An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain [Abridged] 1789738946, 9781789738940

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An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain

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An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain SATYA R. CHAKRAVARTY Indian Statistical Institute, India

PALASH SARKAR Indian Statistical Institute, India

United Kingdom – North America – Japan – India – Malaysia – China

Emerald Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2020 Copyright © 2020 Emerald Publishing Limited Reprints and permissions service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. Any opinions expressed in the chapters are those of the authors. Whilst Emerald makes every effort to ensure the quality and accuracy of its content, Emerald makes no representation implied or otherwise, as to the chapters’ suitability and application and disclaims any warranties, express or implied, to their use. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-78973-894-0 (Print) ISBN: 978-1-78973-893-3 (Online) ISBN: 978-1-78973-895-7 (Epub)

Satya R. Chakravarty dedicates this book to his granddaughter, Anvi Ananyo Chakravarty (Gini). Palash Sarkar dedicates this book to his mother.

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Contents

Preface

xiii

Part I Derivatives, Options and Stochastic Dominance Chapter 1 Background and Preliminaries

3

Chapter 2 Valuation of Cash Flows and Fixed Income Securities: An Abridged Analysis 2.1 Introduction 2.2 Net Present Value 2.3 Bond as a Fixed Income Security

7 7 7 9

Chapter 3 A Brief Analytical Exposition of Markets for Options 3.1 Introduction 3.2 Payoff and Profit Functions of Some Standard Options 3.3 Options as Hedging Strategies 3.4 Forward and Futures Contracts

11 11

Chapter 4 The 4.1 4.2 4.3

19 19 19

Binomial Model: A Simplified Analysis Introduction Formal Framework Valuation of Options in the Cox–Ross–Rubinstein Model

Chapter 5 Brownian Motion, Itˆo Lemma and the Black–Scholes–Merton Model 5.1 Introduction 5.2 Preliminaries

11 14 17

22

25 25 25

viii

Contents 5.3 5.4 5.5 5.6

Itoˆ Lemma, Distribution of Stock Price and Price of a Forward Contract The Black–Scholes–Merton Partial Differential Equation Black–Scholes–Merton Pricing Formulae The Greek Letters

27 29 30 32

Chapter 6 Exotic Options: An Illustrative Presentation 6.1 Introduction 6.2 Asian Options 6.3 Binary or Digital Options 6.4 Barrier Options

35 35 35 37 38

Chapter 7 An Abbreviated Theoretical Treatment of Stochastic Dominance Relations 7.1 Introduction 7.2 First-order Stochastic Dominance 7.3 Second-order Stochastic Dominance

41 41 41 43

Part II Algorithmic Issues Chapter 8 Option Pricing Using Finite Difference Method 8.1 Crank-Nicolson Method 8.2 American Options

49 50 53

Chapter 9 Option Pricing Using Monte Carlo Methods 9.1 Simulation of Wiener Process 9.2 Simulating Itoˆ Stochastic Differential Equation 9.3 Valuing European Options 9.4 Valuing American Options 9.5 Monte Carlo Integration

57 57 58 59 59 61

Chapter 10 Determining Stochastic Dominance Relations

63

Chapter 11 Trading: Background Notions and Market Microstructure 11.1 Trading Systems

67 68

Contents 11.2 11.3 11.4 11.5 11.6 11.7

Some Relevant Notions Order Book Order Order Matching Algorithms Algorithmic Trading Efficient Market Hypothesis

Chapter 12 Algorithmic Trading Strategies 12.1 Time Weighted Average Price 12.2 Volume Weighted Average Price 12.3 Percentage of Volume 12.4 Participation of Weighted Price 12.5 Bertsimas–Lo Dynamic Programming Strategy 12.6 Implementation Shortfall 12.7 Almgren–Chriss Efficient Trading Frontier

ix

69 72 73 74 75 77 79 79 80 81 81 81 84 85

Chapter 13 Portfolio Optimisation 13.1 Markowitz Portfolio Optimisation 13.2 Kelly Criterion 13.3 Universal Portfolios

91 92 101 103

Chapter 14 Measures of Risk 14.1 VaR and CVaR 14.2 Sharpe Ratio 14.3 Copula

109 109 112 115

Chapter 15 High-frequency Trading 15.1 Market Making 15.2 Exploiting Limit Order Book 15.3 Mean Reversion and Pairs Trading 15.4 Arbitrage 15.5 Market Manipulation

119 120 120 121 121 125

x

Contents

Part III Blockchain and Cryptocurrency Chapter 16 Background Concepts for Blockchain 16.1 Cryptography 16.2 Distributed Computing

129 129 133

Chapter 17 Introduction to Blockchain 17.1 Transactions 17.2 Blocks of Transactions 17.3 Public Ledger as a Blockchain 17.4 Distributed Public Ledger 17.5 Permissionless versus Permissioned Blockchain

137 137 137 138 140 143

Chapter 18 Cryptocurrency: Basics 18.1 Owner 18.2 Transactions 18.3 Cryptocurrency Addresses 18.4 Recapitulation 18.5 Creation of a Block via Proof of Work 18.6 Block Reward and Creation of Cryptocurrency 18.7 Hash Rate 18.8 Updating the Difficulty Parameter 18.9 Controlling the Rate of Money Creation 18.10 Choosing between Competing Blocks 18.11 Confirmation of Transaction 18.12 No Double Spending 18.13 The 51% Attack

145 145 146 148 148 149 150 151 152 153 153 154 154 155

Chapter 19 Cryptocurrency: Further Issues 19.1 Mining Pools 19.2 Change of Rules 19.3 Forks 19.4 Value of a Cryptocurrency 19.5 Cryptocurrency Exchange 19.6 Cryptocurrency Community 19.7 Stablecoin 19.8 Criticisms of Cryptocurrencies

157 157 157 158 160 160 161 162 162

Contents 19.9 19.10 19.11 19.12 19.13

Government Regulations Central Bank–issued Digital Currency Lightning Network Sidechain Proof of Stake

xi

163 164 165 166 166

Chapter 20 Examples of Cryptocurrencies 20.1 Bitcoin 20.2 Ethereum and Smart Contracts 20.3 Ripple and Payment Systems

169 169 171 173

Chapter 21 Applications of Blockchain 21.1 Fintech Applications 21.2 Logistics Management 21.3 Supply Chain Management 21.4 Governance

177 177 178 178 179

References

181

Index

185

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Preface

Our motivation for writing this book is to provide a broad-based and accessible introduction to three of the presently most important areas of computational finance, namely, option pricing, algorithmic trading and blockchain. To the best of our knowledge, no other book in the market provides such a coverage. It is our hope that the book will be useful to senior undergraduates, graduates and MBA students, as well as researchers and practitioners. In its first part, the book reflects option pricing in different frameworks. A simple treatment of assessment of cash flows and fixed security derivatives is also presented. Finally, the problem of asset ranking is addressed in this part of the book. In a broad sense, the second part of the book covers algorithmic issues related to finance. The first three chapters of the second part addresses some computational issues related to the theory discussed in the first part. The rest of the second part, consisting of five chapters, discusses approaches for algorithmic trading, portfolio optimisation and risk management. The third part of the book is devoted to blockchain and cryptocurrency. A fairly detailed introduction to both of these topics is presented along with various applications of blockchain to financial and other applications. The wide coverage of the book and its authentic and well-expressive presentation make the book quite up-to-date from both theoretical and practical sides, and highly reactive to the problems of recent concern. We believe that in satisfying its objectives, our book offers a unique perspective to contemporary aspects of finance in a lucid manner for senior undergraduates, graduates, MBA students and regulators. Further, the book can serve as a useful reference for basic theory to practitioners in the area. Much of finance today involves a fair amount of mathematics. In the book, we have tried to find a balance between having too much and too little mathematics. Detailed mathematical derivations have been given in some cases, helpful aids have been provided in some other cases so that a reader can complete the derivations, and for cases where the proofs takes us too far away from the discussion at hand, the proofs have been omitted. In our opinion, the first two parts of the book can be understood by somebody having a college-level introduction to calculus, linear algebra, probability and statistics. The mathematical requirement for the third part is markedly different from that of the first two parts. It is due to this reason we have tried to keep the mathematical description of the third part to a bare minimum. Several chapters of the book have been used for offering a course on the theory of finance to Master of Science in Quantitative Economics students at the Indian Statistical Institute, Kolkata, India, in the Fall Semester of 2019. The materials were well received by

xiv

Preface

the students. We express our sincere gratitude to the students for their enthusiasm and helpful comments in the process of direct interactions. We thank Pinaki Sarkar for providing several suggestions to improve the coverage of the book. We also thank Sanjay Bhattacherjee for reading and commenting on some chapters of the third part. Satya R. Chakravarty and Palash Sarkar Kolkata, India December 2019

Part I Derivatives, Options and Stochastic Dominance

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Chapter 1

Background and Preliminaries The financial industry is enormously important to state, national and world economies. This industry relies extensively on mathematical modeling of underlying instruments. Computational techniques become helpful in designing related algorithms that enable us to understand how markets function and also lend themselves to highly relevant research problems. To understand the application of a specific computational technique to the particular financial instrument, it becomes indispensable to have a clear perception of the underlying theory. However, because of vastness of the theoretical literature, some selection becomes necessary. We choose the highly attractive field of option pricing, a core task of computational finance and risk analysis. An option is a contract that gives the holder the right, but not an obligation, to buy or sell an asset at a pre-agreed price, the strike price, on or before the date of expiry, the maturity date. The broad field of option pricing is quite ambitious and diverse enough to call for a wide range of computational tools. Confining mostly to option pricing enables us to have a more coherent and comprehensive textbook, to a large extent, and avoids being distracted away from computational issues. An option is a standard example of a derivative, a financial instrument whose value relies on one or more assets that are usually referred to as underlying assets. Generally, it takes the form of a contract to buy or sell an asset or item like commodity, property, etc. at the strike price, on or before the expiration date. Other examples of a derivative include bonds, futures contracts, forward contracts and swaps. (For detailed discussions see, among others, Jarrow and Turnbull, 2000.) The financial market in which derivatives are traded may be designated as the derivatives market. We assume a perfect market in the sense that there are no costs of transactions, no restrictions on short sales and existence of a common borrowing and lending rate. Short selling is a business tactic that involves borrowing an asset and selling it immediately, repurchasing the asset (hopefully at a lower price) and returning it to the lender to close the process. (Chapters 11 and 15 of this monograph provide further discussions along this line.) A derivatives market can be partitioned into two subgroups, one in which derivatives are traded in an organised exchange market where maintenance of market price and all transparencies are provided. An example of an exchangetraded derivative is an option. Under a futures contract, an exchange-traded An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain, 3–5 Copyright © 2020 Emerald Publishing Limited All rights of reproduction in any form reserved doi:10.1108/978-1-78973-893-320201001

4

An Introduction to Algorithmic Finance, Trading and Blockchain

agreement, the buyer is accountable to buy the underlying asset at a prearranged price at a future date. The seller makes the commitment to hand over the asset at the settled price and date. Another market-traded derivative is a bond, a debt security issued by government and corporate sectors to raise funds for various purposes including expansion of one or more sectors, infrastructural improvements and payment of existing debts. A financial market in which trading of bonds takes place is known as a bond or credit or debt market. A trading in the other market is of over-the-counter (OTC) type. This type of off exchange trading takes place directly between the traders without supervision of an exchange. An example of an OTC derivative is a forward contract under which two parties make an agreement to buy or sell an asset at a designated date on a promised price. A swap is an off exchange-traded derivative under which a financial instrument is exchanged between the parties concerned at a prespecified time. A highly significant component of the derivatives market is risk. Risks can be of various types such as asset risk, interest rate risk, foreign exchange risk, credit risk, commodity risk and so on. While asset risk arises from volatility in asset prices, interest rate risk refers to the chance that variations in rate of interest may negatively affect an investment. Likewise, foreign exchange risk emerges from fluctuations in exchange rate between two different currencies. On the other hand, credit risk indicates the possibility of a lender’s loss of principal and interest if a borrower fails to make committed payments. Commodity risk is related to the apprehension of loss that may arise because of oscillation in a commodity price in the future. But risks may also bring about unexpected benefit. Investors wish to make risky investments with the expectation of making profits in the future. Given that there can be alternative notions of risk, use of appropriate tools for the purpose of risk management becomes essential. Different forms of derivatives become useful in this situation. More precisely, derivatives are financial instruments for administering financial risks. They transfer different forms of financial risks to derivatives market. They are financial securities for hedging or bordering risks in the sense of protecting or at least reducing risks. The basic principle underlying asset pricing is the existence of non-arbitrage. According to an arbitrage opportunity, a trader can take advantage of price imbalance to extract rapid profit without any risk. Consequently, the non-arbitrage assumption means the rule of a single price. (Further discussions on arbitrage are available in Chapters 11 and 15 of this monograph.) Since in the first part of the monograph we will be mainly concerned with options, a brief, rigorous and authoritative discussion on options and related phenomena is presented in Chapters 3–6. For the sake of completeness, an abridged analysis of valuation of cash flows and fixed income securities is presented in Chapter 2. Given that any investment in a risky asset includes the chance of a loss, it is quite likely that an investor will look into the problem of risk management in a portfolio, a composition of assets held by the investor. The investor often may be confronted with the necessity of ordering distributions of random returns on various financial assets or combinations of them on the basis of his preferences,

Background and Preliminaries

5

represented by a von Neumann–Morgenstern utility function. This issue has been addressed in several pioneering contributions to the literature. A quite general approach to the resolution of the problem is the rule of stochastic dominance that enables us to order the distributions of random rates of return on assets for large classes of utility functions. No specific knowledge about the form of utility function is necessary. A concise illustrative introduction to the concept of stochastic dominance is presented in Chapter 7.

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Chapter 2

Valuation of Cash Flows and Fixed Income Securities: An Abridged Analysis 2.1 Introduction In order to judge profitability of an investment, say, a firm’s investment in a project, it becomes necessary to compare present values of cash inflows (benefits) and outflows (costs). The net present value (NPV) criterion that looks at the difference between present values of benefits and costs is a standard tool for this type of analysis. Alternatives to this are benefit–cost ratio and internal rate of return (IRR), a discount rate that makes NPV equal to zero. In this chapter, we present a brief discussion on these criteria. A brief discussion on bond as a particular type of constant earning security is also presented.

2.2 Net Present Value When a firm has decided to start a project, this decision will have some important implications on the firm’s financial position for one or more periods. The firm has to obtain an overall indicator that will judge whether it will become better off or worse off when the project has been undertaken. The NPV method, which is employed to determine current values of all inflows and outflows associated with an investment, is an appropriate tool for this purpose. The risk-free interest rate r over the lifespan of the project is assumed to remain constant. We denote money inflows (benefits) and outflows (costs) occurring in period t by Bt and Ct, respectively, where t 5 0,1…,T. Here, t 5 0 and t 5 T represent, respectively, the current period and final period for which the investment affects benefits and costs. While C0 indicates the size of the initial investment, for t $ 1, Ct may be regarded as the maintenance cost. It is natural to assume that C0 . 0 and Ct $ 0 for t $ 1. On the other hand, B0 is likely to be 0, but from t 5 1 onwards, Bts are non-negative. Then the discounted present value (DPV) of the inflows

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8

An Introduction to Algorithmic Finance, Trading and Blockchain T

comes to be +

t50

Bt . ð1 1 rÞt

T

Similarly, the DPV of the outflows becomes +

t50

Ct . ð1 1 rÞt

Then the NPV of the project can now be defined as Bt 2 Ct t: t ¼ 0 ð1 1 rÞ T

NPV ¼ +

(2.1)

The project should be undertaken or rejected according as NPV . 0 or NPV , 0. The firm is indifferent between the two options if NPV 5 0. For NPV to be an appropriate guideline for launching a new project, it becomes necessary to get correct estimates of Bt and Ct in different periods. If investment amounts differ across projects, then the NPV criterion cannot be used to compare them. Also the lifespan of the projects have to be the same for the purpose of comparison. A firm can also use the benefit–cost ratio as a yardstick to judge whether a project should be undertaken or not. It is formally defined as T

B ¼ C

+

Bt ð1 1 rÞt

+

Ct ð1 1 rÞt

t¼0 T t¼0

:

(2.2)

Evidently, B/C is greater or less than 1 if and only if the NPV is positive or negative. Higher values of both the criteria NPV and B/C are better for the investor. However, they may rank alternative projects in different directions. The final evaluator we use in the current context is the IRR, the rate of interest rate that makes the NPV of the project equal to 0. Since interest rate can be interpreted as time value of money, IRR is the interest rate that an investor can expect from his investment. If IRR is higher than the rate of discounting then the project should be undertaken. IRR can be implicitly defined as T

+

Bt 2 Ct

t ¼ 0 ð1 1 IRRÞ

t

¼ 0:

(2.3)

In general, there are T roots of this equation. In such a case, the highest root whose value exceeds the discount rate may be used as IRR. In case there are mutually several exclusive projects, then the firm should choose the one that produces the highest IRR. When the discount rate varies over time, the use of the IRR criterion as an evaluation principle for selection of a project may not be feasible. Thus, while the concern of NPV is project surplus, IRR is concerned with break-even money flow. Of these two criteria, NPV is more popular as a project selection yardstick because it is easy to understand in terms of profitability of a project. (For further discussions, see Bierwag, 1987 and Damodaran, 2010.)

Valuation of Cash Flows and Fixed Income Securities

9

2.3 Bond as a Fixed Income Security There are financial instruments that assure to pay a fixed amount of money annually up to certain years, say, T. Such an instrument is known as a fixed income security. A standard example is an annuity that pays A dollars each year for T years. The DPV of an instrument of this category is T

DPV ¼ +

A

t ¼ 1 ð1 1 rÞ

t;

(2.4)

where r is the annual risk-free interest rate. An example of an annuity is a lender’s fixed annual earnings from his borrower. An annuity is known as perpetuity or a consol if it guarantees to
pay  A dollars annually for an infinite period. The DPV of a consol comes to be

A r

. Governments and corporate sectors issue bonds to

borrow money with the objective of increasing liquid funds. Let Q be the face or principal value of a bond. It is the bond’s price, par value; when the bond is first issued. This is the amount paid to the holder by the borrower on maturity of the bond. In exchange, the bond issuer is required to pay a fixed amount of money, say A dollars (the coupon) for a certain number of years, say T. At the terminal or maturity period T, the borrower returns the face value along with this period’s coupon to the holder of the bond. Such bonds are fixed income derivatives. The combined DPV of bond coupons and face value is given by T

A Q : t1 ð1 1 rÞ ð1 1 rÞT t¼1

DPVB ¼ +

(2.5)

If annual coupon value is available as a percentage, say, c% of the face value of the bond, then the DPV becomes T

cQ Q ; t1 ð1 1 rÞ ð1 1 rÞT t¼1

DPVB ¼ +

(2.6)

where 0 , c , 1 is a constant. We refer to this constant as coupon rate. Evidently, given other things (maturity period, risk factors, etc.), a buyer will not distinguish between a currently available bond and new bonds if the former is sufficiently discounted to yield the same DPV as that yielded by the latter. Suppose an existing bond is currently being traded at a price H. Given that A, Q, T and H are known, we can determine the rate of interest for which the DPV of the bond given by Equation (2.5) equals its price H. Formally, we need to determine the discount rate r that solves the equation T

A Q : t1 ð1 1 rÞ ð1 1 rÞT t¼1

H ¼ +

(2.7)

The value of r that solves this equation is known as the yield to maturity on the bond if the investor has decided to hold the bond until maturity. It is an indicator of the return on the bond. By definition, it is an IRR of an investment in a bond. The above equation has T many solutions of which only one solution is relevant

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An Introduction to Algorithmic Finance, Trading and Blockchain

to the bond holder. All the other roots of the equation are either unimportant or imaginary. In case annual coupon value is given as a percentage of the face value, formula (2.7) becomes cQ 1 H ¼ 12 r ð1 1 rÞT

! 1

Q ð1 1 rÞT

;

(2.8)

which, on simplification, becomes H c 1 ¼ 12 Q r ð1 1 rÞT

! 1

1 ð1 1 rÞT

:

(2.9)

This formula enables us to compare the coupon rate with the yield to maturity. If the two rates coincide, then the face value coincides with the bond price (H 5 Q). A bond of this type is called a par bond. Equivalently, one says that the bond is at par. If c . r, then Q , H. In such a case the bond is known as a premium bond. Investors will pay a high price for a premium bond to get a higher coupon rate. Finally, the bond is referred to as a discount bond if c , r, so that the inequality Q . H holds. Lower coupon rate in this case is compensated by a lower or discounted bond price. So investors will pay for such a bond. (Further discussions on these issues are available in Damodaran, 2010 and Chakravarty, 2013.)

Chapter 3

A Brief Analytical Exposition of Markets for Options 3.1 Introduction A financial derivative that takes the form of a contract between a buyer and a seller providing the buyer the right but not the obligation to buy or sell an asset on or before the expiry date at the contracted price, referred to as the strike price or exercise price, is known as an option. It represents a claim for an underlying asset where the price of the asset is the subject of the claim. In exchange of writing the contract the price received by the seller is called the premium of the option. The purpose of this chapter is to furnish a brief analytical exposition of different types of exchange-traded options, their usefulness and payoff and profit functions associated with them. There is no obligation on the part of the buyer to exercise the right. So, one choice for the buyer is to allow the contract to expire. The non-obligatory right component of an option on the part of the buyer makes it different from forward and futures undertakings at the basic level. While a forward contract is an off exchange-traded obligatory contract between a buyer and a seller, to buy or sell an asset at a prearranged future time, at a prefixed delivery price; a futures contract is an obligatory exchange-traded contract between a buyer and a seller to buy or sell an asset at a predetermined delivery price during a specified time period in the future. In Section 3.2 of this chapter we provide an analytical exposition of some standard options, their usefulness, and payoff and profit functions. Our discussion in this section and in the following sections will be brief because we do not wish to present a detailed survey of the materials; instead we want to give some idea about the concepts. Section 3.3 presents a short discussion on the role of options as hedging instruments. A short discussion on forward and futures contracts is presented in Section 3.4.

3.2 Payoff and Profit Functions of Some Standard Options The two major variants of options are call and put options. A call option gives its holder the right to buy the underlying asset by the expiry or maturity date at the

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An Introduction to Algorithmic Finance, Trading and Blockchain

strike price. In contrast, for a put option the right for selling the underlying asset can be utilized by the expiry date at the strike price. Options can be of European or American type depending on the flexibility of exercising the right. While a European option can be exercised only at the date of maturity, an American option can be exercised at any date on or before the date of maturity. Thus, for a European option the holder has exactly one degree of freedom with respect to utilizing the right, for an American option such degrees of freedom are much higher. A financial derivative that guarantees its holder the right of getting involved into transaction (buying or selling), but not the obligation, at a pre-agreed price within a given time period, is known as a plain vanilla option. It can be a call or a put option without any special aspect. It is evident that the following four types of transactions can take place in a market for options: (1) buying call options, (2) selling call options, (3) buying put options and (4) selling put options. If a market participant writes or sells an option then he is known to possess a short position in the market, whereas if a participant buys an option he is known to maintain a long position in the market. Let us consider a European call option. Let ST denote the price of the underlying asset at the maturity date T. We denote the strike price, the price at which the option can be exercised, by K. If ST . K holds, that is, when the option expires in-the money, the buyer will exercise the right and buy the asset at the price K from the option seller although the asset’s current market price is ST. This is as per the contract. If ST 5 K holds the option is said to expire at-themoney. In this case the trader is indifferent between exercising the right or not. Hence for ST $ K, the payoff associated with a long position in a European call option turns out to be (ST 2 K). On the other hand, when the option expires outof-the-money, that is, when ST , K materializes, the payoff becomes zero, since the right is not exercised. Using more compressed notation, we can write the payoff JECL ðST ; KÞ from a long position in a European call option as JECL ðST ; KÞ 5 maxðST 2 K; 0Þ 5 2 minðK 2 ST ; 0Þ. Now, let us analyze the issue from the seller’s perspective. If ST # K holds the buyer will not exert the right at the date of expiration. Consequently, the buyer’s payoff is zero. In contrast, if the opposite inequality ST . K materializes, the right is exerted and the seller needs to purchase the asset at its current market price ST and sell it to the buyer at the strike price K. Combining these two expressions, we can write the payoff function JECS ðST ; KÞ of a trader for a short position in a European call option as JECS ðST ; KÞ 5 minðK 2 ST ; 0Þ 5 2 maxðST 2 K; 0Þ. In exchange of issuing an option the writer receives an income, the current price of the option, from the buyer. This is known as option premium (premium, for short). While the payoff function is the finance analogue of revenue function, the premium may be regarded as a contract fee, on payment of which the option gets validated. The buyer has to pay this fixed amount to the writer irrespective of whether the right is exercised or not. Hence it gets incorporated in the calculation of the profit function. Throughout the chapter we will use the terms ‘premium’ and ‘contract fee’ synonymously.

A Brief Analytical Exposition of Markets for Options

13

For the holder of a European put option it will be worthwhile to exercise his right if the option expires in-the-money, that is, if ST , K holds. This is because in this situation if the right is exerted, as per the agreement, the holder has to supply the asset at the price ST to the writer which he can do after buying it from the market at ST. Since in exchange he receives K, his positive payoff from the deal becomes (K 2 ST). The trader becomes indifferent between exercising the right or not if the put option expires at-the-money (ST 5 K). For the option expiring out-of-the money (ST . K), the right will not be exercised and the resulting payoff becomes 0. Consequently, the payoff function corresponding to a long position in a European put option turns out to be JEPL ðST ; KÞ 5 maxðK 2 ST ; 0Þ 5 2 minðST 2 K; 0Þ. A parallel argument can be employed to establish that the payoff from a short position in a European put option is JEPS ðST ; KÞ 5 minðST 2 K; 0Þ 5 2 maxðK 2 ST ; 0Þ. If we denote the premium of a European call option by ac , then the profit function pECL ðac ; ST ; KÞ associated with the option is given by pECL ðac ; ST ; KÞ ¼ maxðST 2 K; 0Þ 2 ac ;

(3.1)

the excess of payoff over the contract fee, the fixed cost incurred by the buyer. Since the seller receives the contract fee, the profit function pECS ðac ; ST ; KÞ for a short position in a European call option is given by pECS ðac ; ST ; KÞ ¼ minðK 2 ST ; 0Þ 1 ac ;

(3.2)

the sum of the seller’s payoff function and the contract fee. Denote the premium, the contract fee, paid by a buyer for a European put option by ap . Then from the definitions of payoff functions for long and short positions in a European put option it follows that the corresponding profit functions for the option are given respectively by pEPL ðap ; ST ; KÞ ¼ maxðK 2 ST ; 0Þ 2 ap ;

(3.3)

pEPS ðap ; ST ; KÞ ¼ minðST 2 K; 0Þ 1 ap :

(3.4)

and The definition of the profit function pECL ða ; ST ; KÞ of a European call option indicates that if a trader expects a chance of a rise in the price of a stock over the strike price, it is highly likely that he will buy the right to purchase the stock at the strike price to exercise the right at the date of maturity. If at the date of expiration the strike price K exceeds the asset price ST, the individual will not exercise the right; the contract becomes worthless and the holder’s loss is simply the premium ac . Evidently, the same amount of loss arises if K 5 ST holds. If 0 , ST 2 K , ac holds, that is, if the asset price is higher than the strike price but their difference is less than the contract fee, the trader should exercise the right because his loss is only a part of the contract fee. The right will certainly be exercised if ST 2 K 2 ac . 0 holds. Likewise, when a trader foresees a possibility of a reduction in a stock price then he may be tempted to write a call option for a profitable deal. This gives a clear motivation for a trader to participate in a market for options. A second motivation can be explained using the definition of c

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An Introduction to Algorithmic Finance, Trading and Blockchain

a European put option. We can deduce that if a trader presupposes that there can be a downturn in the price of an asset, he may be willing to purchase a put option to sell the asset for a profitable transaction. Analogously, if an asset price is anticipated to take an upturn, a trader can write a put option with the presumption of a beneficial execution. In other words, speculators who consciously take risks by gambling on future events can use options with the expectation of deriving benefits from changes in asset prices. If assets are priced inconsistently, arbitrageurs can use options to make riskless profits, since arbitrage is a trading strategy that takes advantage of inconsistent pricing. Finally, hedgers invest in asset markets to reduce/eliminate risks. We elaborate this issue in the next section. Since a long position in a call option and a short position in a put option become profitable under an increase in the asset price, they may be referred to as optimistic or bullish positions. Similarly, we may regard a long position in a put option and a short position in a call potion as pessimistic or bearish positions since they emerge as beneficial for a downward trend in the asset price.

3.3 Options as Hedging Strategies Investors may often require options for the purpose hedging, a strategy to reduce or evade risk. More precisely, hedging is a tool to guard the downturn of return from one investment by the upturn of return from another. In other words, hedging is a risk reduction/elimination strategy. Thus, a hedging involves at least two financial instruments that are highly negatively correlated. When it is possible to shield an investment or a portfolio, a dashboard of two or more investments, against all losses with the help of hedging, then we refer to the situation as perfect hedging. Hence, for perfect hedging a list of investments has to be combined in appropriate proportions so that the overall risk gets eliminated. The brief illustrational discussion presented in this section will enable us to understand the role of options as hedging strategies. Consider a portfolio consisting of long positions in one unit of the underlying asset and a European put option with the strike price being near the current price of the asset. It is referred to as a protective put because the put option may safeguard any downfall in the asset price. Denote the contract fee paid for purchasing the put option by ap and the current asset price by S0. Then the corresponding profit function is given by p pPP ðap ; S0 ; S
T ; KÞ ¼ ST 2 S0 1 maxðK 2 ST ; 0Þ 2 a p 2 a 2 S0 1 ST if ST $ K; ¼ 2 ap 2 S0 1 K if ST , K:

(3.5)

If the right is exercised (ST , K), the maximum loss is the excess of the sum of the purchase price of the asset S0 and option premium ap over the strike price K. Thus, if K is near S0, the loss will be around ap . In contrast, if the asset price goes on rising so that the right is not exercised (ST $ K), the asset price can guard any downward movement in put option’s payoff. A loss in profit resulting from a steep increase in the asset price when a trader has a short position in a European call option can be encountered if combined with this short position the trader has a long position in one unit of the underlying

A Brief Analytical Exposition of Markets for Options

15

asset. This portfolio is known as a covered call because of capability of the long position in the underlying asset to cover a loss that may arise due to possession of a short position in the European call option when the asset price goes up. Using the notation already introduced, the trader’s profit function at the date of expiry can be defined as pCL ðac ; S0 ; ST
; KÞ ¼ ST 2 S0 2 maxðST 2 K; 0Þ 1 ac ac 2 S0 1 K if ST $ K; ¼ ac 2 S0 1 ST if ST , K:

(3.6)

While, in the two transaction strategies defined above, the capacity of a long position in one unit of the underlying asset combined with two different forms of European option is explained, it may often be worthwhile to design a portfolio combining the same type of options, such as two or more call options or two or more put options so that only options are employed as hedging tools. To illustrate the usefulness of only option-based hedging tools, suppose a trader with a long position in a European call expects big movements in the price of the underlying asset during the lifespan of the option. However, he is unsure about the direction of movements. An appropriate hedging instrument here is to combine the long call with a long put on the same underlying asset with the same strike price and maturity period as the call. This neutral option strategy is known as bottom straddle or straddle purchase. The payoff function of the strategy is max(ST 2 K, 0) 2 min(ST 2 K, 0). This function has a V- shaped graph where the centre of the graph occurs at the position K on the axis representing the price of the asset as a function time. The profit function of the strategy is given by pBS ðac ; ap ; ST ; KÞ
¼ maxðST 2 K; 0Þ 2 minðST 2 K; 0Þ 2 ac 2 ap ST 2 K 2 ac 2 ap if ST $ K; ¼ K 2 ST 2 ac 2 ap if ST , K:

(3.7)

This strategy results in a loss if the asset maturity price is close to the strike price. However, a substantial profit materializes under big jumps in the asset price in either direction. Evidently, originally if the trader has a long position in a European put, under apprehensions of big movements in the asset price he can combine the put with long position in a call on the same underlying asset with the same strike price and maturity date to arrive at the profit function (3.7). On the other hand, a top straddle or straddle write consists of short positions in a European call and a put on the same underlying asset with the strike price and maturity date being the same. In this case big jumps in asset price result in a substantial loss but if the asset price on maturity is close to the strike price profit is likely to be significant. The profit function associated with a top straddle takes on the form pTS ðac ; ap ; ST ; KÞ 5 minðK 2 ST ; 0Þ 1 minðST 2 K; 0Þ 1 ac 1 ap , which becomes ðK 2 ST Þ 1 ac 1 ap if ST $ K holds. On the other hand, if ST , K materializes, pTS ðac ; ap ; ST ; KÞ becomes ðST 2 KÞ 1 ac 1 ap . A third concrete example of an only option-based tool is a spread strategy that contains two or more options of the same type such as puts or calls with clashing positions in the sense of long position in some and short position in the others. If the strategy consists of options with differing positions, relying on the same underlying asset, having the same maturity date but different strike prices, then it

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An Introduction to Algorithmic Finance, Trading and Blockchain

is known as a price-spread strategy. If instead, the maturity dates are different but the strike prices are the same, then the strategy is referred to as a calendar-spread strategy (or time-spread strategy.) A price-spread scheme is called a bullish or a bearish strategy according as an increase or a reduction in the price of the underlying asset becomes beneficial for the holder. An example of the former is a portfolio composed of a long position in a European call and a short position in another European call with a higher strike price. Let K1 and K2 stand for the two different strike prices and the respective contract fees are denoted by ac1 and ac2 respectively, where K1 , K2, ac2 , ac1 . Since the exercise price for the first call is lower, its premium, the seller’s issuing charge, is likely to be higher. Then profit function of the portfolio is given by maxðST 2 K1 ; 0Þ 2 ac1 1 ðminðK2 2 ST ; 0Þ 1 ac2 Þ 5 maxðST 2 K1 ; 0Þ 2 ac1 2 maxðST 2 K2 ; 0Þ 1 ac2 which can be written more clearly as 8 c c < a2 2 a1 if ST # K1 ; pBPS ðST ; K1 ; K2 Þ ¼ ac2 2 ac1 1 ST 2 K1 if K1 , ST # K2 ; : c a2 2 ac1 1 K2 2 K1 if ST . K2 :

(3.8)

If ST # K1 holds no right is exercised. If K1 , ST # K2 materializes while the right for the long call is exercised it is not done for the short call. Both the rights are exercised if ST . K2 holds. If the price of the underlying asset rises up to K2, the strike price of the put, the profit rises, after which the profit becomes steady for any additional increase in the asset price. If the asset price reduces, profit reduces and the profit downfall continues up to K1, the strike price of the call, and the profit loss becomes insensitive to any further reduction in the asset price. Consequently, a bullish price-spread strategy borders a trader’s upward and downward risks. A bearish call price spread strategy is a trading strategy involving two European call options with differing strike prices, where the long call has a higher strike price but lower premium than its short counterpart. We can as well amalgamate a bullish price-spread strategy and a bearish price-spread strategy to constitute a hedging instrument. Popularly known as butterfly price-spread strategy, it is a portfolio containing two European long calls and two European short calls of the same underlying asset, with the strike prices of the former two options being K1 and K3, K1 , K3, and the latter two options have the same strike price K2 at the middle of first two, that is, K2 5 ðK1 1 K3 Þ=2. Usually K2 is near the current price of the underlying asset. The contract fees associated with the first two options are denoted respectively by ac1 and ac3 , and for the third and fourth options the common premium ac2 lies below the middle for the first two, that is, 2ac2 , ac1 1 ac3 . Then the profit function of the tool can be expressed as pBS ðST ; K1 ; K2 ; K3 Þ 5 ðmaxðST 2 K1 ; 0Þ 2 ac1 Þ 1 ðmaxðST 2 K3 ; 0Þ 2 ac3 Þ 1 ð2       minðK2 2 ST ; 0Þ 1 2ac2 ¼ maxðST 2 K1 ; 0Þ 2 ac1 1 maxðST 2 K3 ; 0Þ 2 ac3    1 2 2 maxðST 2 K2 ; 0Þ 1 2ac2 ; which in its more explicit form becomes

A Brief Analytical Exposition of Markets for Options 8 2ac 2 ac 2 ac if S # K ; T 1 2 1 3 > > < 2ac 2 ac 2 ac 1 S 2 K if K , S # K ; T 1 1 T 2 2 1 3 pBS ðST ; K1 ; K2 ; K3 Þ ¼ c c c > 2a 2 a 2 a 1 K 2 S if K , S # K 3 T 2 T 3; > 2 1 3 : 2ac2 2 ac1 2 ac3 if ST $ K3 :

17

(3.9)

As a hedging instrument, a butterfly price-spread strategy is suitable if large deviations of the asset price from its current level are unlikely. The maximum loss associated with the strategy is the net premium amount 2ac2 2 ac1 2 ac3 . The maximum profit that the strategy can generate is 2ac2 2 ac1 2 ac3 1 K2 2 K1 which arises if the asset price ST coincides with K2. Since the holder of an American option possesses the freedom of exercising the right at any time t before the date of expiry T and also at T itself, the following boundary conditions must hold respectively for payoff functions of American call and put options: JACL ðSt ; KÞ $ maxðST 2 K; 0Þ, JAPL ðSt ; KÞ $ maxðK 2 ST ; 0Þ for all t # T, where the underlying asset price St at t # T is arbitrary.

3.4 Forward and Futures Contracts A forward contract is a personal agreement between two parties to buy or sell an asset (the underlying asset) at a mutually-agreed future date T (the delivery date) at a pre-agreed price K (the delivery price). This off exchange-traded contract does not involve payment of any contract fee, although the seller is obliged to deliver the asset at the delivery date and the holder is obliged to buy it. Thus, nonpayment of a contract fee and obligations on the parts of the buyer and seller make a forward contract different from a standard option. One major objective of a person to buy a forward contract is to avoid any risk of variation in an asset price in the future. A farmer may sign an agreement with a wholesaler worrying that his crop price may reduce in the future. Other examples include electricity, natural gas, beef and oil. At the starting point, the contract has a value zero to both the parties of the transaction. A trader can initiate a contract at any time point t , T. Consequently, the delivery price will depend on the period of contract t and we can replace the delivery price K of the contract by a more concrete notation PFOC(t, T). The payoff functions of the buyer and seller at the time of delivery are given respectively by ST 2 PFOC(t, T) and PFOC(t, T) 2 ST. Under perfectness of the market with non-arbitrage, the price PFOC(t, T) of the contract is given by St erðT 2 tÞ , where St is the price of the asset at time t and r is the risk-free common lending and borrowing rate of interest. Like a forward contract, a futures contract is an obligatory agreement between the buyer and the seller of an underlying asset to buy and supply the asset at a predetermined price K, the delivery price, at a pre-agreed date T, the delivery date. However, unlike a forward contract, a futures contract is an exchangetraded contract. Examples of the underlying asset include stocks, foreign currencies, timber, commodities, metals (e.g., gold, diamond, silver), pearls and grains.

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An Introduction to Algorithmic Finance, Trading and Blockchain

We write PFUC(t, T) for the price of a futures contract when the period of contract is t. The two parties of a futures contract can observe gains and losses at specified n time points t1,t2,…,tn 5 T, normally consecutive days, where t1 , t2, … , tn 5 T. At the end of a trading period tj11, the holder of a contract will receive the amount of change in the price PFUC(tj11, T) 2 PFUC(tj, T), known as marking to market, if it is positive, and will need to pay if it is negative, where j 5 1,2,…,n 2 1. The opposite payments situation arises from a trader with a short futures position. The following restrictions are laid down: (1) PFUC(T, T) 5 ST, where ST is the asset price at the expiry date of the contract, and (2) at each of the n time points the contract fee is nil.

Chapter 4

The Binomial Model: A Simplified Analysis 4.1 Introduction The binomial option valuation model is an extremely simple model for pricing of options. The model traces the evolution of the underlying asset price in discrete time units. The price of the underlying asset, instead of changing in a continuous manner, takes a leap to one of the two different new values, up and down, at the next time point. Consequently, the valuation of options relies on an iterative process, where for each iteration the only two possible outcomes are up and down. For each iteration, the model uses the same probabilities for up and down. The model is analytically simple and easy to understand. The simple structure of the model becomes helpful in administering the valuation of options under the possibility of early exercise.

4.2 Formal Framework In this section, we briefly analyse the Cox–Ross–Rubinstein (1979) binomial model of stock price evolution process stating the assumptions explicitly. The process has a finite number of times t0 , t1 , … , tT. Starting from the time t0, it terminates at the time tT, and any two consecutive time points are equidistant, that is, ti 2 ti21 is a positive constant, say Dt, where i 5 1,2,…,T. Thus, the time steps iDt, i 5 1,2,…,T, correspond, respectively, to the times t1, t2,…,tT. Let Si denote the price of the underlying asset at time ti. Assume that S0, the price of the asset at the initial period t0, is known. We make the following assumptions at the outset. (1) For any i 5 0,2,…,T 2 1, at time ti, Si either increases to Siu or decreases to Sid at time ti11, where the unknown parameters u and d satisfy the constraint 0 , d , 1 , u. Further, for any i 5 0,2,…,T 2 1, at ti the probability of Si increasing to Siu is pu, 0 # pu # 1 and the probability of Si reducing to Sid is pd 5 1 2 pu, where pu (hence pd) are not known. (2) The underlying asset is nondividend paying during the lifespan of the option. (3) There are no transaction costs. (4) The rate of interest r is risk free throughout the lifetime of the option. (5) The risk-neutral valuation principle for the pricing and hedging of financial derivatives applies. The risk-neutral valuation criterion is a process of valuation under which the expected rate of return on all assets is the risk-free rate r. The associated An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain, 19–23 Copyright © 2020 Emerald Publishing Limited All rights of reproduction in any form reserved doi:10.1108/978-1-78973-893-320201004

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An Introduction to Algorithmic Finance, Trading and Blockchain

probabilities employed for calculating the expected value are known as risk-neutral probabilities, and the implied probability measure is known as a risk-neutral probability measure. Consequently, in a risk-neutral financial framework, an investor cannot claim any extra return above the risk-free rate r, which is as well the spontaneous rate of return on a stock. The present value of any cash flow stream is the discounted expected value at the risk-free rate. A market model has a riskneutral probability measure if and only it does not provide any scope for arbitrage. (See Sundaram,1997; Bjork, 2004 and Chakravarty, 2013 for rigorous discussions on the relationship between non-arbitrage and risk-neutral valuation.) An implication of assumption (1) is that the direction of change in Si is independent of direction of changes in Sj, where j 5 0,1,…,i 2 1. To understand this more explicitly, we consider the two-period evolution process. Note that at the time step Dt, the asset price S1 can take on values S10 5 S0 d and S11 5 S0 u with probabilities pd and pu, respectively. Thus, S1 is stochastic although S0 is nonstochastic. (More generally, for any 1 # i # T, Si is stochastic.) Next, at time step 2Dt, the three possible values that S2 can assume are S20 5 S0 d 2 , S21 5 S0 ud and S22 5 S0 u2 , with the respective binomial probabilities being p2d , 2pupd and p2u . This clearly indicates that the direction of change in S1 is independent of direction of changes in S0. Evidently, this is true for all i 5 1,2,…,T 2 1. Thus, starting with a given initial price S0 of the underlying asset, a tree, known as binomial tree, of all possible asset prices for the remaining T periods can be generated. In the general setup, at time step iDt, the sequence of j prices is given by Sij 5 S0 d i 2 j uj , i2j j ðiÞ ! with the corresponding binomial probability sequence being ðjÞ !ði 2 jÞ !pd pu ; where j 5 0,1,2,…,i; i 5 1,2,…,T and for any positive integer k, (k)! means 1.2.3….(k 2 1).k. In view of assumption (1), it follows that

JðSi 1 1 Þ ¼ pu Si u 1 ð1 2 pu ÞSi d;

(4.1)

where J stands for the expectation operator. Now, the risk-neutral valuation principle demands that the price of the underlying asset at time ti equals the expected value of its price at time ti11, discounted by the risk-free interest rate r. Given that ti11 2 ti 5 Dt for all i 2 {0,1,2,…,T 2 1}, this formally means that Si 5 exp(2rDt)J(Si11), where ‘exp’ denotes the exponential transformation. Hence JðSi 1 1 Þ ¼ Si expðrDtÞ:

(4.2)

We can get an estimate of pu in terms of the parameters u, d and r by equating (4.1) with (4.2). Since (4.2) holds under risk neutrality, equality between (4.1) and (4.2) holds under the implicit assumption that the probability pu is a risk-neutral probability. That is, pu in (4.1) is a risk-neutral probability that matches (4.2). The value of pu that solves equations (4.1) and (4.2) is given by pu ¼

erDt 2 d : u2d

(4.3)

The Binomial Model: A Simplified Analysis

21

The restriction 0 #pu # 1 holds if and only if d # expðrDtÞ # u:

(4.4)

Evidently, under positivity of r and Dt, exp(rDt) . 1. One characteristic of the Cox–Ross–Rubinstein binomial tree is that the variances of the discrete and continuous stock price processes are equated to determine u, d and pu. In the continuous case, it is assumed that, given Si, Si11 follows a lognormal distribution with parameters r and s2, where s represents volatility, which is assumed to be constant (see Seydel, 2012, and Chapter 5 of this monograph). Then the first two moments of the lognormal distribution in terms of the notation used here are J(Si11) 5 Si exp(rDt) and JðSi21 1 Þ 5 Si2 exp½ð2r 1 s2 ÞDt so that the variance VarðSi 1 1 Þ 5 JðSi21 1 Þ 2 ðJðSi 1 1 ÞÞ2 becomes
   VarðSi 1 1 Þ ¼ Si2 expð2rDtÞ exp s2 Dt 2 1 :

(4.5)

In the discrete framework, the variance comes to be     2 VarðSi 1 1 Þ ¼ J Si21 1 2 ðJðSi 1 1 ÞÞ ¼ pu ðSi uÞ2 1 ð1 2 pu ÞðSi dÞ2 2 Si2 ðpu u 1 ð1 2 pu ÞdÞ2 : (4.6)

Equation (4.5), when equated erDt 5 pu u 1 ð1 2 pu Þd from (4.3), gives

with

equation

(4.6),

along

with


   exp 2r 1 s2 Dt ¼ pu u2 1 ð1 2 pu Þd 2 ;

from which we deduce that pu ¼


   exp 2r 1 s2 Dt 2 d 2 : u2 2 d 2

(4.7)

By equating the expressions for pu given by (4.3) and (4.7), we derive the following expression for (u 1 d): u1d ¼


   exp 2r 1 s2 Dt 2 d 2 : expðlDtÞ 2 d

(4.8)

The left-hand side of the equation (4.8) involves the two unknowns u and d, whereas its right-hand side depends on the three known parameters r, Dt and s, and the unknown number d. Once we determine u and d from (4.8), we can get the value of pu from (4.3) or (4.7). But (4.8) alone is not sufficient to deduce values of u and d. We need one more equation. One convenient selection in this situation is u×d ¼ 1:

(4.9)

This equation does not contradict the assumption 0 , d , 1 , u. The following quadratic equation emerges when d 5 1/u from (4.9) is plugged into (4.8): u2 2 2Cu 1 1 ¼ 0;

(4.10)

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An Introduction to Algorithmic Finance, Trading and Blockchain

where C ¼


   1
expð 2 rDtÞ 1 exp r 1 s2 Dt : 2

(4.11)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi The two solutions that solve (4.10) are u 5 C 6 C 2 2 1. Now, by the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi requirements that 0 , d , 1 , u and u×d 5 1, it follows that u 5 C 1 C 2 2 1 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 5 C 2 C 2 2 1. Evidently, these values of u and d determine the probabilities for up-move and down-move of the underlying asset price under risk-neutral valuation. More explicitly, substituting these values of u and d into (4.3), we derive the value of the corresponding risk-neutral probability of up-move, which, when subtracted from 1 gives the risk-neutral probability of down-move. The constant length Dt . 0 of the time intervals [ti, ti11] should be chosen such that pu , 1 is ensured.

4.3 Valuation of Options in the Cox–Ross–Rubinstein Model In the binomial option valuation model of Cox–Ross–Rubinstein, the payoff function of an option, as usual, is assumed to depend on the strike price K and the price of the underlying asset at the expiry period. Then the prices of a European call and put options in the Cox–Ross–Rubinstein model are given, respectively, by   PjECT ¼ max STj 2 K; 0 ;

(4.12)

  PjEPT ¼ max K 2 STj ; 0 ;

(4.13)

where j 5 0,1,2,…,T. Under risk-neutrality condition, we have i h 11 j Pji ¼ expð 2 rDtÞ pu Pji 1 1 1 ð1 2 pu ÞPi 1 1 ;

(4.14)

11 j 11 th where Pji 1 1 is the price of the option when the stock price is Si 1 1 , the (j 1 1) j possible value of the stock price at time ti11 and so on. Since we know the value of PT for j 5 0,1,2,…,T, from the payoff function, it is possible to deduce the values of Pji recursively for each i 5 0,1,2,…,h; h , T, for obtaining the current value of the option P0T . This reasoning applies to both call and put options. Since for American options early exercise is allowed, the reasoning will be somewhat different. There is no change in the basic structure of the binomial tree. It becomes essential at each time step to verify whether early exercise is profitable. More precisely, at each time step the holder has to decide whether to go for an early exercise of the option or to hold it further. We can evaluate PjT from the payoff function and look back along the branches of the tree to determine the value of the option. It is then essential to decide on the course of action – early exercise or retaining the option.

The Binomial Model: A Simplified Analysis

23

The two binomial equations in (4.12) and (4.13) will now be revised as follows: io n   h 11 j ; PjACi ¼ max max Sij 2 K; 0 ; expð 2 rDtÞ pu Pji 1 1 1 ð1 2 pu ÞPi 1 1

(4.15)

io n   h 11 j PjAPi ¼ max max K 2 Sij ; 0 ; expð 2 rDtÞ pu Pji 1 : 1 1 ð1 2 pu ÞPi 1 1

(4.16)

11 j If in PjACi , maxðSij 2 K; 0Þ . expð 2 rDtÞ ½pu Pji 1 1 1 ð1 2 pu ÞPi 1 1 , then exercising the right at time step iDt is a profitable decision, otherwise it is better to hold the option for next time steps. Similarly, we can interpret the formula in (4.16). (For further discussions, see Wilmott, Howison, & Dewynne, 1995.)

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Chapter 5

Brownian Motion, Itoˆ Lemma and the Black–Scholes–Merton Model 5.1 Introduction The discrete time models for the pricing of derivatives are simple and easy to understand. However, they have some disadvantages as well. The range over which asset price moves is restricted. This in turn restricts the time points at which price movements occur. The length of a trading interval in such a stock price evolution process is assumed to be of some reasonable length and the number of trading intervals is limited. In a practical situation this may not be so. In a perfectly competitive framework where all information are readily accessible, largeness of number of trading intervals and smallness of length of an interval are quite likely so that trading takes place virtuously continuously. In fact, in the limiting situation where the length of the trading interval becomes infinitesimally small, the trading process turns out to be continuous, on which the majority of option theory relies. In this chapter, we present a concise annotation of the celebrated continuous time model, the Black–Scholes (1973)–Merton (1973) model. The principal idea that regulates the Black–Scholes–Merton model is hedging to wipe out risk. The model establishes that under continuous adjustment of the proportion of the stock with option in a portfolio, it is possible to make the portfolio riskless.

5.2 Preliminaries Since essential to the Black–Scholes–Merton differential equation is a Brownian motion, we begin our presentation with the following definition. Definition 5.1: A continuous time stochastic process fWt jt2½0; ‘Þg, where the subscript t denotes time, is said to follow a Weiner process (standard Brownian motion) under fulfilment of the following conditions: (1) W0 5 0, (2) For each t . 0, ðWt 2 W0 Þ 5 Wt , the increment over the interval [0, t], follows normal distribution with mean 0 and variance t. Symbolically, Wt ;Nð0; tÞ: An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain, 25–34 Copyright © 2020 Emerald Publishing Limited All rights of reproduction in any form reserved doi:10.1108/978-1-78973-893-320201005

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(3) For any 0 # t1 , t2 , Wt2 2 Wt1 ;Nð0; ðt2 2 t1 ÞÞ, which means that {Wt} maintains stationary increments. (4) {Wt} possesses independent increments, that is, for any 0 # t1 # t2 # t3 ⋯⋯ # tn , the non-overlapping increments Wt2 2 Wt1 , Wt3 2 Wt2 ; ⋯⋯; Wtn 2 Wtn 2 1 are independent normally distributed random variables with a common mean of 0 and variances (t2 2 t1), (t3 2 t2), …, (tn22 2 tn21) and (tn 2 tn21), respectively. We now show the relevance of a Brownian motion to the evolution of a stock price. Let St stand for the price of the stock under consideration at time t. The rate of return associated with the stock price St over the short time interval ½t; t 1 Dt is defined as DSt St 1 Dt 2 St ¼ ; St St

(5.1)

t where Dt . 0 is assumed to be small. In the limit as Dt→0, DS St becomes the t . Now, assume that the stock price S spontaneous rate of return dS t evolves over St time in a continuous manner because of bids and counterbids in the financial market. For a stock with price St and expected rate of return m, the change in St during the next tiny time period dt can be decomposed into two components:

t (1) An anticipated deterministic component dSt 5 mSt dt, equivalently, dS St 5 mdt. (2) A stochastic and unexpected component, which reflects the random changes in stock price in response to external effects such as unexpected upturn and downturn in stock prices.

Given that the changes in a stock price are characterised by an uncertain constituent, we can plausibly assume that the variability in proportionate return during a tiny time period dt is independent of the stock price. This enables us to argue that the standard deviation of the rate of return in a tiny time period dt is proportional to the rate of return. The unexpected segment of the return equals t dSt 5 sSt dWt , or dS St 5 sdWt , where s, the volatility of the stock price, is the standard deviation of the rate of return. It is a measure of the extent to which uncertainty exists in the future movements of the stock price. The Weiner increment dWt is a normal variable with mean 0 and variance dt. t If we combine the stochastic component dS St 5 sdWt with the anticipated deterministic component, the overall change in relative stock price becomes dSt ¼ mdt 1 sdWt : St

(5.2)

We can rewrite equation (5.2) as dSt ¼ mSt dt 1 sSt dWt :

(5.3)

If at least one term in a differential equation is a stochastic process, then it is known as a stochastic differential equation. Thus, equation (5.3) is a stochastic

Brownian Motion, Itˆo Lemma and the Black–Scholes–Merton Model

27

differential equation. If a stock price process fulfils equation (5.3), then we say that it follows a ðm; sÞ-geometric Brownian motion. The drift and volatility components m and s rely on proportionate changes on the stock prices from time to time. The Black–Scholes–Merton pricing formulae are dependent on several assumptions which, in view of our earlier discussions, are self-explanatory. (1) The stock price St is assumed to follow the geometric Brownian motion defined by (5.3), where both m and s are treated as constants. (2) The stock does not pay any dividend throughout the lifetime of the option. (3) The trading does not involve any transaction cost. (4) The risk-free interest rate r over the lifespan of the option remains constant. (5) Opportunities of riskless arbitrage are ruled out. (6) There are no restrictions on short selling.

5.3 Itˆo Lemma, Distribution of Stock Price and Price of a Forward Contract We begin by observing that the stochastic differential equation given by (5.3) drops out as a particular case of the following more general process dSt ¼ h1 ðSt ; tÞdt 1 h2 ðSt ; tÞdWt ;

(5.4)

under the specifications that h1 ðSt ; tÞ 5 mSt and h2 ðSt ; tÞ 5 sSt , where h1 and h2 are two real-valued functions and dWt ;Nð0; dtÞ. A process that verifies equation (5.4) is known as an Itoˆ process, where h1(St, t) is the drift function and h2(St, t) is the volatility for an increment in St. It is as well referred to as the Itˆo stochastic differential equation (see Oksendal, 2003). The derivations of the prices of derivative securities we are going to consider in this chapter rely on the Itoˆ Lemma, which we state below for the sake of completeness. Itˆo Lemma: Let {St} follow an Itoˆ process, that is, dSt 5 h1(St, t)dt 1 h2(St, t) dWt, where dWt ;Nð0; dtÞ, h1(St, t) and h2(St, t) are two-coordinated real-valued functions. Let q(x, t) be a two-coordinated real-valued function which is assumed to be twice continuously differentiable. Then q(St, t) follows an Itoˆ process and
dqðSt ; tÞ ¼

 ∂q ∂q 1 ∂2 q ∂q h1 1 1 ðh2 Þ2 dt 1 h2 dWt ; 2 ∂x ∂t 2 ∂x ∂x

(5.5)

where the derivatives of q and the coefficient functions h1 and h2 depend on (St, t). It is assumed further that dt×dt 5 dt×dWt 5 dWt×dt 5 0, dWt×dWt 5 dt. Assuming that St follows the geometric Brownian motion, a particular case of the Itoˆ process, let us substitute x 5 St, h1 ðSt ; tÞ 5 mSt and h2 ðSt ; tÞ 5 sSt into (5.5), to get " dqðSt ; tÞ ¼

# ∂q ∂q 1 ∂2 q 2 2 ∂q mSt 1 1 s S sSt dWt : t dt 1 ∂St ∂t 2 ∂St2 ∂St

(5.6)

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Equation (5.6) analytically describes the process followed by q(St, t). A standard example of q(St, t) is the price of a derivative security. As a first application of the lemma, let q(St, t) 5 ln(St), where ln stands for logarithm. Then ∂qðSt ; tÞ 1 ∂2 qðSt ; tÞ 1 ∂qðSt ; tÞ ¼ 0: ¼ ; ¼ 2 2; ∂St St ∂St2 St ∂t

Plugging these values into (5.6) we get " dðln St Þ ¼

# 1 1 1 1 mSt 2 × 2 s2 St2 dt 1 sSt dWt St 2 St St 

¼

m2

 s2 dt 1 sdWt : 2

From the above equation it follows that Z

t

 dðlnðSz ÞÞ ¼

m2

s2 2

Z

t

Z dz 1

t

sdWz ;

 0 s2 which, on simplification, gives lnðSt Þ 2 lnðS0 Þ 5 m 2 2 t 1 sðWt 2 W0 Þ. As 0

0

condition (1) in Definition 5.1, W0 5 0. Hence lnðSt Þ 5   2 lnðS0 Þ 1 m 2 s2 t 1 sWt , where S0 is the stock price at the initial period. This per

indicates that lnðSt Þ;N

   2 lnðS0 Þ 1 ðm 2 s2 tÞ; s2 t . In other words, St follows 

lognormal distribution with parameters

  2 lnðS0 Þ 1 m 2 s2 t and s2 t. The

density function of the stock price St is then given by  i2 3  2 h 2 ln St 2 ln S0 2 m 2 s2 t 1 5; f ðSt ; t; S0 ; m; sÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi exp4 2 2s2 t St 2ps2 t

(5.7)

where the abbreviation ‘exp’ represents the exponential transformation. This establishes that if {ln St} follows geometric Brownian motion, then St follows a lognormal distribution. As a second example, let qðSt ; tÞ 5 St erðT 2 tÞ , the price of a forward contract when the period of initiation is t. Here T denotes the period of maturity of the 2 t ;tÞ rðT 2 tÞ ∂ qðSt ;tÞ 5 e , 5 0 and contract and r is the risk-free rate of interest. Then ∂qðS 2 ∂St ∂St ∂qðSt ;tÞ rðT 2 tÞ . Substituting these values into (5.6), we get ∂t 5 2 rSt e  h i  d St erðT 2 tÞ ¼ mSt erðT 2 tÞ 2 rSt erðT 2 tÞ dt 1 sSt erðT 2 tÞ dWt :

Since qðSt ; tÞ 5 St erðT 2 tÞ 5 PFOC ðt; TÞ, the price of a forward contract, we can rewrite the above equation as dðPFOC ðt; TÞÞ ¼ ðm 2 rÞPFOC ðt; TÞdt 1 sPFOC ðt; TÞdWt :

(5.8)

Brownian Motion, Itˆo Lemma and the Black–Scholes–Merton Model

29

Equation (5.8) establishes that the forward price process follows a geometric Brownian motion with an expected growth rate of ðm 2 rÞ.

5.4 The Black–Scholes–Merton Partial Differential Equation As a final illustration of the lemma, let q(St, t) be the price of an arbitrary derivative security that depends on St, the current price of the stock, and t, the time of commencement of the contract. We now design a dynamic strategy that exactly replicates the instantaneous changes in the price of the derivative security. A replicating strategy of assets is a scheme that exactly reproduces the cash flows of one or more stochastic scenarios. If a trader is required to trade continuously to hold this replicating scheme, then it possesses a dynamic character. The choice of the scheme is guided by the requirement that the market risk is completely eliminated. It is assumed to fulfil the self-financing property in the sense that there can be no inflow or outflow of money during its lifespan. Any rearrangement in the strategy at time t can be made using just the money which comes from the initial investment and the gain up to time t. Equivalently, we say that it is a closed strategy. The two instruments involved in the construction of the strategy are the stock, the underlying asset, and a risk-free bond. (See also Baz & Chacko, 2008.) If Yt stands for the amount of investment in the derivative security at time t, then this amount enables a trader to purchase qðSYtt;tÞ units of the derivative security. The form of dq(St, t) given by equation (5.6), when substituted into dYt 5 qðSYtt;tÞ dqðSt ; tÞ, gives the following more explicit form of the instantaneous change: dYt ¼

# " Yt ∂q ∂q 1 ∂2 q 2 2 Yt ∂q mSt 1 1 s S sSt dWt : t dt 1 ∂t 2 ∂St2 qðSt ; tÞ ∂St qðSt ; tÞ ∂St

(5.9)

Alternatively, suppose out of the total of available money Yt to be  amount  St ∂qðSt ;tÞ invested, the trader has decided to put qðSt ;tÞ ∂St Yt amount in the stock and   t ;tÞ Yt in the risk-free bond. As the sum of these two the amount 1 2 qðSStt;tÞ ∂qðS ∂St investments equals Yt, if one of two components becomes negative, it can be regarded as short selling that security and spending the corresponding revenue generated on the other security. This investment strategy consists of   S t

1 2 qðS t ;tÞ

∂qðSt ;tÞ ∂St

∂qðSt ;tÞ St qðSt ;tÞ ∂St

St

Yt

Yt

units of the stock and units of the bond, where Bt stands for the Bt value of the bond at time t. Since the bond is risk free it follows that dBt 5 rBt dt: The dynamics of the investment strategy then turns out to be dYt ¼

St ∂qðSt ;tÞ qðSt ;tÞ ∂St

St

Yt dSt 1



1 2 qðSStt;tÞ

∂qðSt ;tÞ ∂St

Bt

 Yt

dBt :

Since the stock price follows a ðm; sÞ-geometric Brownian motion, we substitute dSt 5 mSt dt 1 sSt dWt from (5.3), and dBt 5 rBt dt; into the above equation, to derive

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dYt ¼

  St ∂qðSt ; tÞ St ∂qðSt ; tÞ St ∂qðSt ; tÞ Yt rdt; Yt mdt 1 Yt sdWt 1 1 2 qðSt ; tÞ ∂St qðSt ; tÞ ∂St qðSt ; tÞ ∂St

which we rewrite as
dYt ¼

   St ∂qðSt ; tÞ St ∂qðSt ; tÞ Yt ∂qðSt ; tÞ Yt r dt 1 Yt m 1 1 2 St sdWt : qðSt ; tÞ ∂St qðSt ; tÞ ∂St qðSt ; tÞ ∂St (5.10)

In the equations (5.9) and (5.10), the Weiner processes are identical and the volatility components of the two investments are the same. Consequently, the risks involved in the two investments are equal. Now, by the non-arbitrage principle, the investments that are subject to equal risk must indicate equal expected return. Consequently, the drift components across the dynamics (5.9) and (5.10) must be equal. This gives # "   
Yt ∂q ∂q 1 ∂2 q 2 2 St ∂qðSt ; tÞ St ∂qðSt ; tÞ Yt r ; mSt 1 1 s S Y m 1 1 2 ¼ t t ∂t 2 ∂St2 qðSt ; tÞ ∂St qðSt ; tÞ ∂St qðSt ; tÞ ∂St

or, # "   
Yt ∂q ∂q 1 ∂2 q 2 2 St ∂qðSt ; tÞ qðSt ; tÞ St ∂qðSt ; tÞ 2 Yt r mSt 1 1 s S Y m 1 ¼ t t ∂t 2 ∂St2 qðSt ; tÞ qðSt ; tÞ∂St qðSt ; tÞ ∂St qðSt ; tÞ ∂St

which on simplification, reduces to 1 2 2 ∂2 q ∂q ∂q s St 2 1 rSt ¼ 0 2 rq 1 2 ∂St ∂St ∂t

(5.11)

This parabolic partial differential equation is known in the theory of finance as the Black–Scholes (1973)–Merton (1973) partial differential equation. Thus, in deriving this partial differential equation, we have noted how a dynamic trading strategy involving the stock and a risk-free bond can be employed. Other trading strategies can be designed to derive this equation. For instance, a dynamic ∂q amount of stocks can be portfolio composed of 2q, short one derivative, and ∂S t designed such that the Wiener process gets eliminated. By the non-arbitrage assumption, the rate of return earned instantaneously by the portfolio must be equal to the rate of return earned if the money put into the portfolio is invested in a risk-free asset. This argument enables us to arrive at the partial differential equation (5.11). A long discussion along this line is beyond the scope of this book. (For details, see Hull, 2014.)

5.5 Black–Scholes–Merton Pricing Formulae While the premium of an option is the price that the buyer pays in exchange of receiving the right assured by the option, the price of an option is what the option should cost currently using all known parameters on which it depends, such as the price of the underlying asset, strike price, time to expiry and volatility. The latter is an estimated value of an option determined by employing an analytical model

Brownian Motion, Itˆo Lemma and the Black–Scholes–Merton Model

31

like the Black–Scholes–Merton model. Since this price is what an option should currently be worth, we apply risk neutral valuation. The fundamental equation (5.11) was derived for a general payoff function q(St, t).We now wish to determine the price of a European call option using (5.11). The call has a payoff function max(ST 2 K, 0). We can solve the partial differential equation (5.11) by setting q(St, t) 5 max(ST 2 K, 0), and the solution will be the price of a European call option on a non-dividend paying stock. The expected value of a call option at the period of maturity T is X½maxðST 2 K; 0Þ. Under risk neutrality, the price or value of the option PEC at period 0 is the discounted present value of this expectation, where the discounting is done with the risk-free rate interest rate r. Symbolically, ^ ½maxðST 2 K; 0Þ, where X ^ stands for the expected value under risk PEC 5 e 2 rT X neutrality. In view of assumption (1) of Section 5.2, we presume that ST follows the ðm; sÞ-geometric Brownian motion defined by (5.3), where both m and s are treated as constants so that ln(ST) is normally distributed with mean   s2 lnðS0 Þ 1 r 2 2 T and variance s2 T. The risk neutrality assumption enables us to replace m by the risk-free rate r. It can then be established that the Black–Scholes (1973)–Merton (1973) price PEC of the call option at period 0, the expected discounted value of the option at the risk-free rate of interest, is ^ ½maxðST 2 K; 0Þ ¼ S0 Fðz1 Þ 2 Ke 2 rT Fðz2 Þ; PEC ¼ e 2 rT X

(5.12)

where z1 ¼

ln

S

0

K

  2 1 r 1 s2 T pffiffiffiffi ; s T

(5.13)

and   2 ln SK0 1 r 2 s2 T pffiffiffiffi pffiffiffiffi : z2 ¼ z1 2 s T ¼ s T

(5.14)

Here F is the standard normal distribution function. More precisely, Z FðtÞ ¼






The integrand

p1ffiffiffiffi exp 2p

2

2 v2


1 v2 pffiffiffiffiffiffi exp 2 dv: 2 2 ‘ 2p t

(5.15)

in FðtÞ is the density function of the standard

normal random variable. Parallel arguments can be employed to demonstrate that the Black–Scholes (1973)–Merton (1973) price of a European put option is given by PEP ¼ Ke 2 rT Fð 2 z2 Þ 2 S0 Fð 2 z1 Þ:

(5.16)

With a substantial increase in the current price of the asset, both Fðz1 Þ and Fðz2 Þ approach 1 and PEC tends to S0 2 Ke2rT, but PEP approaches 0. If volatility

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An Introduction to Algorithmic Finance, Trading and Blockchain

approaches 0, the price of the call option approaches max(S0 2 Ke2rT, 0). In contrast, the price of the put option approaches max(Ke2rT 2 S0, 0). The derivation of the Black–Scholes–Merton partial differential equation given by (5.11) assumed at the outset of the stock is non-dividend paying (assumption (2) in Section 5.2). Suppose that the stock pays dividends at a constant rate Dy, which we refer to as the dividend yield. The usual stochastic process model then gets revised as dSt 5 mSt dt 1 sSt dWt 2 Dy St dt. One can now easily S

2 ln K0 1 r 2 Dy 1 s2 T p ffiffiffi and the modified verify that z1 in (5.13) gets modified as z1 5 s T pffiffiffiffi value of z2 is obtained by subtracting s T from the modified value of z1. By substituting these values of z1 and z2 in (5.12) and (5.16), we can derive the prices of the two options when the stock becomes one of dividend paying category. The formal relationship between the prices of a European call option and a European put option under ceteris paribus assumptions is known as put-call parity. By ceteris paribus assumptions, we mean here that the parameters on which the prices of the two options depend are identical. (For detailed discussion, see Chapter 15 of this monograph.)

5.6 The Greek Letters A Greek letter in the theory of finance indicates sensitivity of the price/value of a derivative with respect to a change in a parameter on which it depends. The five parameters on which the price of a European option relies are the stock price, the time to maturity, the strike price, the risk-free interest rate and the volatility. The Greek letters, popularly known as ‘Greeks’, are used by traders for the purpose of hedging the risks associated with options and portfolios. Below we present the definitions and values of five Greeks associated with European call and put options (Table 5.1). To understand the implications of changes in the option price with respect to a change in one or more parameters, suppose a trader’s objective is to invest X units of money in w1 units of a risk-free bond B, one long unit of a put option and w2 units of the stock, the underlying asset, where the choices of w1 and w2 are guided by the requirement that change in X should be negligible for a small change in the stock price. Now, the value of the portfolio consisting of the financial instruments ∂X EP 5 w2 1 ∂P considered is given by X 5 w1B 1 w2S0 1 PEP. Then ∂S ∂S0 5 0 w2 1 DeltaP 5 w2 1 DeltaC 2 1, where DeltaP and DeltaC stand, respectively, for ∂X Deltas of a put and a call. Our requirement can now be executed by setting ∂S 5 0. 0 Consequently, if the trader invests w2 5 1 2 DeltaC units in the stock and C ÞS0 2 PEP Þ w1 5 ðX 2 ð1 2 Delta units in the bond, the change in the total value of the B portfolio will be very minor for a small change in the stock price. This is Delta hedging, a hedging strategy modelled to keep the value of a portfolio of derivatives insensitive to small variations in the price of the underlying asset. Such a policy becomes useful when the trader apprehends a small reduction in the stock price.

Name

Delta Gamma

Definitions (European Call Option) ∂PEC ∂S0

5 Fðz1 Þ . 0
 z21 ∂2 PEC 1 5 S spffiffiffiffiffiffiffi exp 2 2 . 0 ∂S 2 2pT 0

0

Theta

∂PEC ∂T

5

1 2 S0 s 2pffiffiffiffiffiffiffi exp 2pT



z2 2 21



Definitions (European Put Option) ∂PEP ∂S0

5 2 Fð 2 z1 Þ 5 Fðz1 Þ 2 1 , 0
 z21 ∂2 PEP 1 5 S spffiffiffiffiffiffiffi exp 2 2 . 0 ∂S02 2pT 0   z21 ∂PEP 1 ffiffiffiffiffiffiffi p ∂T 5 2 S0 s 2 2pT exp 2 2 1 rK expð 2 rTÞFð 2 z2 Þ (either sign)

2 rK expð 2 rTÞFðz2 Þ , 0

Vega Rho

∂PEC ∂s ∂PEC ∂r

pffiffiffi T 5 S0 pffiffiffiffi exp 2p




z2 2 21



∂PEP ∂s

.0

5 TK expð 2 rTÞFðz2 Þ . 0

∂PEP ∂r

pffiffiffi T 5 S0 pffiffiffiffi exp 2p




z2 2 21

 .0

5 2 TK expð 2 rTÞFð 2 z2 Þ , 0

Brownian Motion, Itˆo Lemma and the Black–Scholes–Merton Model

Table 5.1. Greek Letters.

33

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Alternatively, if the trader invests X units of money in w1 units of the risk-free bond B, w2 units of the stock and possesses a short position in a call option, then the value of his portfolio becomes V 5 w1B 1 w2S0 2 PEC. In such a case, Delta ∂V EC hedging requires ∂S 5 w2 2 ∂P ∂S0 5 w2 2 DeltaC 5 0, which gives w2 5 DeltaC. 0

C S0 Þ Hence the trader invests w1 5 ðX 2 Delta units in the bond. The premium B received from the sale of the call option can be invested such that the trader can meet his obligations at the time of maturity of the option. If the value of a portfolio remains unchanged under small variations in the value of the underlying asset, then it is called Delta neutral. That is, the sum of Deltas across constituents of a portfolio is 0. In other words, Delta neutrality of a portfolio refers to its immunity with respect to minor variations in the price of the underlying asset. A trading strategy composed of one unit of the stock and some amalgamation of long and short calls and puts on the same underlying asset can make the sum of the Deltas across the components of the strategy equal to 0. Thus, the trader can have a long put (call) to hedge a long call (put) for downward (upward) asset price changes. While Delta neutrality is useful in the case of insignificant movements of the underlying asset price, Gamma neutrality becomes helpful for wider movements in the asset price. As time passes the value of a call option is likely to reduce. This is indicated by its Theta. Vega relates to the volatility of the asset price. It is quite important for bottom straddle strategies since big jumps in the asset price are likely to increase volatility. Rho generally has minor impact on option value. We may now investigate the relationship among Greeks using the Black– Scholes–Merton partial differential equation (see Lee, Finnerty, Lee, Lee, & Wort, 2013). Let q in equation (5.11) denote the value of a portfolio dependent on the underlying asset. We can then rewrite the equation in terms of the Greeks as follows:

1 2 2 s St ðGammaÞ 1 rSt ðDeltaÞ 1 Theta ¼ rq: 2

(5.17)

Now, consider two Delta neutral portfolios having the same value q. Given Delta neutrality and exogenous feature of r, it follows that 1 2 2 1 s St ðGamma1 Þ 1 Theta1 ¼ s2 St2 ðGamma2 Þ 1 Theta2 ; 2 2

(5.18)

where Gammai and Thetai stand, respectively, for the Greeks Gamma and Theta of portfolio i; 1 # i # 2: Now, suppose the price of the underlying asset remains unchanged. Given other things, if Thetai changes, then Gammai must change accordingly for the relation (5.18) to hold, where 1 # i # 2: This clearly establishes a relation between the Greeks Gamma and Theta under certain minor conditions. Hedging against any of the components of a portfolio that depend on asset price, time to maturity, volatility and risk-free interest rate needs knowledge on the asset and another derivative. By maintaining appropriate harmony between the underlying asset and other derivatives, hedgers can wipe out risks associated with changes in such components of the portfolio.

Chapter 6

Exotic Options: An Illustrative Presentation 6.1 Introduction Exotic options are nonstandard financial derivatives in that characteristics like dates of maturity, payoffs, assets on which their values rely, the strike or prearranged prices, etc. make them different from standard plain vanilla options, e.g. European and American options. Various complex features of such off exchange-traded financial instruments make them more suitable for hedging and risk management. Since our chapter is illustrative, we choose three common types of exotic options, namely, Asian, barrier and binary or digital options, and analyse their properties. The style of exercise of all options is assumed to be of European type so that their prices become variants of Black–Scholes–Merton prices.

6.2 Asian Options Asian options are one of the most frequently bartered over-the-counter options of non-vanilla categories. It is often referred to as the average option since its value relies on the average stock price instead of the spot price of the stock. It is a pathdependent derivative because it depends on the entire history of the underlying asset price starting from its inception. The premium of an Asian option is lower than that of a comparable European or an American option. The reason behind this is that the volatility associated with the average value of the underlying asset is likely to be lower than the volatility of the asset. This characteristic of the option makes it less risky. An Asian option is commonly traded in currency and commodity markets. While with an increase in the stock price the holder of a European call may be able to make large amount of money, this is ruled out for the holder of an Asian call since the latter relies on the average stock price. This is a negative feature of an Asian option. To determine the payoff function of an Asian option, it is necessary to calculate the average stock price, where the averaging principle may be of arithmetic or geometric type. We consider a finite number of equidistant time steps, t1 , … , tm indicating that (ti 2 ti21) takes on the same value for An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain, 35–39 Copyright © 2020 Emerald Publishing Limited All rights of reproduction in any form reserved doi:10.1108/978-1-78973-893-320201006

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i 5 2,…,m. For a finite positive time series St1 ; St2 ; …; Stm of stock prices, the
 1 m 1 m 5 exp geometric mean is defined as ð∏m S Þ lnð∏ S Þ 5 i51 ti i 5 1 ti m ! exp

1 m

m

+ lnðSti Þ , where ‘exp’ and ‘ln’ stand, respectively, for the exponential

i51

and logarithmic transformations. When the observations on the stock price are sampled continuously over the non-degenerate interval [0, T], the continuous analogue to the geometric average defined above comes to be
 R  G 5 exp 1 T lnðSt Þdt . Here we are going to adopt the geometric averaging S T 0 rule since underlying asset prices have a lognormal distribution when the price follows a (m, s)-geometric Brownian motion, where m and s are, respectively, the drift and volatility components of the motion. Given that risk neutrality holds, we can replace m by the risk-free interest rate r. (See Section 4.2 of Chapter 4 and Section 5.5 of Chapter 5.)  G associated with the stock price St varying Definition 6.1: For an average S over the continuous non-degenerate interval [0, T], strike price K, Asian options have the following payoff functions at the date of expiry T:  G 2 K; 0Þ, (1) Average price call: maxðS  G ; 0Þ. (2) Average price put: maxðK 2 S The functions (1) and (2), the Asian counterparts of a European call and a put payoff functions, respectively, are obtained from these two latter payoffs by G. substituting the maturity period asset price ST by the average price S  Since in a geometric average Asian call option, the role of S G is the same as that of ST in a European call, the price of an average call option for a nondividend paying stock can be derived straightway from the Black–Scholes– Merton pricing formula for a European call (see Kemna & Vorst, 1990). Now, as in Section 5.5 of Chapter 5, we write F for the standard normal distribution function. Then the explicit analytical form of the price can be written as PGAC ¼ S0 expððb 2 rÞTÞFðz1 Þ 2 Ke 2 rT Fðz2 Þ   b ¼ expð 2 rTÞ Eðgeometric average until TÞFðz1 Þ 2 KFðz2 Þ ;

(6.1)

b indicates that expectation is taken using a risk-neutral probability where E S   s 2 ln

0 K

1

b1

G 2

T

pffiffiffi , measure, ‘exp’ stands for exponential transformation, z1 5 sG T S   s 2
 G 0 pffiffiffiffi ln K 1 b 2 2 T 2 pffiffiffi z2 5 z1 2 sG T 5 , b 5 12 r 2 s6 , sG 5 psffiffi3 and S0 is the initial s T G

price of the stock. Using similar arguments, one can demonstrate that the price of a geometric average put option is given by PGAP ¼ K expð 2 rTÞFð 2 z2 Þ 2 S0 expððb 2 rÞTÞFð 2 z1 Þ:

Exotic Options: An Illustrative Presentation

37

For a stock with constant dividend yield Dy, in the two price formulae stated   2 2 above, b 5 12 r 2 s6 needs to be modified as b 5 12 r 2 Dy 2 s6 .

6.3 Binary or Digital Options Essential to the notion of a binary or digital option is a binary variable that takes on two specific values depending on whether a particular situation occurs or not. Depending on the nature of settlement, such options can generally be classified into two categories: cash-or-nothing and asset-or-nothing. A cash-or-nothing call/ put option pays lump sum cash M, say, if the option expires in-the-money, defined in the respective way. If it expires out-of-the money, it pays nothing. Thus, while for the holder of a call option the cash receipt M materialises if the asset price exceeds the strike price K at the date of maturity, the holder of a put option receives the cash payment if the opposite inequality holds at the maturity date. (See Section 3.2 of Chapter 3.) The discontinuous payoff functions of cash-or-nothing call option can, therefore, be defined as follows:  JCNC ðST ; KÞ ¼

M if ST . K; 0 if ST # K:

(6.2)

The discontinuous payoff function for the corresponding put option is given by  JCNP ðST ; KÞ ¼

M if ST , K; 0 if ST $ K:

(6.3)

The definitions of their counterparts for asset-or-nothing options are as follows:  JANC ðST ; KÞ ¼  JANP ðST ; KÞ ¼

ST if ST . K; 0 if ST # K:

(6.4)

ST if ST , K; 0 if ST $ K:

(6.5)

One distinguishing feature of a cash-or-nothing option from other options is that the value of the cash payment is determined at the outset; it is independent of the amount by which the asset price deviates from the strike price. A second differing characteristic is that at the maturity period the necessity of payment of strike price does not arise; it works like a boundary point only. Binary options involve limited risks and are suitable for short-term trading. They can be used for betting on upturn and downturn movements of asset prices as cheaper alternatives to standard vanilla options. For asset-or-nothing call/put options, the payment equals to the asset price at the date of expiration. Other characteristics of these options are similar to those of cash-or-nothing options. The Black–Scholes–Merton pricing formulae can be adjusted to determine the prices of cash-or-nothing options. The formulae for the prices of cash-or-nothing

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An Introduction to Algorithmic Finance, Trading and Blockchain

call and put options for a non-dividend paying stock are given, respectively, by PCNC 5 exp(2rT)MF(z2) and PCNP 5 exp(2rT)MF(2z2), where S   2  ! ln K0 1 r 2 s2 T pffiffiffi . On the other hand, the respective formulae for the z2 5 s T prices of asset-or-nothing call and put options are PANC 5 S0F(z1) and PANP 5 S   2  ! ln K0 1 r 1 s2 T pffiffiffi . (See Baz & Chacko, 2008, for further S0F(2z1), where z1 5 s T discussions.)

6.4 Barrier Options A second example of a path-dependent option is a barrier option. While for an Asian option the value depends on the average asset price, for a barrier option it is contingent on whether the path of the asset has reached a second strike price, known as the barrier or trigger B. Such options appear in four types: up-and-in, up-and-out, down-and-in and down-and-out. An up-and-in (up-and-out) option is a derivative that becomes effective (ineffective) when the underlying asset price increases to the barrier. In contrast, a downand-in (down-and-out) option becomes operative (inoperative) if the barrier is attained by decreasing the price of the underlying asset. Thus, we can categorise these four options into two broader subgroups – Ups (the asset price starts from a level below the barrier and increases up to it) and Downs (the asset starts from a level above the barrier and reduces down to it). They can also be categorised as knockout or extinguishable options that cease to be active when the asset price hits the barrier and knock-in or lightable options that become active as soon as the asset price breaches the barrier. Therefore, the barrier can be above or below the asset price at the time of initiation of the contract. Barrier options, which are less expensive than standard options, can be useful as hedging instruments when the investor has potential idea about the movement of the asset price. To determine the payoff structure of the barrier options, let SM and Sm denote, respectively, the maximum and minimum values that the asset price St can take over the interval (0, T). Now, for an up-and-in call option if the barrier B . S0 is reached during the life of the option, then it becomes a standard call and its maturity payoff is JUIC ðST ; K; BÞ 5 maxðST 2 K; 0Þ 3 I ðSM $ BÞ, where I(SM $ B) is an indicator function which equals 1 if SM $ B, 0, otherwise. For the put option, the maturity payoff is JUIP ðST ; K; BÞ 5 maxðK 2 ST ; 0Þ 3 IðSM $ BÞ. For an up-and-out call option, the relationship is reversed: it remains a standard call option unless the spot asset price breaches the barrier so that its maturity payoff comes to be JUOC ðST ; K; BÞ 5 maxðST 2 K; 0Þ 3 IðSm , BÞ, where I(Sm , B) is an indicator function that takes on the value 1 if Sm , B, 0, otherwise. For the put, this payoff is JUOP ðST ; K; BÞ 5 maxðK 2 ST ; 0Þ 3 I ðSm , BÞ. For down-and-out and down-and-in options, the barrier is set below the initial price of the underlying asset, B , S0. A down-and-in call option

Exotic Options: An Illustrative Presentation

39

becomes a standard call if the asset price St reaches the barrier B sometime during the life of the option. Consequently, its payoff at maturity is JDIC ðST ; K; BÞ 5 maxðST 2 K; 0Þ 3 I ðSm # BÞ, where I(Sm # B) is an indicator function which equals 1 if Sm # B, 0, otherwise. For the corresponding put option, the payoff function is JDIP ðST ; K; BÞ 5 maxðK 2 ST ; 0Þ 3 I ðSm # BÞ. For the remaining member of the subgroup Downs, the down-and-out option, the opposite relationship holds so that the call and put payoff functions are, respectively, JDOC ðST ; K; BÞ 5 maxðST 2 K; 0Þ 3 IðSM . BÞ and JDOP ðST ; K; BÞ 5 maxðK 2 ST ; 0Þ 3 I ðSM . BÞ, where the indicator function I(SM . B) takes on the value 1 or 0 according as SM . B or SM # B. Each of an up-and-out and down-and-out option becomes a plain vanilla option unless the stock price touches the barrier. To evaluate an up-and-out call option, as before, we assume risk-neutral valuation of the stock price. The price of the option will depend on the regular Black–Scholes–Merton parameters S0, K, r, T, s as well as on the barrier level B. The prices of barrier options can be calculated using the observation that the sum of values of call options of knock-in and knockout varieties equals the value of a plain vanilla call option, given that the strike prices, barriers and maturity periods are the same. We may refer to this result as ‘in-out’ parity. The formulae for prices of a down-and-in and a down-and-out call options for a non-dividend paying stock are given, respectively, by
PDOC ðST ; K; BÞ ¼ PEC ðST ; KÞ 2
PDIC ðST ; K; BÞ ¼

B ST

B ST

q

q


PEC


PEC

 B2 ;K ; ST

 B2 ;K ; ST

(6.6)

where q 5 2r s22s . Evidently, PDOC(ST, K, B) 1 PDIC(ST, K, B) 5 PEC(ST, K). A similar relation holds between up-and-in and an up-and-out call options. (Exotic options are analysed in greater detail in Clewlow and Strickland, 1997; Zhang, 1998; Baz and Chacko, 2008 and Hull, 2014.) 2

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Chapter 7

An Abbreviated Theoretical Treatment of Stochastic Dominance Relations 7.1 Introduction Consider a person who is exploring the possibility of investing in one of the two prospects or assets, say stocks, A and B, whose rates of returns (returns, for short) are uncertain. It is natural for the investor to choose the stock whose expected return is higher. A risk-averse investor also becomes biased towards the choice of an asset whose expected return is not associated with high risk. One proposition to the resolution to this problem is the mean-variance approach (see Markowitz, 1959; Tobin, 1958), where the variance of returns is taken as an indicator of risk. In the well-known Markowitz model of portfolio management, the quadratic utility function, the mean return minus a positive constant multiple of the variance, is taken as the expected utility function of an investor. In this model, a riskaverse investor’s objective becomes designing a portfolio to maximise expected return for a given level of risk or to minimise risk when the expected return is given. But the Markowitz approach, although quite appealing, relies on a specific expected utility function. Furthermore, two investments cannot be unambiguously ranked if one of them has a higher expected return accompanied by a higher variance. Alternative weaker concepts, with extensive applications, that can rank uncertain prospects have been suggested in the literature. They can be defined without having knowledge about an investors’ trade-off between return and risk, as represented by his utility function. These ranking criteria, the stochastic dominance relations, are the subject of this chapter.

7.2 First-order Stochastic Dominance For each prospect, the uncertain return is assumed to follow a continuous type distribution. Consider two prospects A and B whose uncertain returns are required to be ordered by the first-order stochastic dominance criterion. Let FA, FB : [a, b] → [0, 1] be the cumulative distribution functions of the uncertain returns associated with the prospects A and B, respectively. The lower and upper limits a and b of the non-degenerate interval [a, b] represent, respectively, the lowest and highest values that the returns can assume. An Introduction to Algorithmic Finance, Algorithmic Trading and Blockchain, 41–45 Copyright © 2020 Emerald Publishing Limited All rights of reproduction in any form reserved doi:10.1108/978-1-78973-893-320201007

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An Introduction to Algorithmic Finance, Trading and Blockchain

For each j 5 A, B, Fj(z) is the cumulative probability that the return on the prospect j does not exceed the level z, that is, Fj(z) 5 Pr(Rj # z), where Pr stands for probability and Rj is the uncertain return on the prospect j. Further, Fj is increasing, continuously differentiable, Fj(a) 5 0 and Fj(b) 5 1 for each j 5 A, B. FA is said to first order stochastic dominate FB, FA fFSD FB , for short, if and only if FA ðzÞ # FB ðzÞ

(7.1)

for all z2½a; b, with , for at least one z2ða; bÞ. Graphically, FA fFSD FB means that the graph of the cumulative distribution function of the prospect B lies nowhere below that of A and above in some places (at least). To state the utility equivalent condition of (7.1), consider the von Neumann–Morgenstern utility function U defined on [a, b], that is, U : ½a; b→ℜ, where ℜ is the real line. Assume that U is twice continuously differentiable. The first- and second-order derivatives of U are denoted by U 9 and U 0 , respectively. It is assumed throughout that U 9 . 0, which means that U is increasing, an assumption that reflects the view that more is preferred to less. Given increasingness, we also assume that U is strictly concave, that is, U 0 , 0. Strict concavity of a person’s utility function is related to his attitude towards risk. An individual is called risk averse if he prefers his current wealth to any fair gamble in which there is a 50–50 chance of losing some amount out of the current wealth or winning the same amount. A person is risk averse if and only if U is strictly concave. This is also necessary and sufficient for the Arrow (1970)-Pratt (1964) absolute and 0 relative coefficients of risk aversion; defined, respectively, as ðzÞ zU 0 ðzÞ and AP ðzÞ 5 2 to be positive, where z2ða; bÞ. A person APA ðzÞ 5 2 U R U 9 ðzÞ U 9 ðzÞ possessing a utility function U satisfying strict convexity ðU 0 . 0Þ is risk lover and these measures are negative for him. While a risk-loving person prefers high variability among levels of returns on a prospect, a risk averter dislikes this. In contrast to these two extreme positions, if the individual’s utility function is linear ðU 0 5 0Þ, then these two coefficients take on the value zero; and the individual is regarded as risk neutral. The following theorem of Hadar and Russell (1969) which identifies the class of utility functions compatible with the ordering FA fFSD FB can now be stated. Theorem 7.1: Let FA, FB : [a, b] → [0, 1] be the distribution functions of the uncertain prospects A and B, respectively. Then the following statements are equivalent: (1) FA fFSD FB , that is, FA first-order stochastic dominates FB. Rb Rb (2) UðzÞdFA ðzÞ . UðzÞdFB ðzÞ for all utility functions U : ½a; b→ℜ that are a

a

increasing, that is, expected utility of the prospect A is higher than that of B for all increasing utility functions. What this theorem says is the following. Of the two prospects A and B characterised with uncertain returns, if the former first stochastic dominates the

An Abbreviated Theoretical Treatment of Stochastic Dominance

43

latter then the former will be preferred to the latter by all individuals who prefer more to less. The converse is also true. Thus, if the stochastic dominance condition (1) in Theorem 7.1 is demonstrated, we do not need to have information on the form of the utility function to judge whether the expected utility of the former is higher than that of the latter. However, although the ordering is transitive and irreflexive, it is not complete. To understand this, note that FA fFSD FB requires that the graph of FA can never lie above that of FB, that is, the two graphs can never intersect. If the two graphs intersect at least once we can get two different utility functions which will rank the prospects A and B in different directions with respect to expected utility values. Thus, fFSD is a partial or quasi-ordering. Examples of utility functions that become congruent with the ordering fFSD are U1(z) 5 ln(z), z . 0, U2(z) 5 1 2 e2z, U3(z) 5 ez, U4 ðzÞ 5 zu , where u . 1 is a constant and U5(z) 5 a 1 bz, where b . 0 and a are constants. The following example, which is a variant of an example given in Chakravarty (2013), will illustrate the fFSD criterion (see also Eichberger & Harper, 1969). Example 7.1: Consider the following distribution functions of the assets A and B, respectively:  GA ðnÞ ¼

1 2 e 2 z ; z $ 0; ; GB ðnÞ ¼ 0; z , 0:



1 2 e 2 cz ; z $ 0; 0; z , 0:

(7.2)

where c . 1 is a constant. It is easy to see that GA(0) 5 GB(0) 5 0, GA ð‘Þ 5 GB ð‘Þ 5 1 and both GA and GB are increasing. Note also that GA(z) , GB(z) for all finite positive return values z. Consequently, GA first-order stochastic dominates GB. Among the examples of five utility functions considered above, while the first two are strictly concave, the next two are strictly convex and the fifth one displays risk neutrality. However, if FA fFSD FB is substantiated then for all of them A is preferred to B by the expected utility criterion. This is because under the firstorder stochastic dominance criterion, the only distinguishing feature between two uncertain prospects is efficiency, that is, desirability for a higher expected utility. Consideration about the decision maker’s attitude towards risks is totally absent. In other words, this principle of ranking prospects does not tell us whether a person is risk lover, risk averter or risk neutral. This issue is addressed by the second-order stochastic dominance rule, which we analyse below.

7.3 Second-order Stochastic Dominance If FA and FB stand, respectively, for the distribution functions of the risky prospects A and B, then FA is said to second-order stochastic dominate FB, FA fSSD FB , for short, if and only if Zz

Zz FA ðnÞdn #

a

FB ðnÞdn a

(7.3)

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An Introduction to Algorithmic Finance, Trading and Blockchain

for all z2½a; b, with , for at least one z. In words, for any z2½a; b, the area underneath the curve of FA is not higher than that underneath the curve of FB, and at some places (at least), the former area is lower. Evidently, FA fFSD FB implies FA fSSD FB but the converse is not true. To see this explicitly, we consider the following example (see Chakravarty, 2013; Eichberger & Harper, 1969). Example 7.2: Assume that the distribution functions of the assets A and B are given, respectively, by: 2

HA ðzÞ ¼ ðz 2 1Þ; 1 # z , 2;

z HB ðzÞ ¼ ; 0 # z # 3: 3

(7.4)

 A(1) 5 HB(0) 5 0 and H  A(2) 5 HB(3) 5 1. Both H  A and HB are Note that H increasing over respective domains. Since their domains are different, they  A as cannot be compared. In order to make the comparison valid, we modify H follows: H A ðzÞ ¼

8