Martingale Methods in Financial Modelling

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1 Marek Musiela Marek Rutkowski Martingale Methods in Financial Modelling Second Edition Springer

2 Table of Contents Preface to the First Edition Preface to the Second Edition V VII Part I. Spot and Futures Markets 1. An Introduction to Financial Derivatives Options Futures Contracts and Options Forward Contracts Call and Put Spot Options One-period Spot Market Replicating Portfolios Maxtingale Measure for a Spot Market Absence of Arbitrage Optimality of Replication Put Option Futures Call and Put Options Futures Contracts and Futures Prices One-period Futures Market Maxtingale Measure for a Futures Market Absence of Arbitrage One-period Spot/Futures Market Forward Contracts Forward Price Options of American Style Universal No-arbitrage Inequalities Discrete-time Security Markets The Cox-Ross-Rubinstein Model Binomial Lattice for the Stock Price Recursive Pricing Procedure CRR Option Pricing Formula 43

3 X Table of Contents 2.2 Martingale Properties of the CRR Model Martingale Measures Risk-neutral Valuation Formula The Black-Scholes Option Pricing Formula Valuation of American Options American Call Options American Put Options American Claim Options on a Dividend-paying Stock Finite Spot Markets Self-financing Trading Strategies Arbitrage Opportunities Arbitrage Price Risk-neutral Valuation Formula Price Systems Completeness of a Finite Market Change of a Numeraire Finite Futures Markets Self-financing Futures Strategies Martingale Measures for a Futures Market Risk-neutral Valuation Formula Futures Prices Versus Forward Prices Discrete-time Models with Infinite State Space Benchmark Models in Continuous Time The Black-Scholes Model Risk-free Bond Stock Price Self-financing Trading Strategies Martingale Measure for the Spot Market Black-Scholes Option Pricing Formula Case of Time-dependent Coefficients Merton's Model Put-Call Parity for Spot Options Black-Scholes PDE A Riskless Portfolio Method Black-Scholes Sensitivities Market Imperfections Numerical Methods A Dividend-paying Stock Case of a Constant Dividend Yield Case of Known Dividends Bachelier Model Bachelier Option Pricing Formula Bachelier's PDE 124

4 Table of Contents XI Bachelier Sensitivities Black Model Self-financing Futures Strategies Martingale Measure for the Futures Market Black's Futures Option Formula Options on Forward Contracts Forward and Futures Prices Robustness of the Black-Scholes Approach Uncertain Volatility European Call and Put Options Convex Path-independent European Claims General Path-independent European Claims Foreign Market Derivatives Cross-currency Market Model Domestic Martingale Measure Foreign Martingale Measure Foreign Stock Price Dynamics Currency Forward Contracts and Options Forward Exchange Rate Currency Option Valuation Formula Foreign Equity Forward Contracts Forward Price of a Foreign Stock Quanto Forward Contracts Foreign Market Futures Contracts Foreign Equity Options Options Struck in a Foreign Currency Options Struck in Domestic Currency Quanto Options Equity-linked Foreign Exchange Options American Options Valuation of American Claims American Call and Put Options Early Exercise Representation of an American Put Analytical Approach Approximations of the American Put Price Option on a Dividend-paying Stock Exotic Options Packages Forward-start Options Chooser Options Compound Options Digital Options,198

5 XII Table of Contents 6.6 Barrier Options Lookback Options Asian Options Basket Options Quantile Options Other Exotic Options Volatility Risk Implied Volatilities of Traded Options Historical Volatility Implied Volatility Implied Volatility Versus Historical Volatility Approximate Formulas Implied Volatility Surface Asymptotic Behavior of the Implied Volatility Marked-to-Market Models Vega Hedging Correlated Brownian Motions Forward-start Options Extensions of the Black-Scholes Model CEV Model Shifted Lognormal Models Local Volatility Models Implied Risk-Neutral Probability Law Local Volatility Mixture Models Advantages and Drawbacks of LV Models Stochastic Volatility Models PDE Approach Examples of SV Models Hüll and White Model Heston's Model SABR Model Dynamical Models of Volatility Surfaces Dynamics of the Local Volatility Surface Dynamics of the Implied Volatility Surface Alternative Approaches Modelling of Asset Returns Modelling of Volatility and Realized Variance Continuous-time Security Markets Standard Market Models Standard Spot Market Futures Market Choice of a Numeraire 291

6 Table of Contents XIII Existence of a Martingale Measure Fundamental Theorem of Asset Pricing Multidimensional Black-Scholes Model Market Completeness Variance-minimizing Hedging Risk-minimizing Hedging Market Imperfections 310 Part II. Fixed-income Markets 9. Interest Rates and Related Contracts Zero-coupon Bonds Term Structure of Interest Rates Forward Interest Rates Short-term Interest Rate Coupon-bearing Bonds Yield-to-Maturity Market Conventions Interest Rate Futures Treasury Bond Futures Bond Options Treasury Bill Futures Eurodollar Futures Interest Rate Swaps Forward Rate Agreements Stochastic Models of Bond Prices Arbitrage-free Family of Bond Prices Expectations Hypotheses Case of Ito Processes Market Price for Interest Rate Risk Forward Measure Approach Forward Price Forward Martingale Measure Forward Processes Choice of a Numeraire Short-Term Rate Models Single-factor Models Time-homogeneous Models Time-inhomogeneous Models Model Choice American Bond Options Options on Coupon-bearing Bonds Multi-factor Models 367

7 XIV Table of Contents State Variables Affine Models Yield Models Extended CIR Model Squared Bessel Process Model Construction Change of a Probability Measure Zero-coupon Bond Case of Constant Coefficients Case of Piecewise Constant Coefficients Dynamics of Zero-coupon Bond Transition Densities Bond Option Models of Instantaneous Forward Rates Heath-Jarrow-Morton Methodology Ho and Lee Model Heath-Jarrow-Morton Model Absence of Arbitrage Short-term Interest Rate Gaussian HJM Model Markovian Case European Spot Options Bond Options Stock Options Option on a Coupon-bearing Bond Pricing of General Contingent Claims Replication of Options Volatilities and Correlations Volatilities Correlations Futures Price Futures Options PDE Approach to Interest Rate Derivatives PDEs for Spot Derivatives PDEs for Futures Derivatives Recent Developments Market LIBOR Models Forward and Futures LIBORs One-period Swap Settled in Arrears One-period Swap Settled in Advance Eurodollar Futures LIBOR in the Gaussian HJM Model Interest Rate Caps and Floors 439

8 Table of Contents XV 12.3 Valuation in the Gaussian HJM Model Plain-vanilla Caps and Floors Exotic Caps Captions LIBOR Market Models Black's Formula for Caps Miltersen, Sandraann and Sondermann Approach Brace, Gatarek and Musiela Approach Musiela and Rutkowski Approach Jamshidian's Approach Properties of the Lognormal LIBOR Model Transition Density of the LIBOR Transition Density of the Forward Bond Price Valuation in the Lognormal LIBOR Model Pricing of Caps and Floors Hedging of Caps and Floors Valuation of European Claims Bond Options Extensions of the LLM Model Alternative Market Models Swaps and Swaptions Forward Swap Rates Swaptions Exotic Swap Derivatives Valuation in the Gaussian HJM Model Swaptions CMS Spread Options Yield Curve Swaps Co-terminal Swap Rates Jamshidian's Approach Valuation of Co-terminal Swaptions Hedging of Swaptions Bermudan Swaptions Co-initial Swap Rates Valuation of Co-initial Swaptions Valuation of Exotic Options Co-sliding Swap Rates Modelling of Co-sliding Swap Rates Valuation of Co-sliding Swaptions Swap Rate Model Versus LIBOR Model Swaptions in the LLM Model Caplets in the Co-terminal Swap Market Model Markov-functional Models Terminal Swap Rate Model 515

9 XVI Table of Contents Calibration of Markov-functional Models Flesaker and Hughston Approach Rational Lognormal Model Valuation of Caps and Swaptions Cross-currency Derivatives Arbitrage-free Cross-currency Markets Forward Price of a Foreign Asset Valuation of Foreign Contingent Claims Cross-currency Rates Gaussian Model Currency Options Foreign Equity Options Cross-currency Swaps Cross-currency Swaptions Basket Caps Model of Forward LIBOR Rates Quanto Cap Cross-currency Swap Concluding Remarks 561 Part III. APPENDICES A. Conditional Expectations 565 B. Itö Stochastic Calculus 569 B.l Itö Integral 569 B.2 Girsanov's Theorem 576 B.3 Ito-Tanaka-Meyer Formula 578 B.4 Laws of Certain Functionals of a Brownian Motion 579 References 583 Index 629

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