Interest Rate Modeling Theory and Practice Second Edition CHAPMAN & HALL/CRC Financial Mathematics Series Aims and scope: The field of financial mathematics forms an ever-expanding slice of the finan- cial 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 aca- demics 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 Equity-Linked Life Insurance Partial Hedging Methods Alexander Melnikov, Amir Nosrati 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, Second Edition Lixin Wu For more information about this series please visit: https://www.crcpress.com/Chapman-and- HallCRC-Financial-Mathematics-Series/book-series/CHFINANCMTH Interest Rate Modeling Theory and Practice Second Edition Lixin Wu CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 (cid:13)c 2019byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-0-8153-7891-4(Hardback) Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Rea- sonable 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 conse- quences of their use. 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For organizations that have been granted a photocopy license by the CCC, a separate system ofpaymenthasbeenarranged. Trademark Notice:Productorcorporatenamesmaybetrademarksorregisteredtrade- marks,andareusedonlyforidentificationandexplanationwithoutintenttoinfringe. Library of Congress Cataloging-in-Publication Data Names:Wu,Lixin,1961-author. Title:Interestratemodeling:theoryandpractice/LixinWu. Description:2ndedition.|BocaRaton,Florida:CRCPress,[2019]| Includesbibliographicalreferencesandindex. Identifiers:LCCN2018050904|ISBN9780815378914(hardback:alk. paper)|ISBN9781351227421(ebook:alk.paper) Subjects:LCSH:Interestrates--Mathematicalmodels.|Interestrate futures--Mathematicalmodels. Classification:LCCHG6024.5.W822019|DDC332.801/5195--dc23 LCrecordavailableathttps://lccn.loc.gov/2018050904 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my parents, To Molly, Dorothy and Derek Contents Preface to the First Edition xv Preface to the Second Edition xix Acknowledgments to the Second Edition xxi Author xxiii 1 The Basics of Stochastic Calculus 1 1.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Simple Random Walks . . . . . . . . . . . . . . . . . . 2 1.1.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Adaptive and Non-Adaptive Functions . . . . . . . . . 6 1.2 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Evaluation of Stochastic Integrals. . . . . . . . . . . . 10 1.3 Stochastic Differentials and Ito’s Lemma . . . . . . . . . . . 11 1.4 Multi-Factor Extensions . . . . . . . . . . . . . . . . . . . . . 16 1.4.1 Multi-Factor Ito’s Process . . . . . . . . . . . . . . . . 16 1.4.2 Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.3 Correlated Brownian Motions . . . . . . . . . . . . . . 17 1.4.4 The Multi-Factor Lognormal Model . . . . . . . . . . 18 1.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 The Martingale Representation Theorem 23 2.1 Changing Measures with Binomial Models . . . . . . . . . . 23 2.1.1 A Motivating Example . . . . . . . . . . . . . . . . . . 23 2.1.2 Binomial Trees and Path Probabilities . . . . . . . . . 26 2.2 Change of Measures under Brownian Filtration . . . . . . . . 29 2.2.1 The Radon–Nikodym Derivative of a Brownian Path . 29 2.2.2 The CMG Theorem . . . . . . . . . . . . . . . . . . . 31 2.3 The Martingale Representation Theorem . . . . . . . . . . . 32 2.4 A Complete Market with Two Securities . . . . . . . . . . . 33 2.5 Replicating and Pricing of Contingent Claims . . . . . . . . 34 2.6 Multi-Factor Extensions . . . . . . . . . . . . . . . . . . . . . 36 vii viii Contents 2.7 A Complete Market with Multiple Securities . . . . . . . . . 37 2.7.1 Existence of a Martingale Measure . . . . . . . . . . . 38 2.7.2 Pricing Contingent Claims. . . . . . . . . . . . . . . . 40 2.8 The Black–Scholes Formula . . . . . . . . . . . . . . . . . . . 41 2.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3 Interest Rates and Bonds 51 3.1 Interest Rates and Fixed-Income Instruments . . . . . . . . . 51 3.1.1 Short Rate and Money Market Accounts . . . . . . . . 51 3.1.2 Term Rates and Certificates of Deposit . . . . . . . . 52 3.1.3 Bonds and Bond Markets . . . . . . . . . . . . . . . . 53 3.1.4 Quotation and Interest Accrual . . . . . . . . . . . . . 55 3.2 Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.1 Yield to Maturity . . . . . . . . . . . . . . . . . . . . 57 3.2.2 Par Bonds, Par Yields, and the Par Yield Curve . . . 59 3.2.3 Yield Curves for U.S. Treasuries . . . . . . . . . . . . 60 3.3 Zero-Coupon Bonds and Zero-Coupon Yields . . . . . . . . . 61 3.3.1 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . 61 3.3.2 Bootstrapping the Zero-Coupon Yields . . . . . . . . . 62 3.3.2.1 Future Value and Present Value . . . . . . . 63 3.4 Forward Rates and Forward-Rate Agreements . . . . . . . . 64 3.5 Yield-Based Bond Risk Management . . . . . . . . . . . . . 65 3.5.1 Duration and Convexity . . . . . . . . . . . . . . . . . 65 3.5.2 Portfolio Risk Management . . . . . . . . . . . . . . . 67 4 The Heath–Jarrow–Morton Model 71 4.1 Lognormal Model: The Starting Point . . . . . . . . . . . . . 72 4.2 The HJM Model . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 Special Cases of the HJM Model . . . . . . . . . . . . . . . . 78 4.3.1 The Ho–Lee Model . . . . . . . . . . . . . . . . . . . . 78 4.3.2 The Hull–White (or Extended Vasicek) Model . . . . 79 4.4 Estimating the HJM Model from Yield Data . . . . . . . . . 82 4.4.1 From a Yield Curve to a Forward-Rate Curve . . . . . 82 4.4.2 Principal Component Analysis . . . . . . . . . . . . . 87 4.5 A Case Study with a Two-Factor Model . . . . . . . . . . . . 92 4.6 Monte Carlo Implementations . . . . . . . . . . . . . . . . . 93 4.7 Forward Prices . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.8 Forward Measure . . . . . . . . . . . . . . . . . . . . . . . . 99 4.9 Black’s Formula for Call and Put Options . . . . . . . . . . . 102 4.9.1 Equity Options under the Hull–White Model . . . . . 103 4.9.2 Options on Coupon Bonds. . . . . . . . . . . . . . . . 106 4.10 Numeraires and Changes of Measure . . . . . . . . . . . . . . 109 Contents ix 4.11 Linear Gaussian Models . . . . . . . . . . . . . . . . . . . . . 110 4.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 Short-Rate Models and Lattice Implementation 119 5.1 From Short-Rate Models to Forward-Rate Models . . . . . . 120 5.2 General Markovian Models . . . . . . . . . . . . . . . . . . . 122 5.2.1 One-Factor Models . . . . . . . . . . . . . . . . . . . . 128 5.2.2 Monte Carlo Simulations for Options Pricing . . . . . 130 5.3 Binomial Trees of Interest Rates . . . . . . . . . . . . . . . . 131 5.3.1 A Binomial Tree for the Ho–Lee Model . . . . . . . . 132 5.3.2 Arrow–Debreu Prices . . . . . . . . . . . . . . . . . . 133 5.3.3 A Calibrated Tree for the Ho–Lee Model. . . . . . . . 135 5.4 A General Tree-Building Procedure . . . . . . . . . . . . . . 138 5.4.1 A Truncated Tree for the Hull–White Model . . . . . 139 5.4.2 Trinomial Trees with Adaptive Time Steps . . . . . . 144 5.4.3 The Black–Karasinski Model . . . . . . . . . . . . . . 145 6 The LIBOR Market Model 149 6.1 LIBOR Market Instruments . . . . . . . . . . . . . . . . . . 149 6.1.1 LIBOR Rates . . . . . . . . . . . . . . . . . . . . . . . 150 6.1.2 Forward-Rate Agreements . . . . . . . . . . . . . . . . 150 6.1.3 Repurchasing Agreement . . . . . . . . . . . . . . . . 152 6.1.4 Eurodollar Futures . . . . . . . . . . . . . . . . . . . . 152 6.1.5 Floating-Rate Notes . . . . . . . . . . . . . . . . . . . 154 6.1.6 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.1.7 Caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.1.8 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.1.9 Bermudan Swaptions . . . . . . . . . . . . . . . . . . . 159 6.1.10 LIBOR Exotics . . . . . . . . . . . . . . . . . . . . . . 160 6.2 The LIBOR Market Model . . . . . . . . . . . . . . . . . . . 162 6.3 Pricing of Caps and Floors . . . . . . . . . . . . . . . . . . . 167 6.4 Pricing of Swaptions . . . . . . . . . . . . . . . . . . . . . . . 168 6.5 Specifications of the LIBOR Market Model . . . . . . . . . . 175 6.6 Monte Carlo Simulation Method . . . . . . . . . . . . . . . . 178 6.6.1 The Log–Euler Scheme . . . . . . . . . . . . . . . . . 178 6.6.2 Calculation of the Greeks . . . . . . . . . . . . . . . . 179 6.6.3 Early Exercise . . . . . . . . . . . . . . . . . . . . . . 180 6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7 Calibration of LIBOR Market Model 189 7.1 Implied Cap and Caplet Volatilities . . . . . . . . . . . . . . 190 7.2 Calibrating the LIBOR Market Model to Caps . . . . . . . . 192