Interest Rate Modeling: Theory and Practice

346 Pages·2009·10.345 MB·English

by  Lixin Wu

Interest Rate
Modeling
Theory and Practice
© 2009 by Taylor & Francis Group, LLC
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CHAPMAN & HALUCRC
Financial Mathematics Series
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Published Titles
American-Style Derivatives: Valuation and Computation, Jerome Detemple
Analysis, Geometry, and Modeling in Finance: Advanced Meiliods in Option Pricing,
Pierre Henry-Labordere
Credit Risk: Models, Derivatives, and Management, Niklas Wagner
Engineering BGM, Alan Brace
Financial Modelling wiili Jump Processes, Rama Cont and Peter Tankov
Interest Rate Modeling: Theory and Practice, Lixin lfu
An Introduction to Credit Risk Modeling, Christian Bluhm, Ludger Overbeck, and
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Introduction to Stochastic Calculus Applied to Finance, Second Edition,
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Portfolio Optimization and Performance Analysis, Jean-Luc Prigent
Quantitative Fund Management, M. A. H. Dempster, Georg Pflug, and Gautam Mitra
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and
Ludger Overbeck
Understandlltg Risk: The Theory and Practice of Financial Risk Management,
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Chapman &H all/CRC FINANCIAL MATHEMATICS SERIES
Interest Rate
Modeling
Theory and Practice
Lixin Wu

eRe Press Is an ImprInt of the
Taylor &: Francs Group, an Informa business
A CHAPMAN & HALL BOOK
© 2009 by Taylor & Francis Group, LLC
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© 2009 by Taylor & Francis Group, LLC
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© 2009 by Taylor & Francis Group, LLC
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Contents
Preface xiii
Acknowledgments xvii
Author xix
CHAPTER 1 The Basics of Stochastic Calculus 1
1.1 BROWNIAN MOTION 2
1.1.1 SimpleRandomWalks 2
1.1.2 BrownianMotion 4
1.1.3 AdaptiveandNon-AdaptiveFunctions 6
1.2 STOCHASTIC INTEGRALS 8
1.2.1 EvaluationofStochasticIntegrals 11
1.3 STOCHASTIC DIFFERENTIALSAND ITO’S LEMMA 13
1.4 MULTI-FACTOR EXTENSIONS 18
1.4.1 Multi-FactorIto’sProcess 19
1.4.2 Ito’sLemma 20
vii
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viii (cid:2) Contents
1.4.3 CorrelatedBrownianMotions 20
1.4.4 TheMulti-FactorLognormalModel 21
1.5 MARTINGALES 22
CHAPTER 2 The Martingale RepresentationTheorem 27
2.1 CHANGING MEASURESWITH BINOMIAL MODELS 28
2.1.1 AMotivatingExample 28
2.1.2 BinomialTreesandPathProbabilities 30
2.2 CHANGE OF MEASURES UNDER BROWNIAN
FILTRATION 34
2.2.1 TheRadon–NikodymDerivativeofaBrownianPath 34
2.2.2 TheCMGTheorem 37
2.3THE MARTINGALE REPRESENTATIONTHEOREM 38
2.4 A COMPLETE MARKETWITHTWO SECURITIES 39
2.5 REPLICATINGAND PRICING OF CONTINGENT
CLAIMS 40
2.6 MULTI-FACTOR EXTENSIONS 43
2.7 A COMPLETE MARKETWITH MULTIPLE SECURITIES 44
2.7.1 ExistenceofaMartingaleMeasure 44
2.7.2 PricingContingentClaims 47
2.8THE BLACK–SCHOLES FORMULA 48
2.9 NOTES 51
CHAPTER 3 Interest Rates and Bonds 59
3.1 INTEREST RATESAND FIXED-INCOME INSTRUMENTS 60
3.1.1 ShortRateandMoneyMarketAccounts 60
3.1.2 TermRatesandCertificatesofDeposit 61
3.1.3 BondsandBondMarkets 62
3.1.4 QuotationandInterestAccrual 64
3.2YIELDS 66
3.2.1 YieldtoMaturity 66
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3.2.2 ParBonds,ParYields,andtheParYieldCurve 69
3.2.3 YieldCurvesforU.S.Treasuries 69
3.3 ZERO-COUPON BONDSAND ZERO-COUPON
YIELDS 70
3.3.1 Zero-CouponBonds 70
3.3.2 BootstrappingtheZero-CouponYields 72
3.3.2.1 FutureValueandPresentValue 73
3.4 FORWARD RATESAND FORWARD-RATE
AGREEMENTS 73
3.5YIELD-BASED BOND RISK MANAGEMENT 75
3.5.1 DurationandConvexity 75
3.5.2 PortfolioRiskManagement 78
CHAPTER 4 The Heath–Jarrow–Morton Model 81
4.1 LOGNORMAL MODEL:THE STARTING POINT 83
4.2THE HJM MODEL 86
4.3 SPECIAL CASES OFTHE HJM MODEL 89
4.3.1 TheHo–LeeModel 90
4.3.2 TheHull–White(orExtendedVasicek)Model 91
4.4 ESTIMATINGTHE HJM MODEL FROMYIELD DATA 94
4.4.1 FromaYieldCurvetoaForward-RateCurve 94

4.4.2 PrincipalComponentAnalysis 99
4.5 A CASE STUDYWITHATWO-FACTOR MODEL 105
4.6 MONTE CARLO IMPLEMENTATIONS 107
4.7 FORWARD PRICES 110
4.8 FORWARD MEASURE 113
4.9 BLACK’S FORMULA FOR CALLAND PUT OPTIONS 116
4.9.1 EquityOptionsundertheHull–WhiteModel 118
4.9.2 OptionsonCouponBonds 122
4.10 NUMERAIRESAND CHANGES OF MEASURE 125
4.11 NOTES 127
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CHAPTER 5 Short-Rate Models and Lattice Implementation 133
5.1 FROM SHORT-RATE MODELSTO FORWARD-RATE
MODELS 134
5.2 GENERAL MARKOVIAN MODELS 137
5.2.1 One-FactorModels 144
5.2.2 MonteCarloSimulationsforOptionsPricing 146
5.3 BINOMIALTREES OF INTEREST RATES 147
5.3.1 ABinomialTreefortheHo–LeeModel 148
5.3.2 Arrow–DebreuPrices 149
5.3.3 ACalibratedTreefortheHo–LeeModel 152
5.4 A GENERALTREE-BUILDING PROCEDURE 156
5.4.1 ATruncatedTreefortheHull–WhiteModel 156
5.4.2 TrinomialTreeswithAdaptiveTimeSteps 162
5.4.3 TheBlack–KarasinskiModel 163
CHAPTER 6 The LIBOR Market Model 167
6.1 LIBOR MARKET INSTRUMENTS 167
6.1.1 LIBORRates 168
6.1.2 Forward-RateAgreements 169
6.1.3 RepurchasingAgreement 171
6.1.4 EurodollarFutures 171
6.1.5 Floating-RateNotes 172
6.1.6 Swaps 174
6.1.7 Caps 177
6.1.8 Swaptions 178
6.1.9 BermudanSwaptions 179
6.1.10 LIBORExotics 179
6.2THE LIBOR MARKET MODEL 182
6.3 PRICING OF CAPSAND FLOORS 187
6.4 PRICING OF SWAPTIONS 188
6.5 SPECIFICATIONS OFTHE LIBOR MARKET MODEL 196
6.6 MONTE CARLO SIMULATION METHOD 200
6.6.1 TheLog–EulerScheme 200
© 2009 by Taylor & Francis Group, LLC
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