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Analytical Finance Volume I The Mathematics of Equity Derivatives, Markets, Risk and Valuation ANALYTICAL FINANCE VOLUME I JAN R. M. RÖMAN Analytical Finance: Volume I Jan R. M. Röman Analytical Finance: Volume I The Mathematics of Equity Derivatives, Markets, Risk and Valuation Jan R. M. Röman Västerås, Sweden ISBN 978-3-319-34026-5 ISBN 978-3-319-34027-2 (eBook) DOI 10.1007/978-3-319-34027-2 Library of Congress Control Number: 2016956452 © The Editor(s) (if applicable) and The Author(s) 2017 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover image © David Tipling Photo Library / Alamy Printed on acid-free paper This Palgrave Macmillan imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To my soulmate, supporter and love – Jing Fang Preface This book is based upon lecture notes, used and developed for the course Analytical Finance I at Mälardalen University in Sweden. The aim is to cover the most essential elements of valuing derivatives on equity markets. This will also include the maths needed to understand the theory behind the pricing of the market instruments, that is, probability theory and stochastic processes. We will include pricing with time-discrete models and models in continuous time. First, in Chap. 1 and 2 we give a short introduction to trading, risk and arbitrage-free pricing, which is the platform for the rest of the book. Then a number of different binomial models are discussed. Binomial models are impor- tant, not only to understand arbitrage and martingales, but also they are widely used to calculate the price and the Greeks for many types of derivative. Binomial models are used in trading software to handle and value several kinds of derivative, especially Bermudan and American type options. We also discuss how to increase accuracy when using binomial models. We continue with an introduction to numerical methods to solve partial differential equations (PDEs) and Monte Carlo simulations. In Chap. 3, an introduction to probability theory and stochastic integration is given. Thereafter we are ready to study continuous finance and partial differential equations, which is used to model many financial derivatives. We focus on the Black–Scholes equation in particular. In the continuous time model, there are no solutions to American options, since they can be exercised during the entire lifetime of the contracts. Therefore we have no well-defined boundary condition. Since most exchange-traded options with stocks as vii viii Preface underlying are of American type, we still need to use descrete models, such as the binomial model. We will also discuss a number of generalizations relating to Black–Scholes, such as stochastic volatility and time-dependent parameters. We also discuss a number of analytical approximations for American options. A short introduction to Poisson processes is also given. Then we study diffusion processes in general, martingale representation and the Girsanov theorem. Before finishing off with a general guide to pricing via Black–Scholes we also give an introduction to exotic options such as weather derivatives and volatility models. As we will see, many kinds of financial instrument can be valued via a discounted expected payoff of a contingent claim in the future. We will denote this expectation E[X(T)] where X(T) is the so-called contingent claim at time T. This future value must then be discounted with a risk-free interest rate, r,to give the present value of the claim. If we use continuous compounding we can write the present value of the contingent claim as r T t XtðÞ ¼eÀ ðÞÀ EX½ðÞ T : In the equation above, T is the maturity time and t the present time. Example: If you buy a call option on an underlying (stock) with maturity T and strike price K, you will have the right, but not the obligation, to buy the stock at time T, to the price K.IfS(t) represents the stock price at time t, the contingent claim can be expressed as X(T) ¼ max{S(T) – K, 0}. This means that the present value is given by r T t r T t XtðÞ ¼eÀ ðÞÀ EX½¼ðÞ T eÀ ðÞÀ E½ maxfg SðÞÀ T K,0 : The max function indicates a price of zero if K S(T). With this condition you can buy the underlying stock at a lower (same) price on the market, so the option is worthless. By solving this expectation value we will see that this can be given (in continuous time) as the Black–Scholes–Merton formula. But generally we have a solution as : ÀrðÞTÀt : XtðÞ ¼SðÞ0 Q1ðÞÀSTðÞ> K e K Q2ðÞSTðÞ> K ; > > where Q1(S(T) K) and Q2(S(T) K) make up the risk neutral probability for the underlying price to reach the strike price K in different “reference systems”. This can be simplified to the Black–Scholes–Merton formula as Preface ix : ÀrðÞTÀt : XtðÞ ¼SðÞ0 NdðÞÀ1 e K NdðÞ2 : Here d1 and d2 are given (derived) variables. N(x) is the standard normal distribution with mean 0 and variance 1, so N(d2) represent the probability for the stock to reach the strike price K. The variables d1 and d2 will depend on the initial stock price, the strike price, interest rate, maturity time and volatility. The volatility is a measure of how much the stock price may vary in a specific period in time. Normally we use 252 days, since this is an approximation of the number of trading days in a year. Also remark that by buying a call option (i.e., going long in the option contract), as in the example above, we do not take any risk. The reason is that we cannot lose more money than what we invested. This is because we have the right, but not the obligation, to fulfil the contract. The seller, on the other hand, takes the risk, since he/she has to sell the underlying stock at price K. So if he/she doesn’t own the underlying stock he/she might have to buy the stock at a very high price and then sell it at a much lower price, the option strike price K. Therefore, a seller of a call option, who have the obligation to sell the underlying stock to the holder, takes a risky position if the stock price becomes higher than the option strike price. Acknowledgements I would like to thank all my students for their comments and questions during my lectures. Special thanks go to Mai Xin who asked me to translate my notes into English. I would also like to thank Professor Dmitrii Silvestrov, who encouraged me to teach Analytical Finance, and Professor Anatoly Malyarenko for his assistance and advice. xi Contents 1 Trading Financial Instruments 1 1.1 Clearing and Settlement 1 1.2 About Risk 7 1.3 Credit and Counterparty Risk 10 1.4 Settlement Risk 12 1.5 Market Risk 12 1.6 Model Risk 13 2 Time-Discrete Models 21 2.1 Pricing via Arbitrage 21 2.2 Martingales 23 2.3 The Central Limit Theorem 25 2.4 A Simple Random Walk 27 2.5 The Binomial Model 29 2.6 Modern Pricing Theory Based on Risk-Neutral Valuation 38 2.7 More on Binomial Models 43 2.8 Finite Difference Methods 65 2.9 Value-at-Risk (VaR) 78 3 Introduction to Probability Theory 91 3.1 Introduction 91 3.2 A Binomial Model 91 3.3 Finite Probability Spaces 93 3.4 Properties of Normal and Log-Normal Distributions 121 xiii xiv Contents 3.5 The Itô Lemma 125 3.6 Stochastic Integration 133 4 Continuous Time Models 145 4.1 Classifications of Partial Differential Equations 145 4.2 Parabolic PDEs 146 4.3 The Black–Scholes–Merton Model 154 4.4 Volatility 177 4.5 Parity Relations 187 4.6 A Practical Guide to Pricing 189 4.7 Currency Options and the Garman–Kohlhagen Model 197 4.8 Analytical Pricing Formulas for American Options 235 4.9 Poisson Processes and Jump Diffusion 240 5 Black–Scholes – Diffusion Models 245 5.1 Martingale Representation 245 5.2 Girsanov Transformation 246 5.3 Securities Paying Dividends 273 5.4 Hedging 284 6 Exotic Options 289 6.1 Contract for Difference: CFD 290 6.2 Binary Options/Digital Options 292 6.3 Barrier Options: Knock-out and Knock-in Options 298 6.4 Lookback Options 311 6.5 Asian Options 312 6.6 Chooser Options 316 6.7 Forward Options 318 6.8 Compound Options: Options on Options 324 6.9 Multi-asset Options 326 6.10 Basket Options 328 6.11 Correlation Options 332 6.12 Exchange Options 332 6.13 Currency-linked Options 333 6.14 Pay-later Options 335 6.15 Extensible Options 336 6.16 Quantos 337 6.17 Structured Products 340 Contents xv 6.18 Summary of Exotic Instruments 345 6.19 Something About Weather Derivatives 346 7 Pricing Using Deflators 351 7.1 Introduction to Deflators 351 8 Strategies with Options 365 8.1 Introduction 365 8.2 Strategies 375 8.3 A Decreasing
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