=1

S

1 2

⎣

i+ 21

j=1

α

⎞1

α

i, j − Si i, j Si+1

α

2

⎠

α

i, j qi+ 1 , j (Si+1 + Si i, j ) 2

1 1 1 α α q2 h (S i, j + Si i, j ) 2 i+ 21 i+ 21 , j ν j i+1 1 αi, j α (Si+1 − Si i, j ) 2

S2

⎤ ⎥ |vi+1, j − vi, j |⎦

⎡

1 ⎛ αi, j αi, j ⎞ 2 N Nν −1 S −1 N ν −1 S  i+ 21 (Si+1 − Si ) ⎢  ⎠ ≤ Ch(τ (u)1,∞ + q1,∞ u∞ ) ⎣ S1 h ν j |v1, j | + ⎝ α α q (S i, j + Si i, j ) j=1 i=1 j=1 i+ 1 , j i+1 2

⎛ ·⎝

N S −1 N ν −1  i=1

α

qi+ 1 , j Si+ 1 h ν j 2

j=1

2

α

i, j Si+1 + Si i, j

αi, j Si+1

α

− Si i, j

(vi+1, j

⎞1 ⎤ 2 2⎠ ⎥ − vi, j ) ⎦

⎡

⎤ ⎞1 ⎛ 2 αi, j α N Nν −1 S −1 N ν −1 S  Si+1 − Si i, j i+ 21 ⎢  ⎠ vh 1,h S ⎥ h ν j |v1, j | + ⎝ = Ch(τ (u)1,∞ + q1,∞ u∞ ) ⎣ S1 ⎦. αi, j α q 1 (Si+1 + Si i, j ) j=1 i=1 j=1 i+ 2 , j

(3.3.11) By a Taylor expansion we have αi, j Si+1 αi, j Si+1

− +

α Si i, j α Si i, j

"αi, j ! "αi, j ! h h 1 + 2x Si 1 − 1 − 2x Si 1 i+ 2 i+ 2 "αi, j ! "αi, j =! h Si h Si 1 + 2S 1 + 1 − 2x§ 1 i+ 2 i+ 2 ! " ! " ! ! h h 1 + αi, j O 2S Si 1 − 1 − αi, j O 2S Si 1 i+ 2 i+ 2 h Si "" ! "" ≤ Cαi, j ! ! =! . h xi h xi S i+ 21 1 + αi, j O 2xi+1/2 + 1 − αi, j O 2xi+1/2

From this estimate, we see that the sum in (3.3.11) can be estimated as N S −1 N ν −1  i=1

α

i, j Si+ 21 Si+1 − Si,α j

q 1 S αi, j j=1 i+ 2 , j i+1

+

α Si i, j

≤C

N S −1 N ν −1  i=1

j=1

h Si

αi, j qi+ 21 , j

≤ C,

since αi, j /qi+ 21 , j = p j . Therefore, ⎛ R3 ≤ Ch(τ (u)1,∞ + q1,∞ u∞ ) ⎝

N S −1 N ν −1  i=1

⎞ |v1, j |h S h ν + vh 1,h S ⎠

j=1

≤ Ch(τ (u)1,∞ + q1,∞ )vh h . Since R3 and R4 are symmetric, by symmetry, we have R4 ≤ Ch(τ¯ (u)1,∞ + q ¯ 1,∞ )vh h .

3.3 Convergence of the FVM

75

Replacing Ri ’s in (3.3.10) with their respective bounds above, we obtain ¯ 1,∞ |B(u h − u I , vh )| ≤ Ch(τ (u)1,∞ + τ¯ (u)1,∞ + u1,∞ + q1,∞ + q ¯ + d1,∞ + d1,∞ + c ˆ 1,∞ + u2,∞ +  f 1 )vh h . Setting vh = u h − u I in the above estimate and using Theorem 3.3.2 we obtain (3.3.9).  We remark that we can replace u I in (3.3.9) with u, as the norm  · h only uses values at the mesh nodes and both u and u I have the save nodal values. The following theorem shows that u h depends continuously on the given data f . Theorem 3.3.4 Let u h be the solution to Problem 3.3.4. There exists a constant C > 0, independent of h and u h , such that u h h ≤ C f 0,h . Proof Setting vh = u h in (3.3.4) and using Cauchy–Schwarz inequality we have Bh (u h , u h ) = (P( f ), P(u h )) =

N S −1 N ν −1  i=1

u i j f i j |Ωi j | ≤ u h 0,h  f 0,h ≤ Cu h h  f 0,h .

j=1

Combining Theorem 3.3.2 and the above inequality yields u h h ≤ C f 0,h .



3.4 Power Penalty Method for Pricing American Options with Stochastic Volatility In this section we shall extend the power penalty method in Chap. 2 to pricing American put options with stochastic volatility.

3.4.1 The Linear Complementarity Problem Let V be the valuation of an American put option on one asset with stochastic volatility. We introduce u = eβ (V0 − V ), where V0 is a sufficiently smooth function satisfying the boundary conditions in (3.1.10) as aforementioned. As in Chap. 2, u is governed by the following Linear Complementarity Problem (LCP). L2D u ≤ f, u − u ∗ ≤ 0, (L2D u − f ) (u − u ∗ ) = 0

(3.4.1)

in Ω × [0, T ) satisfying u(S, ν, t) = 0 for (S, ν, t) ∈ ∂Ω × (0, T ) and u(S, ν, T ) = g(S, ν), where L2D is defined in (3.1.12) and u ∗ is a given lower bound on u. In what follows, we assume, as in (2.1.8), u ∗ = eβt (V0 (S, ν) − V ∗ (S)), where V ∗ is the Vanilla payoff function defined by V ∗ (S) = max{0, K − S}.

76

3 Options on One Asset with Stochastic Volatility

1 1 Let K = {v ∈ H0,w (Ω) : v ≤ u ∗ }, where H0,w (Ω) is defined Sect. 3.1.2. Similarly to Theorem 2.1.1, we have that following variational inequality which is the variational form of (3.4.1).

Problem 3.4.1 Find u(t) ∈ K such that, for all v ∈ K ,   ∂u − , v − u + B(u, v − u; t) ≥ ( f, v − u) ∂t

(3.4.2)

for t ∈ (0, T ) a.e., where B(u, v; t) is a bilinear form defined in (3.1.15). The unique solvability of Problem 3.4.1 is given in the following theorem. Theorem 3.4.1 Problem 3.4.2 has a unique solution. Proof In the proof of Theorem 3.1.1, we have shown B(φ, φ; t) ≥ Cφ21,w and 1 (Ω). From [3, Lemma 1 & TheB(φ, ψ; t) ≤ Mφ1,w ψ1,w for any φ, ψ ∈ H0,w orem 1.33], we see that Problem 3.4.2 has a unique solution. 

3.4.2 The Penalty Method and Convergence Following the idea in Sect. 2.2.1, we approximate (3.4.1) by the following equation: L2D u λ + λ[u λ − u ∗ ]+ = f, (S, ν, t) ∈ Ω × [0, T ) 1/κ

(3.4.3)

satisfying the following boundary and payoff conditions u λ (S, ν, t) = 0 for (S, ν) ∈ ∂Ω and u λ (S, ν, T ) = u ∗ (S, , ν, T ),

(3.4.4)

where λ > 1 and κ > 0 are parameters. The variational problem corresponding to (3.4.3)–(3.4.4) is given below. 1 1 (Ω) such that, for all v ∈ H0,w (Ω), Problem 3.4.2 Find u λ (t) ∈ H0,w

 −

 ! " ∂u λ 1/κ , v + B(u λ , v; t) + λ [u λ − u ∗ ]+ , v = ( f, v), t ∈ (0, T ) a.e.. ∂t (3.4.5)

For this problem, we have the following theorem. 1 Theorem 3.4.2 Problem 3.4.2 has a unique solution in H0,w (Ω).

The proof is identical to that of Theorem 3.4.2, and this it is omitted here. Before further discussion, we first introduce a space. Let H be a Hilbert space. we use L p (0, T ; H (I )) to denote the space defined by L p (0, T ; H ) = {v : v((·, ·, t)) ∈ H a.e. in (0, T ); v((·, ·, t), t) H ∈ L p ((0, T ))},

3.4 Power Penalty Method for Pricing American Options …

for a p ∈ [1, ∞], equipped with the norm v L p (0,T ;H ) = Using this space, we have the following lemma.

77

! T 0

p

v(·, t) H dt

"1/ p .

Lemma 3.4.1 Let u λ be the solution to Problem 3.4.2. If u λ ∈ L p (Ω × (0, T )) with p = 1 + 1/κ, then there exists a constant C > 0, independent of u λ and λ, such that C , λκ C ≤ κ/2 . λ

[u λ − u ∗ ]+  L p (Ω×(0,T )) ≤ 1 [u λ − u ∗ ]+  L ∞ (0,T ;L 2 (Ω) + [u λ − u ∗ ]+  L 2 (0,T ;H0,w (Ω)

(3.4.6) (3.4.7)

Proof Let C is a generic positive constant, independent of u λ and λ. We introduce φ(S, ν, t) := [u λ − u ∗ ]+ . Note u λ ≤ u ∗ is satisfied on ∂Ω. This is because, when determine the boundary conditions for V , we either choose V (S, ν, t) = V ∗ (S, t) for S = 0 and S = Smax , or solve the 1D LCP (2.1.7) in Sect. 2.1.2 at ν = νmin and 1 (Ω) a.e. in (0, T ). Replacing v with φ in (3.4.5) gives νmax . Thus φ(·, ·, t) ∈ H0,w   ∂u λ − , φ + B(u λ , φ; t) + λ(φ 1/k , φ) = ( f, φ), a.e. in (0, T ). ∂t ∗

Subtracting −( ∂u , φ) + B(u ∗ , φ; t) from both sides of the above equation and then ∂t integrating the resulting equation from t to T , we have 

T t

   T  T ∂(u λ − u ∗ ) − B(u λ − u ∗ , φ; τ )dτ + λ (φ 1/κ , φ) , φ dτ + ∂τ t t  T ∗   T  T ∂u ( f, φ)dτ + B(u ∗ , φ; τ )dτ = , φ dτ − ∂t t t t  T 1/q  T 1/ p q p ≤  f (τ ) L q (Ω) dτ φ(τ ) L p (Ω) dτ t

t



T

+β

eβτ (V0 − V ∗ , φ(τ ))dτ −

t



T

B(u ∗ (τ ), φ(τ ); τ )dτ,

(3.4.8)

t

where q = 1 + κ so that 1/ p + 1/q = 1. In the above we used Hölder’s inequality. ∗ Note ∂(u λ∂t−u ) · φ = ∂φ · φ. Using integration bu parts, we have ∂t 

T t

   T ∂(u λ − u ∗ ) ∂φ − , φ dτ = (φ(t), φ(t)) − (− , φ)dτ, ∂τ ∂τ t

since φ(·, ·, T ) = 0. Thus,  t

T

  ∂φ 1 1 − , φ dτ = (φ(t), φ(t) = φ(t)2L 2 (Ω) . ∂τ 2 2

(3.4.9)

78

3 Options on One Asset with Stochastic Volatility

From the above estimate, (3.4.8) and (3.1.17), we get  T 1/ p  T  T 1 p p φ21,w dτ + λ φ L p (Ω) dτ ≤ C φ L p (Ω) dτ (φ, φ) + C 2 t t t  T  T   +β eβτ V0 − V ∗ , φ dτ − B(u ∗ , φ)dτ. (3.4.10) t

t

Let us consider the last two integrals in (3.4.10). Since V0 (S, ν) and V ∗ (S) are ¯ using Hölder inequality we have bounded on Ω, 

T

e

βτ



∗





T

V0 − V , φ dτ ≤ C

t

t



 Ω

T

φdΩdτ ≤ C t

1/ p p φ L p (Ω) dτ

. (3.4.11)

From (3.1.15) and u ∗ given in Sect. 3.4.1, we see that

  ˆ ∗ , φ) −B(u ∗ , φ; τ ) = − A∇u ∗ + bu ∗ , ∇φ + (cu  

= eβt (A∇V0 + bV0 , ∇φ) + (cV ˆ 0 , φ) − A∇V ∗ + bV ∗ , ∇φ − (cV ˆ ∗ , φ) . (3.4.12) Let Ω1 = {(S, ν) ∈ Ω : S < K } and Ω2 = {(S, ν) ∈ Ω : S > K }. From the definition of V ∗ in Sect. 3.4.1, we have that ∇V ∗ = (K , 0) when (S, ν) ∈ Ω1 and (0, 0) when (S, ν) ∈ Ω2 . Therefore, integrating by parts gives 

 K (a11 , a21 ) · ∇φd Sdν = K φ(a11 , a21 ) · nds Ω1 ∂Ω1     ∂a11 ∂a21 + φdΩ ≤ C φdΩ (3.4.13) −K ∂S ∂ν Ω1 Ω

 A∇V ∗ , ∇φ =



1 in (0, T ), because a11 , a12 ≥ 0 and φ ≥ 0. Note u ∗ ∈ H 1 (Ω) and φ ∈ H0,w (Ω). Using integration by parts again, we have

  ˆ ∗ , φ) = ∇ · (bu ∗ ), φ + (cu ˆ ∗ , φ) ≤ C −(bu ∗ , ∇φ) + (cu

 Ω

φdΩ.

(3.4.14)

By a similar argument for (3.4.13) and (3.4.14), we are able to show  ˆ 0 , φ) ≤ C (A∇V0 + bV0 , ∇φ) + (cV

Ω

φdΩ.

(3.4.15)

Integrating (3.4.12) from t to T and using (3.4.13)–(3.4.15), we have 

T

− t

∗

B(u (τ ), φ(τ ); τ )dτ ≤ C



 Ω

φ(S, τ )dΩ ≤ C t

T

1/ p p φ(τ ) L p (Ω) dτ

.

3.4 Power Penalty Method for Pricing American Options …

79

Replacing the last two terms in (2.2.10) with (3.4.11) and the above upper bound respectively, we obtain 

1 (φ(t), φ(t)) + 2



T

φ21,w dτ + λ

t



T

p

φ L p (Ω) dτ ≤ C

t

T

t

1/ p p

φ L p (Ω) dτ (3.4.16)

for all t ∈ [0, T ) a.e., which implies 

T

λ t

Thus

! t

T

 p φ(τ ) L p (Ω) dτ

p

φ(τ ) L p (Ω) dτ 

T

t

"1−1/ p

≤

T

≤C t C , λ

1/ p p φ(τ ) L p (Ω) dτ

.

from which we obtain 1/ p

p φ(τ ) L p (Ω) dτ

C

≤

λ1/( p−1)

=

C , λκ

(3.4.17)

since p = 1 + 1/κ. Thus, we have proved (3.4.6). From (3.4.16) and (3.4.17), we have 1 (φ(t), φ(t)) + 2

 t

T

 φ(τ )21,w dτ ≤ C

T t

1/ p p

φ(τ ) L p (Ω) dτ

≤

C . λκ

Thus, from the above inequality, we finally have  (φ(t), φ(t))

1/2

+ t

T

1/2 φ(τ )21,w dτ

≤

C λκ/2

for all t ∈ [0, T ). Thus, we have proved (3.4.7).



Using Lemma 3.4.1, we are able to prove the following convergence result. Theorem 3.4.3 Let ∂u be sufficiently smooth and the assumptions in Lemma 3.4.1 ∂t be fulfilled. The solutions u and u λ to Problems 3.4.1 and 3.4.2 respectively satisfy 1 u − u λ  L ∞ (0,T ;L 2 (Ω)) + u − u λ  L 2 (0,T ;H0,w (Ω)) ≤

C , λκ/2

where C is a positive constant, independent of u, u λ and λ. 1 Proof Note φ := [u λ − u ∗ ]+ ∈ H0,w (Ω). We decompose u − u λ as

u − u λ = u − u ∗ − (u λ − u ∗ ) = u − u ∗ + [u λ − u ∗ ]− − [u λ − u ∗ ]+ =: rλ − φ, (3.4.18) where [u λ − u ∗ ]− = − min{u λ − u ∗ , 0} and rλ = u − u ∗ + [u λ − u ∗ ]− . From the definition of φ, we see (φ α , [u λ − u ∗ ]− ) = [u − u ∗ ]α+ [u λ − u ∗ ]− ≡ 0 for α > 0.

80

3 Options on One Asset with Stochastic Volatility

It is also easy to show that rλ = 0 on ∂Ω. In fact, when (S, ν) ∈ ∂Ω, u − u ∗ = u λ − u ∗ ≤ 0, and so rλ = u − u ∗ + [u − u ∗ ]− = u − u ∗ − (u − u ∗ ) = 0,. Thus, rλ 1 ∈ H0,w (Ω), since all terms in rλ are in Hw1 (Ω). Also, from (3.4.18) we see that u − rλ = u λ − φ ≤ u ∗ , since φ ≥ 0. Thus, u − rλ ∈ K . From (3.4.18) and Lemma 3.4.1, we see that to establish an upper bound for u − u λ , we need only to determine an upper bound for rλ . Choosing v = u − rλ in (3.4.2) and v = rλ in (3.4.5), we have   ∂u − , −rλ + B(u, −rλ ; t) ≥ ( f, −rλ ), ∂t   ∂u λ − , rλ + B(u λ , rλ ; t) + λ(φ 1/κ , rλ ) = ( f, rλ ). ∂t Adding up the above inequality and equality gives   ∂(u λ − u) − , rλ + B(u λ − u, rλ ; t) + λ(φ 1/κ , rλ ) ≥ 0. ∂t

(3.4.19)

Note φ · [u λ − u ∗ ]− = 0. Using the definition of rλ , we have (φ 1/κ , rλ ) = (φ 1/κ , u − u ∗ + [u λ − u ∗ ]− ) = (φ 1/κ , u − u ∗ ) ≤ 0,

(3.4.20)

since φ ≥ 0 and u − u ∗ ≤ 0. Therefore, combining (3.4.19) and (3.4.20) gives   ∂(u − u λ ) − , rλ + B(u − u λ , rλ ; t) ≤ 0. ∂t Using the decomposition of u − u λ in (3.4.18), we have from the above inequality     ∂rλ ∂φ − , rλ + B(rλ , rλ ; t) ≤ − , rλ + B(φ, rλ ; t). ∂t ∂t Note that rλ (T ) = 0, since u, u ∗ and u λ satisfy the payoff condition g(S). Integrating the above from t to T and using (3.4.9) and Cauchy–Schwarz inequality, we obtain  T 1 (rλ (t), rλ (t)) + B(rλ (τ ), rλ (τ ); τ )dτ 2 t   T  T ∂φ(τ ) − , rλ (τ ) dτ + ≤ B(φ(τ ), rλ (τ ); τ )dτ ∂τ t t   T  T ∂rλ (τ ) φ(τ ), dτ + B(φ(τ ), rλ (τ ); τ )dτ ≤ (φ(t), rλ (t)) + ∂τ t t 1 1 ≤ φ L ∞ (0,T ;L 2 (I )) rλ  L ∞ (0,T ;L 2 (Ω)) + Cφ L 2 (0,T ;H0,w (Ω)) r λ  L 2 (0,T ;H0,w (Ω))

3.4 Power Penalty Method for Pricing American Options …

 + t

T

81

  ∂rλ (τ ) φ(τ ), dτ t ∈ (0, T ). ∂t

(3.4.21)

Using (2.1.8) and the definition of rλ we have   ∂u(τ ) ∂u ∗ (τ ) ∂u(τ ) ∂rλ (τ ) = φ(τ ) − = φ(τ ) − φ(τ )βeβτ (V0 − V ∗ ). φ(τ ) ∂τ ∂τ ∂τ ∂τ Thus, using (3.4.6), we obtain    T  T  T   ∂rλ (τ ) ∂u(τ ) φ(τ ), φ(τ ), dτ = dτ − β eβτ φ(τ ), V0 − V ∗ dτ ∂τ ∂τ t t t 0 0  0 ∂u 0 C 0 + V0 − V ∗  L q (Ω) ≤ κ , ≤ Cφ L p (Ω) 0 0 ∂t 0 q λ L (Ω)

where p = 1 + 1/κ and q = κ + 1. Combining the above upper bound with (3.4.21) and using (3.4.7), we obtain 1 (rλ (t), rλ (t)) + 2



T t

B(rλ (τ ), rλ (τ ); τ )dτ

C ≤ φ L ∞ (0,T ;L 2 (Ω)) rλ  L ∞ (0,T ;L 2 (Ω)) + Cφ L 2 (0,T ;H 1 (Ω)) rλ  L 2 (0,T ;H 1 (Ω)) + κ 0,w 0,w λ "! " ! ≤ C φ L ∞ (0,T ;L 2 (Ω)) + φ L 2 (0,T ;H 1 (Ω)) rλ  L ∞ (0,T ;L 2 (Ω)) + rλ  L 2 (0,T ;H 1 (Ω)) 0,w 0,w ! "   −κ −k/2 −k rλ  L ∞ (0,T ;L 2 (Ω)) + rλ  L 2 (0,T ;H 1 (Ω)) + λ . + Cλ ≤ C λ 0,w

On the other hand, from (3.1.17) and the definition of  ·  L p (0,T ;H ) , we have from the above estimate, "2 ! 1 rλ  L ∞ (0,T ;L 2 (Ω)) + rλ  L 2 (0,T ;H0,w (Ω))   1 rλ 2L ∞ (0,T ;L 2 (Ω)) + rλ 2L 2 (0,T ;H 1 (Ω)) ≤C 0,w 2  T 1 ≤ (rλ (t), rλ (t)) + B(rλ (τ ), rλ (τ ); τ )dτ 2 t ! "   −κ 1 . ≤ C λ−κ/2 rλ  L ∞ (0,T ;L 2 (Ω)) + rλ  L 2 (0,T ;H0,w (Ω)) + λ

(3.4.22)

1/2 −κ This is of the form! y 2 ≤ Cρ " y + Cρ with ρ = λ , which can be rewritten as 2 (y − 21 Csρ 1/2 )2 ≤ C + C4 ρ. Rearrange this inequality gives y ≤ Cρ 1/2 . (Recall that C is a generic positive constant.) Thus, applying this analysis to (3.4.22) yields

rλ  L ∞ (0,T ;L 2 (Ω)) + rλ  L 2 (0,T ;H01 (Ω)) ≤

C . λκ/2

82

3 Options on One Asset with Stochastic Volatility

Using the triangle inequality, (3.4.7) and the above inequality, we have from (2.2.17), 1 u − u λ  L ∞ (0,T ;L 2 (Ω)) + u − u λ  L 2 (0,T ;H0,w (Ω)) ≤ 1 1 rλ  L ∞ (0,T ;L 2 (Ω)) + rλ  L 2 (0,T ;H0,w (Ω)) + φ L ∞ (0,T ;L 2 (Ω)) + φ L 2 (0,T ;H0,w (Ω))

≤

C . λκ/2

Thus, we have proved (2.2.16).



To conclude this section, we comment that (3.4.3)–(3.4.4) can be solved numerically by a combination of the FVM in Sect. 3.2 and the damped Newton algorithm in Sect. 2.3.2.

3.5 A Numerical Example Consider the following test problem.

Fig. 3.3 Cross-sections of the computed option value at various time points

3.5 A Numerical Example

83

Example 3.1 A European call option governed by (3.1.9)–(3.1.11) with g3 (S) = max{0, S − K }, g1 (t) = 0, and g2 (t) = Smax − K . The parameters are Smax = 100, νmin = 0.01, νmax = 1, T = 1, r = 0.1, ρ = 0.9, σν = 1, μν = 0 and K = 50. We choose a uniform partition (3.2.1) with N S = 50 and Nν = 50, and (0, T ) is also partitioned uniformly into 50 sub-intervals. The boundary conditions G 1 (S, t) and G 2 (S, t) in (3.1.11) are determined numerically by solving the constant volatility counterpart of this example using the 1D FVM in Sect. 1.3 on the aforementioned uniform mesh. The cross-sections of the numerical solution to Example 3.1 at various time points are depicted in Fig. 3.3.

References 1. Cox J (1975) Notes on option pricing I: constant elasticity of diffusions. Unpublished draft, Stanford University 2. Garman M (1976) A general theory of asset valuation under diffusion state processes. Working Paper No. 50, University of California, Berkeley 3. Haslinger J, Miettinen M (1999) Finite element method for hemivariational inequalities. Kluwer Academic Publisher, Dordrecht 4. Hull J (2015) Options, future, and other derivatives, 9th edn. Pearson, Boston 5. Hull J, White A (1987) The pricing of options on assets with stochastic volatilities. J Financ 42:281–300 6. Heston SI (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6:327–343 7. Huang C-S, Hung C-H, Wang S (2006) A fitted finite volume method for the valuation of options on assets with stochastic volatilities. Computing 77:297–320 8. Huang C-S, Hung C-H, Wang S (2010) On convergence of a fitted finite-volume method for the valuation of options on assets with stochastic volatilities. IMA J Numer Anal 30:1101–1120 9. Kufner A (1985) Weighted sobolev spaces. Wiley Inc, New York 10. Varga RS (1962) Matrix iterative analysis. Prentice-Hall, Englewood Cliffs

Chapter 4

Options on One Asset Revisited

Abstract In this chapter we propose a superconvergent Finite Volume Method (FVM) based on that in Sect. 1.3 for the nonlinear penalized Black–Scholes equation governing the valuation of European and American options on one asset. Unlike the FVM in Sect. 1.3, we construct an un-symmetric dual mesh using a set of judiciously chosen points. We show that the approximate flux at these points has a 2nd-order truncation error, instead of the 1st-order one in the FVM in Chap. 1. Thus, the resulting FVM has a higher order accuracy at these points, which are called superconvergent points. Numerical results are presented to demonstrate our theoretical findings. Keywords Option pricing · Finite volume method · Superconvergence · Truncation error

4.1 The Unsymmetric Finite Volume Method We revisit the FVM in Sect. 1.3 for (2.2.1). Omitting the subscript λ we rewrite (2.2.1) as the following form: −

∂u(S, t) ∂[Sρ(u(S, t))] − + c(t)u(S, t) + ϕ(S, t, u(S, t)) = f (S, t) (4.1.1) ∂t ∂S

for (S, t) ∈ I × [0, T ) satisfying appropriate payoff and boundary conditions as in 1/k (2.2.2), where ρ(u) is the flux defined in (1.3.2) and ϕ(u) = λ[u − u ∗ ]+ with u ∗ given in (2.1.8). We divide I = (0, Smax ) into N sub-intervals Ii := (Si , Si+1 ), i = 0, 1, . . . , M − 1, satisfying 0 = S0 < S1 < · · · < S N = N . Let h i =Si+1 − Si and h= max0≤i≤M−1 h i . Unlike the dual mesh in Sect. 1.3.1 consisting of the mid-points of {Ii }, we define a general dual mesh with nodes Si− 21 , i = 0, 1, . . . , N + 1, such that S−1/2 = S0 < S 21 < S1 < S1+ 21 < · · · < S N −1 < S N − 21 < S N = S N + 21 . The node Si+ 21 is to be determined later for each feasible i. Integrating (4.1.1) over (Si− 21 , Si+ 21 ), we have, for i = 1, 2, . . . , N − 1, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 S. Wang, The Fitted Finite Volume and Power Penalty Methods for Option Pricing, SpringerBriefs in Mathematical Methods, https://doi.org/10.1007/978-981-15-9558-5_4

85

86

4 Options on One Asset Revisited

 −

Si+ 1 2

Si− 1

Si+ 1 ∂u d S − [Sρ(u)] S 21 + i− 2 ∂S

2



Si+ 1 2



Si+ 1

(cu + ϕ(u))d S =

Si− 1

2

f (S)d S.

Si− 1

2

2

Using the one-point quadrature rule, we approximate the above equation by −

  ∂u i li − Si+ 21 ρ(u)| Si+ 1 − Si−1/2 ρ(u)| Si− 1 + (ci u i + ϕi (u i ))li = f i li (4.1.2) 2 2 ∂t

for i = 1, 2, . . . , N − 1, where li =Si+ 21 − Si− 21 , ci = c(Si , t), ϕi (u i ) = ϕ(Si , t, u i ), f i = f (Si ), and u i is an approximation to u(xi , t) to be determined. Following the analysis in Sect. 1.3.1, we consider the following two cases. Case I. i ≥ 1. We consider the following 2-point boundary value problem:   aSw + bi+1/2 w = 0, x ∈ Ii , w(xi ) = u i , w(xi+1 ) = u i+1 .

(4.1.3)

Let αi = bi+ 21 /a. Solving (4.1.3) analytically yields, for S ∈ Ii , ρi := bi+1/2

αi xi+1 u i+1 − xiαi u i ρi u i+1 − u i , w= − αi (Si Si+1 )αi S −αi . αi αi xi+1 − xi bi+ 21 Si+1 − Siαi

(4.1.4)

Case II. i = 0. On I0 , (4.1.3) becomes degenerate at S = 0, and (4.1.3) is not uniquely solvable. We look into the asymptotic behaviour of ρ0 as S0 → 0+ . S

−α0

α

S 0 u −u

When α0 < 0, we rewrite (4.1.4), for any S0 > 0, as ρ0 = b 21 0 −α01 α01 0 . Taking S0 S1 −1 the limit, we have lim S0 →0+ ρ0 = b 12 u 0 . Similarly, when α0 > 0, from (4.1.4) we have lim S0 →0+ ρ0 = b 21 u 1 . Combining these two situations we obtain ρ0 = b 21

1 − sign(b 21 ) 2

u 0 + b 12

1 + sign(b 21 ) 2

u1,

(4.1.5)

since α0 = b 12 /a and b 21 have the same sign pattern. Replacing the fluxes in (4.1.2) with their respective approximations defined in (4.1.4) and (4.1.5), we have the following semi-discretized system: −

∂u i li + ei,i−1 u i−1 + ei,i u i + ei,i+1 u i+1 + ϕi (u i )li = 0, ∂t

(4.1.6)

where b1+ 1 S1+ 21 S2α1 S1 1 − sign(b 21 ) b 21 , e1,2 = − α21 , 2 2 S2 − S1α1 α1 S1 1 + sign(b 21 ) b1+ 21 S1+ 21 S1 = b 21 + c1l1 , + 2 2 S2α1 − S1α1

e1,0 = −

(4.1.7)

e1,1

(4.1.8)

4.1 The Unsymmetric Finite Volume Method

87

α

ei,i−1 = −

i−1 bi− 21 Si− 21 Si−1

α

α

i−1 Si i−1 − Si−1

α

ei,i =

bi− 21 Si− 21 Si i−1 α

α

i−1 Si i−1 − Si−1

+

, ei,i+1 = −

bi+ 21 Si+ 21 Siαi αi Si+1 − Siαi

αi bi+ 21 Si+ 21 Si+1 αi Si+1 − Siαi

+ ci li ,

,

(4.1.9) (4.1.10)

for i = 1, 2, 3, . . . , N − 1. Thus, (4.1.6) is a tri-diagonal nonlinear system in u(t) := (u 1 (t), u 2 (t), . . . , u M−1 (t)) if we take into consideration of u 0 (t) = 0 = u N (t). We write (4.1.6) as the following matrix form: −

∂u i (t) li + E i (t)u(t) + ϕi (u i (t))li = f i (t), ∂t

(4.1.11)

for i = 1, 2, . . . , N − 1, where E 1 = (e11 (t), e12 (t), 0, . . . , 0), E N −1 = (0, . . . , 0, e N −1,N (t), e N −1,N −1 (t)), E i = (0, .., 0, ei,i−1 (t), ei,i (t), ei,i+1 (t), 0, . . . , 0), i = 2, 3, . . . , N − 2, with ei, j ’s defined in (4.1.7)–(4.1.10). K be a set of points in [0, T ] satisfying T = For a given positive integer K , let {tk }k=0 t0 > t1 > · · · > t K = 0. On this time-partition, we apply the implicit time-stepping method with a splitting parameter θ ∈ [1/2, 1] to (4.1.11) to yield u ik+1 − u ik −Δtk

li + θ [E ik+1 uk+1 + ϕi (u ik+1 )li ] + (1 − θ )[E ik uk + ϕi (u ik )li ] = (θ f ik+1 + (1 − θ ) f ik )li

for k = 0, 1, . . . , K − 1 and i = 1, 2, . . . , N − 1 with the terminal condition u0 = (u ∗ (S1 ), u ∗ (S2 ), . . . , u ∗ (S N −1 )) , where Δtk = tk+1 − tk < 0, E ik = E i (tk ), f ik = f i (tk ), and uk denotes the approximation to u(tk ). The above discrete equation can also be rewritten as the following matrix form: (G k + θ E k+1 )uk+1 +θ (uk+1 ) = θ f k+1 + (1 − θ ) f k + [G k − (1 − θ )E k ]uk − (1 − θ ) (uk ), k = 0, 1, . . . , N − 1.

(4.1.12)

When θ = 1/2, the time-stepping scheme becomes that of the Crank–Nicolson and when θ = 1, it is the backward Euler scheme. Both of the two cases are unconditionally stable, and they are second and first order accurate in |Δtk |, respectively. We show in the following theorem that, when |Δtk | is sufficiently small, G k + θ E k+1 in (4.1.12) is an M-matrix. Theorem 4.1.1 For any given k = 1, 2, . . . , K − 1, if |Δtk | is sufficiently small, then G k + θ E k+1 in (4.1.12) is a positive-definite M-matrix. Proof The proof is identical to that of Theorem 1.3.1. The positive-definiteness of G k + θ E k+1 is obvious when |Δtk | sufficiently small. This is because G k = |Δt1 k | I , where I denotes the identity matrix. 

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4 Options on One Asset Revisited

4.2 Determination of Superconvergent Points In what follows, we suppress the time variable t. The discussion below follows that [1, 2]. for any feasible i, we expand b on Ii as b(S) = pi + qi S + O(h i ), where pi and qi c satisfy b(Si ) = pi + qi Si and qi = b (Si ). Assume ρ(u) is given by   aSu  + ( p1 + qi S)u = g(S), S ∈ Ii ,

(4.2.1)

where g(S) is unknown. We assume g is continuously differentiable and consider the following interpolation problem: find w(S) such that   aSw + ( pi + qi S)w = 0, S ∈ Ii , w(Si ) = u(Si ), w(Si+1 ) = u(Si+1 ). (4.2.2) Let v := u − w. The difference between (4.2.2) and (4.2.1) is (aSv + ( pi + qi S)v) = g(S), S ∈ Ii , v(Si ) = 0 = v(Si+1 ).

(4.2.3)

Integrating the equation in (4.2.3) and dividing the resulting equation by aS yield v +



 1 p¯ i + q¯i v = (G(S) + C1 ), S aS

(4.2.4)

where G = g(S)d S, C1 is an additive constant, p¯ i = pi /a and q¯i = qi /a. The p¯i ¯ S = S p¯i eq¯i S , and thus integrating factor of the above equation is μ(S) = e ( S +q)d the general solution to (4.2.4) is v(S) = S −αi e−q¯i S



S

t p¯i eq¯i t

Si

 G(t) + C1 dt + C2 dt, at

(4.2.5)

where C2 is also an additive constant. Using the boundary conditions in (4.2.3), we S 1 have from (4.2.5) C2 = 0 = Si i+1 t p¯i eq¯i t G(t)+C dt, from which we obtain at 

Si+1

t αi −1 eq¯i t G(t)dt = −C1

Si

Thus, C1 = −Q −1 (Si , Si+1 ) as



Si+1

t p¯i −1 eq¯i t dt =: −C1 Q(Si , Si+1 ).

Si

Si+1

t p¯i −1 eq¯i t G(t)dt, and so, we can write G(S) − C1  Si+1 1 (4.2.6) G(S) + C1 = (G(S) − G(t)) t αi −1 eq¯i t dt Q(Si , Si+1 ) Si Si

for S ∈ Ii . Replacing G(t) − G(S) in (4.2.6) with its Taylor expansion G(t) − G(S) = g(S)(t − x) + g  (S)(t − S)2 + · · · , we have

4.2 Determination of Superconvergent Points

89

 Si+1   1 t p¯i −1 eq¯i t g(S)(t − S) + g  (x)(t − S)2 + · · · dt Q(Si , Si+1 ) Si  Si+1 g(S) t p¯i −1 eq¯i t (t − S)dt + g  (S)O(h i2 ) = − Q(Si , Si+1 ) Si g(S) [R(Si , Si+1 ) − S Q(Si , Si+1 )] + g  (S)O(h i2 ) (4.2.7) =: − Q(Si , Si+1 )

G(S) + C1 = −

Si+1

with R(Si , Si+1 ) =

Si

t p¯i eq¯i t dt. Therefore, if we choose Si+ 21 :=

R(Si , Si+1 ) , Q(Si , Si+1 )

(4.2.8)

then the first term on the RHS of (4.2.7) becomes zero. In this case, from (4.2.4) and (4.2.7) we see that [aSv + bi+ 21 v] Si+ 1 = [aSv + ( pi + qi S)v] Si+ 1 + [(Si+ 21 − ( pi + qi S))v] Si+ 1 2

2

2

= G(Si+ 21 ) + C1 + b (Si )O(h i2 ) = g  (Si+ 21 )O(h i2 ) + C1 + b (Si )O(h i2 ), (4.2.9) where b (Si )O(h i2 ) is the remainder of the Taylor’s expansion. In computation, Q(Si , Si+1 ) and R(Si , Si+1 ) can be calculated using an appropriate numerical quadrature rule. However, using the expansion eq¯i t = eq¯i Si [1 + q¯i (t − Si )] + O(h i2 ), we may represent Q(Si , Si+1 ) and R(Si , Si+1 ) as follows. Q(Si , Si+1 ) = e

q¯i Si

R(Si , Si+1 ) = eq¯i Si



 1 − q¯i Si p¯i q¯i p¯i p¯i +1 p¯i +1 (Si+1 − Si ) + 1(Si+1 − Si ) + O(h i3 ), p¯ i p¯ i +

 1 − q¯i Si p¯i +1 q¯i p¯ +1 p¯i +2 p¯ +2 (Si+1 − Si i ) + (Si+1 − Si i ) + O(h i3 ). p¯ i + 1 p¯ i + 2

Using these and (4.2.8), we rewrite Si+ 21 as Si+ 21 =

p¯i +1 p¯ +1 p¯i +2 p¯ +2 1−q¯i Si i (Si+1 − Si i ) + p¯iq¯+2 (Si+1 − Si i ) p¯i +1 p¯i p¯ p¯i +1 p¯ +1 1−q¯i Si (Si+1 − Si i ) + p¯q¯i i+ 1(Si+1 − Si i ) p¯i

+ O(h i2 ).

(4.2.10)

Equation (4.2.10) provides a breakpoint for constructing the dual mesh if we omit O(h i2 ), which may also depends on t. At this point, the approximate flux has a 2nd-order truncation error.

90

4 Options on One Asset Revisited

4.3 Superconvergent Points When b Is Independent of S When b is constant, pi = b and qi = 0, Thus, (4.2.10) becomes Si+ 21 := when α := 

b a

α+1 − Siα+1 α Si+1 , i = 1, 2, . . . , N − 1, α α + 1 Si+1 − Siα

(4.3.1)

= 0, −1. If α = 0 or −1, using the 2nd integral in (4.2.7), we have

Si+1

t α−1 (t − S)dt =

Si

Si+1 − Si − x ln SSi+1 , α = 0, i x ln SSi+1 − (S − S ), α = −1. i+1 i Si Si+1 i

Thus, we have Si+ 21 =

Si Si+1 (ln SSi+1 )/ h i , α = −1, i h i / ln SSi+1 , α=0 i

i = 1, 2, . . . , N − 1.

(4.3.2)

When h i is sufficiently small, using a symbolic computation package such as Maple or Mathematica, we can expand Si+ 21 as the following Taylor polynomial: Si+ 21

1 α − 1 h i2 α − 1 h i3 = Si + h i + − +O 2 12 Si 24 Si2



h i4 Si3

 , i = 1, 2, . . . , N − 1.

(4.3.3) From (4.3.3), we see that Si+ 21 an increasing function of αi . Also, using (4.3.1), it is easy to verify that limαi →−∞ Si+ 21 = Si and limαi →∞ Si+ 21 = Si+1 . A special case of Si+1 (α) with Si = 1 and Si+1 = 2 is depicted in Fig. 4.1.

Fig. 4.1 The superconvergence point in (1, 2) as a function of αi

4.3 Superconvergent Points When b Is Independent of S

91

Remark 4.3.1 Note that when i = 0, S0α (respective S0α+1 ) is singular when α < 0 (respectively α + 1 < 0). In this case, we simply choose S 21 = h 0 /2. This will not affect our overall error if we choose h 0 = O(h 2 ).

4.4 Local Error Estimates at the Superconvergent Points The following theorem establishes error bounds for the interpolated flux and solution. Theorem 4.4.1 Let u be the solution to (4.1.1) with the boundary and terminal conditions (2.2.2), and w be the solution to (4.2.2). If u is three times continuously differentiable at Si+ 21 , then, for any i = 1, 2, . . . , N − 1, we have (4.4.1) ρi (u(Si+ 21 )) − ρi (w(Si+ 21 )) = g  (Si+ 21 )O(h i2 ),  2  2  δ h i δi u(Si+ 21 ) − w(Si+ 21 ) = g(Si+ 21 )O i + g  (Si+1 21 )O , (4.4.2) Si Si where δi = Si+ 21 − Si and g is the function defined in (4.2.1). Proof Since ρi (v) = ρi (u) − ρi (w), (4.2.9) implies (4.4.1), because b (S) = 0. To prove (4.4.2), we set S = Si+ 21 in (4.2.5). Expand G(t) − C1 in the resulting expression as a Taylor’s series at Si+ 21 with a remainder and using (4.4.1), we have v(Si+ 1 ) = S −α1 i+ 2

2

= S −α1 i+ 2



 t α−1  (G(Si+ 1 ) + C1 ) + g(Si+ 1 )(t − Si+ 1 ) + O ((t − Si+ 1 )2 ) dt 2 2 2 2 a Si   Si+ 1  2 αi −1 t dt g  (Si+ 1 )O (h i2 ) + g(Si+ 1 )O (δi ) + O (δi2 ) Si+ 1 2

2

2

Si

 1 − (Si /Si+ 1 )α 2 = g(Si+ 1 )O (δi ) + g  (Si+ 1 )O (h i2 ) , 2 2 α 

where δi = Si+1/2 − Si . Using the Taylor’s expansion we have from (4.4.3)  u(Si+ 21 ) − w(Si+ 21 ) = v(Si+ 21 ) = g(Si+ 21 )O This is (4.4.2), and we have proved the theorem.

δi2 Si

(4.4.3)

1−(Si /Si+ 1 )α α



2

=

δi Si

+ g  (Si+ 21 )O

+ O(δi2 /Si2 ), 

h i2 δi Si

 . 

Using Theorem 4.4.1, we estimate the error between the flux ρ(u) and the interpolant of ρi (w) in the following theorem. Theorem 4.4.2 Assume that ρ(u) defined in (1.3.2) has 2nd-derivative with respect to u which is bounded on I¯. If the conditions in Theorem 4.4.1 are fulfilled, the following estimate holds.

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4 Options on One Asset Revisited

|ρ(u(x)) − Π (ρi (w(x))| ≤ Ch i2 for S ∈ [Si− 21 , Si+ 21 ], where w and ρi (w) are defined in Theorem 4.4.1, C is a positive constant, independent of h i and Π denotes the usual linear interpolation operator defined on [Si− 21 , Si+ 21 ] such that ρi (w(Si± 21 )) = Π (ρi (w(Si± 21 )). Proof From Theorem 4.4.1, we have |ρ(u(S)) − Π (ρi (w(S))| ≤ |ρ(u(S)) − Π (ρi (u(S)))| + |Πρ(u i (S)) − Π (ρi (w(S))| ≤ O(h i2 ) + χi− 1 (S)|ρ(u i (Si− 1 )) − (ρi (w(Si− 1 ))| 2

2

2

+ χi+ 1 (S)|ρ(u i (Si+ 1 )) − (ρi (w(Si+ 1 ))| ≤ O(h i2 ), 2

2

2

where χi± 21 (x) denote the usual linear basis (hat) functions associate with Si± 21 used in interpolation. These functions are in between 0 and 1. Thus, we have proved this theorem.  When i is close to 0, O(Si ) = O(h i ) so that δi2 / h i = O(h i ). If we choose a mesh satisfying h i = O(h), then (4.4.2) only provides an O(h) error for u(Si+ 21 ) − w(Si+ 21 ). Therefore, a non-uniform or graded mesh needs to be used near S = 0 so that h i = O(h 2 ) when i is close to 0.

4.5 Numerical Experiments We now demonstrate numerically that the points in (4.3.1), (4.2.10) and (4.3.2) are superconvergent points. Example 4.1 Consider approximating the flux aSu  + bu = G(S) on the interval I = [S1 , S2 ] by a constant defined in (4.1.4), where u(S) = e S . To demonstrate the superconvergence at the points, we consider the following two different sets of coefficients. Case 1. a = 0.5, b = −1.5, S1 = 1 and S2 = 1.2. This is the case that both a and b are constant, and ρ(u) = e S (aS + b). Using (4.3.1), we find that the superconvergent point S1+1/2 ≈ 1.033607. The exact and approximate fluxes are plotted in Fig. 4.2a in which we also circle the point (S1+1/2 , ρ(u(S1+1/2 ))). As predicted, the approximation defined in (4.1.4) achieves a high-order accuracy at S1+1/2 . Case 2. a = 0.5, b = 1 + S 2 , S1 = 0.2 and S2 = 0.25. In this case the exact flux is ρ(u) = e S (aS + 1 + S 2 ). Since b is non-constant, we approximate it by b ≈ p + q S with p = 1 − Si2 and q = 2S1 . The superconvergent point S1+1/2 is approximated by (4.2.10). As in Case 1, we plot the fluxes, as well as (S1+1/2 , ρ(u(S1+1/2 ))), in Fig. 4.2b, from which we see that the approximate flux also has a high-order accuracy at S1+1/2 .

4.5 Numerical Experiments

93

Fig. 4.2 Computed and exact fluxes for Example 4.1 using two different data sets

Example 4.2 We consider pricing the European put option governed by (1.2.3)– (1.2.4) with K = 50 and T = 1. Other parameters are σ = 0.4, r = 0.03, and D = 0. We choose Smax = 100 and g1 (t) = K exp(t − T ), g2 (t) = 0, g3 (S) = max{0, K − S}. We use this example to demonstrate the rates of convergence of the numerical method. For positive integers N and M, we choose two different meshes: (1) uniform mesh with Si = i Smax /N and |Δtk | = k/M; (2) spatially graded mesh Si = (i/N )2 Smax and |Δtk | = k/M as in [2], for a feasible pair (i, k). We use the numerical solution from (4.1.12) with θ = 1/2 on the graded mesh with N = 2560 = M as the ‘exact’ or reference solution. Using this reference solution, we compute the maximum errors at t = 0 in u h and ρ(u h ) on various graded and uniform meshes. Figure 4.3a contains the maximum flux errors in the numerical solutions at the points given in (4.3.1) using the graded and uniform meshes. For comparison, we also plot y = h 2 and y = h in Fig. 4.3a with h = 1/N = 1/M, from which we see the rates of convergence for

Fig. 4.3 Computed maximum errors for Example 4.2 on different meshes: a errors at dual mesh points; b errors at primal mesh points

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4 Options on One Asset Revisited

the solutions on graded meshes are roughly of the order O(h 2 ), while those of the solutions on uniform meshes have an order of O(h). To further demonstrate the FVM, we solve Example 4.2 on the graded meshes, and calculate the errors at the primal mesh nodes {Si } on the cross-section t = 0. Figure 4.3b contains the maximum errors in the solutions. From this figure, we see the rates of convergence of ρ(u h ) are of order O(h), while those for u h are of order O(h 2 ).

References 1. Angermann L, Wang S (2019) A super-convergent unsymmetric finite volume method for convection diffusion equations. J Comput Appl Math 358:179–189 2. Wang S, Zhang S, Fang Z (2015) A superconvergent fitted finite volume method for Black Scholes equations governing European and American option valuation. Numer Methods Part Differ Equ 31:1190–1208