MATHEMATICAL FINANCE Theory, Modeling, Implementation Christian Fries University of Frankfurt Department of Mathematics Frankfurt, Germany BICENTENNIAL BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
MATHEMATICAL FINANCE Theory, Modeling, Implementation Christian Fries University of Frankfurt Department of Mathematics Frankfurt, Germany This Page Intentionally Left Blank BICENTENNIAL BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
MATHEMATICAL FINANCE
THE WILEY BICENTENNIAL-KNOWLEDGE FOR GENERATIONS Gach generation has its unique needs and aspirations. When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future. Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born. Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how. For 200 years, Wiley has been an integral part of each generation s journey, enabling the flow of information and understanding necessary to meet their needs and fulfill their aspirations. Today, bold new technologies are changing the way we live and learn. Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities. Generations come and go, but you can always count on Wiley to provide you the knowledge you need, when and where you need it! PRESIDENT AND CHIEF EXECUTIVE OFFICER CHAIRMAN OF THE BOARD
MATHEMATICAL FINANCE Theory, Modeling, Implementation Christian Fries University of Frankfurt Department of Mathematics Frankfurt, Germany BICENTENNIAL BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
Copyright 0 2007 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., I 1 1 River Street, Hoboken, NJ 07030, (201) 748-601 I, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this hook, they make no representations or warranties with respect to the accuracy or completeness of the contents of this hook and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall he liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic format. For information ahout Wiley products, visit our web site at www.wiley.com. Wiley Bicentennial Logo: Richard J. Pacific0 Library of Congress Cataloging-in-Publication Data: Mathematical finance : theory, modeling, implementation / Christian Fries p. cm. Published simultaneously in Canada. Includes bihliographical references and index. ISBN 978-0-470-04722-4 (cloth : alk. paper) 1. Derivative securities-prices-mathematical models. 2 Securities-Mathematical models. 3. Investments-Mathematical models. I. Title. HG6024.A3F75 2007 332.601 5195-dc22 2007011325 Printed in the United States of America. 1098765432 I
Nowadays people know the price of everything and the value of nothing. Oscar Wilde The Picture of Dorian Gray [38] Typeset by the author using TeXShop for Mac 0s XTM. Drawings by the author using OmniGraffle for Mac 0s XTM. Charts created using JavaTM code by the author. Version 1.5.12. Build 20070702. V
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Picture Credits All figures are @ copyright Christian Fries except the special section icons (see Section 1.3.3) licensed through istockphoto.com. Note This book is also available in German. See http://www.christian-fries.de/finmath/book vii
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Acknowledgment I am grateful to Andreas Bahmer, Hans-Josef Beauvisage, Michael Belledin, Dr. Urs Braun, Oliver Dauben, Peter Dellbrugger, Dr. Jorg Dickersbach, Dr. Holger Dietz, Sinan Dikmen, Dr. Dirk Ebmeyer, Fabian Eckstadt, Dr. Lydia Fechner, Christian Ferber, Frank Genheimer, Dr. Gido Herbers, Dr. Ansgar Jiingel, Dr. Jorg Kampen, Dr. Christoph Kiehn, Dr. Christoph Kiihn, Dr. Jiirgen Linde, Markus Meister, Dr. Sean Matthews, Dilys and Bill McCann, Michael Paulsen, Matthias Peter, Dr. Erwin Pier-Ribbert, Rosemarie Philippi, Dr. Radu Tunam, Frank Ritz, Marius Rott, Oliver Schluter, Thomas Schwiertz, Arndt Unterweger, Benedikt Wilbertz, Andre Woker, Polina Zeydis, and Jorg Zinnegger. Their support and their feedback as well as the stimulating discussions we had contributed significantly to this work. I am most grateful to my wife and my family. I thank them for their continuous support and generous tolerance. ix
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Contents 1 Introduction 1 1.1 Theory. Modeling. and Implementation... 1 1.2 Interest Rate Models and Interest Rate Derivatives... 1 1.3 AboutThisBook... 3 1.3.1 How to Read This Book... 3 1.3.2 Abridged Versions... 3 1.3.3 Special Sections... 4 1.3.4 Notation... 4 1.3.5 Feedback... 5 1.3.6 Resources... 5 I Foundations 7 2 Foundations 9 2.1 Probability Theory... 9 2.2 Stochastic Processes... 18 2.3 Filtration... 20 2.4 Brownian Motion... 22 2.5 Wiener Measure, Canonical Setup... 24 2.6 It8Calculus... 25 2.6.1 It8 Integral... 28 2.6.2 It6Process... 30 2.6.3 It6 Lemma and Product Rule... 32 2.7 Brownian Motion with Instantaneous Correlation... 36 xi
2.8 Martingales... 2.8.1 Martingale Representation Theorem... 2.9 Change of Measure... 2.10 Stochastic Integration... 2.1 1 Partial Differential Equations (PDEs)... 2.1 1.1 Feynman-KaE Theorem... 2.12 List of Symbols... 3 Replication 3.1 Replication Strategies... 3.1.1 Introduction... 3.1.2 Replication in a Discrete Model... 3.2 Foundations: Equivalent Martingale Measure... 3.2.1 Challenge and Solution Outline... 3.2.2 Steps toward the Universal Pricing Theorem... 3.3 Excursus: Relative Prices and Risk-Neutral Measures... 3.3.1 Why relative prices?... 3.3.2 Risk-Neutral Measure... 38 38 39 44 46 46 48 49 49 49 53 58 58 61 70 70 72 II First Applications 73 4 Pricing of a European Stock Option under the Black-Scholes Model 5 Excursus: The Density of the Underlying of a European Call Option 6 Excursus: Interpolation of European Option Prices 6.1 No-Arbitrage Conditions for Interpolated Prices... 6.2 Arbitrage Violations through Interpolation... 6.2.1 Example 1 : Interpolation of Four Prices... 6.2.2 Example 2: Interpolation of Two Prices... 6.3 Arbitrage-Free Interpolation of European Option Prices... 7 Hedging in Continuous and Discrete Time and the Greeks 7.1 Introduction... 7.2 Deriving the Replications Strategy from Pricing Theory... 7.2.1 Deriving the Replication Strategy under the Assumption of a Locally Riskless Product... 75 81 83 83 85 85 87 89 93 93 94 96 xii
7.2.2 Black-Scholes Differential Equation... 7.2.3 Derivative V(t) as a Function of Its Underlyings Si(r).. 7.2.4 Example: Replication Portfolio and PDE under a Black- Scholes Model... 7.3 Greeks... 7.3.1 Greeks of a European Call-Option under the Black- 7.4 Scholes Model... Hedging in Discrete Time: Delta and Delta-Gamma Hedging... 7.4.1 Delta Hedging... 7.4.2 Error Propagation... 7.4.3 Delta-Gamma Hedging... 7.4.4 Vega Hedging... 7.5 Hedging in Discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method)... 7.5.1 Minimizing the Residual Error at Maturity T... 7.5.2 Minimizing the Residual Error in Each Time Step... 97 97 99 102 103 103 105 106 109 113 113 115 117 111 Interest Rate Structures. Interest Rate Products. and Analytic Pricing Formulas 119 Motivation and Overview... 121 8 Interest Rate Structures 123 8.1 Introduction... 123 8.1.1 Fixing Times and Tenor Times... 124 8.2 Definitions... 124 8.3 Interest Rate Curve Bootstrapping... 130 8.4 Interpolation of Interest Rate Curves... 130 8.5 Implementation... 131 9 Simple Interest Rate Products 133 9.1 Interest Rate Products Part 1: Products without Optionality... 133 9.1.1 Fix. Floating. and Swap... 133 9.1.2 Money Market Account... 140 9.2 Interest Rate Products Part 2: Simple Options... 142 9.2.1 Cap. Floor. and Swaption... 142 9.2.2 Foreign Caplet. Quanto... 144 10 The Black Model for a Caplet 147... Xlll