Interest rate modeling: theory and practice

519 Pages·2019·7.95 MB·English

by  Wu, Lixin

Interest Rate
Modeling
Theory and Practice
Second Edition
CHAPMAN & HALL/CRC
Financial Mathematics Series
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Equity-Linked Life Insurance
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An Introduction to Computational Risk Management of Equity-Linked
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Interest Rate Modeling
Theory and Practice, Second Edition
Lixin Wu
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Interest Rate
Modeling
Theory and Practice
Second Edition
Lixin Wu
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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
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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