Computational Finance Using C and C# Quantitative Finance Series Aims and Objectives • • • • • • Books based on the work of financial market practitioners and academics Presenting cutting-edge research to the professional/practitioner market Combining intellectual rigour and practical application Covering the interaction between mathematical theory and financial practice To improve portfolio performance, risk management and trading book performance Covering quantitative techniques Market Brokers/Traders; Actuaries; Consultants; Asset Managers; Fund Managers; Regulators; Central Bankers; Treasury Officials; Technical Analysis; and Academics for Masters in Finance and MBA market Series Titles Computational Finance Using C and C# The Analytics of Risk Model Validation Forecasting Expected Returns in the Financial Markets Corporate Governance and Regulatory Impact on Mergers and Acquisitions International Mergers and Acquisitions Activity Since 1990 Forecasting Volatility in the Financial Markets, Third Edition Venture Capital in Europe Funds of Hedge Funds Initial Public Offerings Linear Factor Models in Finance Computational Finance Advances in Portfolio Construction and Implementation Advanced Trading Rules, Second Edition Real R&D Options Performance Measurement in Finance Economics for Financial Markets Managing Downside Risk in Financial Markets Derivative Instruments: Theory, Valuation, Analysis Return Distributions in Finance Series Editor: Dr Stephen Satchell Dr Satchell is Reader in Financial Econometrics at Trinity College, Cambridge; Visiting Professor at Birkbeck College, City University Business School and University of Technology, Sydney He also works in a consultative capacity to many firms, and edits the journal Derivatives: use, trading and regulations and the Journal of Asset Management Computational Finance Using C and C# Derivatives and Valuation SECOND EDITION George Levy AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an Imprint of Elsevier Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, UK 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Copyright © 2016, 2008 Elsevier Ltd All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-803579-5 For information on all Academic Press publications visit our website at https://www.elsevier.com/ Publisher: Nikki Levy Acquisition Editor: J Scott Bentley Editorial Project Manager: Susan Ikeda Production Project Manager: Julie-Ann Stansfield Designer: Mark Rogers Typeset by Focal Image (India) Pvt Ltd Dedication To my parents Paul and Paula and also my grandparents Friedrich and Barbara Contents Preface Overview of Financial Derivatives Introduction to Stochastic Processes 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 Brownian Motion A Brownian Model of Asset Price Movements Ito’s Formula (or Lemma) Girsanov’s Theorem Ito’s Lemma for Multi-Asset GBM Ito Product and Quotient Rules in Two Dimensions 2.6.1 Ito Product Rule 2.6.2 Ito Quotient Rule Ito Product in n Dimensions The Brownian Bridge Time Transformed Brownian Motion 2.9.1 Scaled Brownian Motion 2.9.2 Mean Reverting Process Ornstein Uhlenbeck Process The Ornstein Uhlenbeck Bridge Other Useful Results 2.12.1 Fubini’s Theorem 2.12.2 Ito’s Isometry 2.12.3 Expectation of a Stochastic Integral Selected Exercises 10 12 12 14 15 15 17 18 20 21 21 22 25 29 29 29 30 31 Generation of Random Variates 3.1 3.2 3.3 3.4 3.5 xvii Introduction Pseudo-Random and Quasi-Random Sequences Generation of Multivariate Distributions: Independent Variates 3.3.1 Normal Distribution 3.3.2 Lognormal Distribution 3.3.3 Student’s t-Distribution Generation of Multivariate Distributions: Correlated Variates 3.4.1 Estimation of Correlation and Covariance 3.4.2 Repairing Correlation and Covariance Matrices 3.4.3 Normal Distribution 3.4.4 Lognormal Distribution Selected Exercises 35 36 40 40 43 44 44 44 45 49 53 56 European Options 4.1 4.2 4.3 Introduction Pricing Derivatives using A Martingale Measure Put Call Parity 4.3.1 Discrete Dividends 57 57 58 58 vii viii Contents 4.4 4.5 4.6 59 60 60 63 65 73 78 82 82 82 85 87 90 Single Asset American Options 5.1 5.2 5.3 5.4 5.5 5.6 4.3.2 Continuous Dividends Vanilla Options and the Black–Scholes Model 4.4.1 The Option Pricing Partial Differential Equation 4.4.2 The Multi-asset Option Pricing Partial Differential Equation 4.4.3 The Black–Scholes Formula 4.4.4 Historical and Implied Volatility 4.4.5 Pricing Options with Microsoft Excel Barrier Options 4.5.1 Introduction 4.5.2 Analytic Pricing of Down and Out Call Options 4.5.3 Analytic Pricing of Up and Out Call Options 4.5.4 Monte Carlo Pricing of Down and Out Options Selected Exercises Introduction Approximations for Vanilla American Options 5.2.1 American Call Options with Cash Dividends 5.2.2 The Macmillan, Barone-Adesi, and Whaley Method Lattice Methods for Vanilla Options 5.3.1 Binomial Lattice 5.3.2 Constructing and using the Binomial Lattice 5.3.3 Binomial Lattice with a Control Variate 5.3.4 The Binomial Lattice with BBS and BBSR Grid Methods for Vanilla Options 5.4.1 Introduction 5.4.2 Uniform Grids 5.4.3 Nonuniform Grids 5.4.4 The Log Transformation and Uniform Grids 5.4.5 The Log Transformation and Nonuniform Grids 5.4.6 The Double Knockout Call Option Pricing American Options using a Stochastic Lattice Selected Exercises 93 93 93 99 108 108 115 123 125 129 129 131 144 152 156 158 165 173 Multi-asset Options 6.1 6.2 6.3 6.4 6.5 6.6 Introduction The Multi-asset Black–Scholes Equation Multidimensional Monte Carlo Methods Introduction to Multidimensional Lattice Methods Two-asset Options 6.5.1 European Exchange Options 6.5.2 European Options on the Maximum or Minimum 6.5.3 American Options Three-asset Options 175 175 176 180 183 183 185 189 193 Contents 6.7 6.8 7.3 7.4 7.5 7.6 Introduction Interest Rate Derivatives 7.2.1 Forward Rate Agreement 7.2.2 Interest Rate Swap 7.2.3 Timing Adjustment 7.2.4 Interest Rate Quantos Foreign Exchange Derivatives 7.3.1 FX Forward 7.3.2 European FX Option Credit Derivatives 7.4.1 Defaultable Bond 7.4.2 Credit Default Swap 7.4.3 Total Return Swap Equity Derivatives 7.5.1 TRS 7.5.2 Equity Quantos Selected Exercises 203 203 204 205 211 216 221 222 223 225 228 228 229 230 230 233 236 C# Portfolio Pricing Application 8.1 8.2 8.3 8.4 8.5 196 198 Other Financial Derivatives 7.1 7.2 Four-asset Options Selected Exercises ix Introduction Storing and Retrieving the Market Data Equity Deal Classes 8.3.1 Single Equity Option 8.3.2 Option on Two Equities 8.3.3 Generic Equity Basket Option 8.3.4 Equity Barrier Option FX Deal Classes 8.4.1 FX Forward 8.4.2 Single FX Option 8.4.3 FX Barrier Option Selected Exercises 239 247 253 254 256 257 262 266 266 267 269 273 A Brief History of Finance 9.1 9.2 9.3 Introduction Early History 9.2.1 The Sumerians 9.2.2 Biblical Times 9.2.3 The Greeks 9.2.4 Medieval Europe Early Stock Exchanges 9.3.1 The Anwterp Exchange 275 275 275 277 278 279 280 280 x Contents 9.4 9.5 9.6 9.7 9.8 A Introduction Gamma Delta Theta Rho Vega The Normal (Gaussian) Distribution The Lognormal Distribution The Student’s t Distribution The General Error Distribution D.4.1 Value of λ for Variance hi D.4.2 The Kurtosis D.4.3 The Distribution for Shape Parameter, a 325 327 328 330 330 331 332 Mathematical Reference E.1 E.2 E.3 E.4 F 315 315 317 317 319 321 321 323 Statistical Distribution Functions D.1 D.2 D.3 D.4 E 307 310 Standard Statistical Results C.1 The Law of Large Numbers C.2 The Central Limit Theorem C.3 The Variance and Covariance of Random Variables C.3.1 Variance C.3.2 Covariance C.3.3 Covariance Matrix C.4 Conditional Mean and Covariance of Normal Distributions C.5 Moment Generating Functions D 301 302 303 303 304 305 Barrier Option Integrals B.1 The Down and Out Call B.2 The Up and Out Call C 281 284 286 289 290 292 296 297 The Greeks for Vanilla European Options A.1 A.2 A.3 A.4 A.5 A.6 B 9.3.2 Amsterdam Stock Exchange 9.3.3 Other Early Financial Centres Tulip Mania Early Use of Derivatives in the USA Securitisation and Structured Products Collateralised Debt Obligations The 2008 Financial Crisis 9.8.1 The Collapse of AIG Standard Integrals Gamma Function The Cumulative Normal Distribution Function Arithmetic and Geometric Progressions Black–Scholes Finite-Difference Schemes 333 333 334 335 Brownian Motion: More Results 348 APPENDIX | H It can be seen from Section 2.4 that the transformation between measures P and Q can be accomplished using k = ν/σ, and that the associated Radon–Nikodym derivative is dQ = exp kWtP − k t dP ν P ν2t = exp W − σ t σ ν2t ν , (H.5.5) = exp X t − σ σ where we have use that fact that under measure probability measure P, we can write WtP = X t /σ Now ¯ P m tX ≤ b, X¯ t ≥ x = E Q I m X¯ ≤b I{ X¯ t ≥x } , (H.5.6) t where I{condition} is an indicator function which takes unit value when condition ¯ is satisfied and zero otherwise – for example, I m X¯ ≤b is one when m tX ≤ b and t ¯ zero when m tX > b However, see for example Baxter and Rennie (1996), we have E Q I ¯ m tX ≤b So substituting for I { X¯ t ≥x } = E P dQ I{ m X ≤b } I{X t ≥x} t dP (H.5.7) dQ from equation (H.5.5) gives dP ν2t νX t E Q I m X¯ ≤b I{ X¯ t ≥x } = E P I{ m X ≤b } I{X t ≥x} exp − t t σ2 2σ (H.5.8) Expressed in terms of the reflected Brownian motion, X tR = 2b − X t , equation (H.5.8) can be written ν(2b − X tR ) ν t ¯ − P m tX ≤ b, X¯ t ≥ x = E P I{2b−X R ≥x } exp t σ2 2σ νX tR 2νb P ν2t = exp E I{2b−X t ≥x} exp − − σ σ 2σ (H.5.9) Since I{2b−X R > x } = I{−2b+X R