An introduction to financial mathematics option valuation, 2nd edition

An Introduction to Financial Mathematics Option Valuation Second Edition CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the financial sector This series aims to capture new developments and summarize what is known over the whole spectrum of this field It will include a broad range of textbooks, reference works and handbooks that are meant to appeal to both academics and practitioners The inclusion of numerical code and concrete real-world examples is highly encouraged Series Editors M.A.H Dempster Centre for Financial Research Department of Pure Mathematics and Statistics University of Cambridge Dilip B Madan Robert H Smith School of Business University of Maryland Rama Cont Department of Mathematics Imperial College High-Performance Computing in Finance Problems, Methods, and Solutions M.A.H Dempster, Juho Kanniainen, John Keane, Erik Vynckier An Introduction to Computational Risk Management of Equity-Linked Insurance Runhuan Feng Derivative Pricing A Problem-Based Primer Ambrose Lo Portfolio Rebalancing Edward E Qian Interest Rate Modeling Theory and Practice, 2nd Edition Lixin Wu An Introduction to Financial Mathematics Option Valuation, Second Edition Hugo D Junghenn For more information about this series please visit: https://www.crcpress.com/Chapmanand-HallCRC-Financial-Mathematics-Series/book-series/CHFINANCMTH An Introduction to Financial Mathematics Option Valuation Second Edition Hugo D Junghenn CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20190207 International Standard Book Number-13: 978-0-367-20882-0 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com TO MY FAMILY Mary, Katie, Patrick, Sadie Contents Preface Basic 1.1 *1.2 1.3 1.4 *1.5 1.6 xi Finance Interest Inflation Annuities Bonds Internal Rate of Exercises 1 10 11 13 Probability Spaces 2.1 Sample Spaces and Events 2.2 Discrete Probability Spaces 2.3 General Probability Spaces 2.4 Conditional Probability 2.5 Independence 2.6 Exercises 17 17 18 21 26 30 31 Random Variables 3.1 Introduction 3.2 General Properties of Random Variables 3.3 Discrete Random Variables 3.4 Continuous Random Variables 3.5 Joint Distributions of Random Variables 3.6 Independent Random Variables 3.7 Identically Distributed Random Variables 3.8 Sums of Independent Random Variables 3.9 Exercises 35 35 37 38 42 44 46 48 48 51 Options and Arbitrage 4.1 The Price Process of an Asset 4.2 Arbitrage 4.3 Classification of Derivatives 4.4 Forwards 4.5 Currency Forwards 4.6 Futures 55 55 56 58 59 60 61 Return vii viii Contents *4.7 4.8 4.9 4.10 4.11 *4.12 4.13 Equality of Forward and Future Call and Put Options Properties of Options Dividend-Paying Stocks Exotic Options Portfolios and Payoff Diagrams Exercises Prices 63 64 67 69 70 73 76 Discrete-Time Portfolio Processes 5.1 Discrete Time Stochastic Processes 5.2 Portfolio Processes and the Value Process 5.3 Self-Financing Trading Strategies 5.4 Equivalent Characterizations of Self-Financing 5.5 Option Valuation by Portfolios 5.6 Exercises 79 79 83 84 85 87 88 Expectation 6.1 Expectation of a Discrete Random Variable 6.2 Expectation of a Continuous Random Variable 6.3 Basic Properties of Expectation 6.4 Variance of a Random Variable 6.5 Moment Generating Functions 6.6 The Strong Law of Large Numbers 6.7 The Central Limit Theorem 6.8 Exercises 91 91 93 95 96 98 99 100 102 The Binomial Model 7.1 Construction of the Binomial Model 7.2 Completeness and Arbitrage in the Binomial Model 7.3 Path-Independent Claims *7.4 Path-Dependent Claims 7.5 Exercises 107 107 111 115 119 121 Conditional Expectation 8.1 Definition of Conditional Expectation 8.2 Examples of Conditional Expectations 8.3 Properties of Conditional Expectation 8.4 Special Cases *8.5 Existence of Conditional Expectation 8.6 Exercises 125 125 126 128 130 132 134 Martingales in Discrete Time Markets 9.1 Discrete Time Martingales 9.2 The Value Process as a Martingale 9.3 A Martingale View of the Binomial Model 9.4 The Fundamental Theorems of Asset Pricing 135 135 137 138 140 ix Contents *9.5 9.6 Change of Probability Exercises 142 144 10 American Claims in Discrete-Time Markets 10.1 Hedging an American Claim 10.2 Stopping Times 10.3 Submartingales and Supermartingales 10.4 Optimal Exercise of an American Claim 10.5 Hedging in the Binomial Model 10.6 Optimal Exercise in the Binomial Model 10.7 Exercises 147 147 149 151 152 154 155 156 11 Stochastic Calculus 11.1 Continuous-Time Stochastic Processes 11.2 Brownian Motion 11.3 Stochastic Integrals 11.4 The Ito-Doeblin Formula 11.5 Stochastic Differential Equations 11.6 Exercises 12 The Black-Scholes-Merton Model 12.1 The Stock Price SDE 12.2 Continuous-Time Portfolios 12.3 The Black-Scholes Formula 12.4 Properties of the Black-Scholes Call Function *12.5 The BS Formula as a Limit of CRR Formulas 12.6 Exercises 183 183 184 185 188 191 194 13 Martingales in the Black-Scholes-Merton Model 13.1 Continuous-Time Martingales 13.2 Change of Probability and Girsanov’s Theorem 13.3 Risk-Neutral Valuation of a Derivative 13.4 Proofs of the Valuation Formulas *13.5 Valuation under P *13.6 The Feynman-Kac Representation Theorem 13.7 Exercises 197 197 201 204 205 208 209 211 14 Path-Independent Options 14.1 Currency Options 14.2 Forward Start Options 14.3 Chooser Options 14.4 Compound Options 14.5 Quantos 14.6 Options on Dividend-Paying 14.7 American Claims 14.8 Exercises 213 213 216 216 218 219 221 224 226 Stocks 159 159 160 164 170 176 180 .. .An Introduction to Financial Mathematics Option Valuation Second Edition CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding... Runhuan Feng Derivative Pricing A Problem-Based Primer Ambrose Lo Portfolio Rebalancing Edward E Qian Interest Rate Modeling Theory and Practice, 2nd Edition Lixin Wu An Introduction to Financial. .. https://www.crcpress.com/Chapmanand-HallCRC -Financial- Mathematics- Series/book-series/CHFINANCMTH An Introduction to Financial Mathematics Option Valuation Second Edition Hugo D Junghenn CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL