Interest Rate Modeling

Transcription

1 Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Interest Rate Modeling Theory and Practice Lixin Wu CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business A CHAPMAN & HALL BOOK

2 Contents Preface > Acknowledgments Author xiii xvii xix CHAPTER 1 The Basics of Stochastic Calculus BROWNIAN MOTION Simple Random Walks Brownian Motion Adaptive and Non-Adaptive Functions STOCHASTIC INTEGRALS Evaluation of Stochastic Integrals STOCHASTIC DIFFERENTIALS AND ITO'S LEMMA MULTI-FACTOR EXTENSIONS Multi-Factor Ito's Process Ito's Lemma 20 Vll

3 viii Contents Correlated Brownian Motions The Multi-Factor Lognormal Model MARTINGALES 22 CHAPTER 2 The Martingale Representation Theorem CHANGING MEASURES WITH BINOMIAL MODELS A Motivating Example - s Binomial Trees and Path Probabilities CHANGE OF MEASURES UNDER BROWNIAN FILTRATION The Radon-Nikodym Derivative of a Brownian Path The CMG Theorem THE MARTINGALE REPRESENTATION THEOREM A COMPLETE MARKET WITH TWO SECURITIES REPLICATING AND PRICING OF CONTINGENT CLAIMS MULTI-FACTOR EXTENSIONS A COMPLETE MARKET WITH MULTIPLE SECURITIES Existence of a Martingale Measure Pricing Contingent Claims THE BLACK-SCHOLES FORMULA NOTES 51 CHAPTER 3 Interest Rates and Bonds INTEREST RATES AND FIXED-INCOME INSTRUMENTS Short Rate and Money Market Accounts Term Rates and Certificates of Deposit Bonds and Bond Markets Quotation and Interest Accrual YIELDS Yield to Maturity 66

4 Contents ix Par Bonds, Par Yields, and the Par Yield Curve Yield Curves for U.S. Treasuries ZERO-COUPON BONDS AND ZERO-COUPON YIELDS Zero-Coupon Bonds Bootstrapping the Zero-Coupon Yields Future Value and Present Value FORWARD RATES AND FORWARD-RATE AGREEMENTS YIELD-BASED BOND RISK MANAGEMENT Duration and Convexity Portfolio Risk Management 78 CHAPTER 4 The Heath-Jarrow-Morton Model LOGNORMAL MODEL: THE STARTING POINT THE HJM MODEL SPECIAL CASES OF THE HJM MODEL The Ho-Lee Model ' The Hull-White (or Extended Vasicek) Model ESTIMATING THE HJM MODEL FROM YIELD DATA From a Yield Curve to a Forward-Rate Curve Principal Component Analysis A CASE STUDY WITH A TWO-FACTOR MODEL MONTE CARLO IMPLEMENTATIONS FORWARD PRICES FORWARD MEASURE BLACK'S FORMULA FOR CALL AND PUT OPTIONS Equity Options under the Hull-White Model Options on Coupon Bonds NUMERAIRES AND CHANGES OF MEASURE NOTES 127

5 x Contents CHAPTER 5 Short-Rate Models and Lattice Implementation FROM SHORT-RATE MODELS TO FORWARD-RATE MODELS GENERAL MARKOVIAN MODELS One-Factor Models Monte Carlo Simulations for Options Pricing BINOMIAL TREES OF INTEREST RATES A Binomial Tree for the Ho-Lee Model Arrow-Debreu Prices A Calibrated Tree for the Ho-Lee Model A GENERAL TREE-BUILDING PROCEDURE A Truncated Tree for the Hull-White Model Trinomial Trees with Adaptive Time Steps The Black-Karasinski Model 163 CHAPTER 6 The LIBOR Market Model LIBOR MARKET INSTRUMENTS LIBOR Rates Forward-Rate Agreements Repurchasing Agreement Eurodollar Futures Floating-Rate Notes Swaps Caps Swaptions Bermudan Swaptions LIBOR Exotics THE LIBOR MARKET MODEL PRICING OF CAPS AND FLOORS PRICING OF SWAPTIONS SPECIFICATIONS OF THE LIBOR MARKET MODEL MONTE CARLO SIMULATION METHOD The Log-Euler Scheme 200

6 Contents XI Calculation of the Greeks Early Exercise CHAPTER 7 Calibration of LIBOR Market Model IMPLIED CAP AND CAPLET VOLATILITIES CALIBRATING THE LIBOR MARKET MODEL TO CAPS CALIBRATION TO CAPS, SWAPTIQNS, AND INPUT CORRELATIONS " CALIBRATION METHODOLOGIES Rank-Reduction Algorithm The Eigenvalue Problem for Calibrating s to Input Prices SENSITIVITY WITH RESPECT TO THE INPUT PRICES NOTES 253 CHAPTER 8 Volatility and Correlation Adjustments ADJUSTMENT DUE TO CORRELATIONS Futures Price versus Forward Price Convexity Adjustment for LIBOR Rates Convexity Adjustment under the Ho-Lee Model An Example of Arbitrage ADJUSTMENT DUE TO CONVEXITY Payment in Arrears versus Payment in Advance Geometric Explanation for Convexity Adjustment General Theory of Convexity Adjustment Convexity Adjustment for CMS and CMT Swaps TIMING ADJUSTMENT QUANTO DERIVATIVES NOTES 284

7 xii Contents CHAPTER 9 AffineTerm Structure Models AN EXPOSITION WITH ONE-FACTOR MODELS ANALYTICAL SOLUTION OF RICCARTI EQUATIONS PRICING OPTIONS ON COUPON BONDS DISTRIBUTIONAL PROPERTIES OF SQUARE-ROOT PROCESSES MULTI-FACTOR MODELS Admissible ATSMs Three-Factor ATSMs SWAPTION PRICING UNDER ATSMs NOTES 315 References 319 Index 327