Martingale Methods in Financial Modelling

Transcription

1 Stochastic Modelling and Applied Probability 36 Martingale Methods in Financial Modelling Bearbeitet von Marek Musiela, Marek Rutkowski 2nd ed Corr. 3rd printing Buch. xvi, 638 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 1268 g Wirtschaft > Unternehmensfinanzen > Finanzierung, Investition, Leasing Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 Contents Preface to the Second Edition Note on the Second Printing Preface to the First Edition v vii ix Part I Spot and Futures Markets 1 An Introduction to Financial Derivatives Options FuturesContractsandOptions ForwardContracts Call and Put Spot Options One-period Spot Market ReplicatingPortfolios Martingale Measure for a Spot Market Absence of Arbitrage Optimality of Replication Change of a Numeraire PutOption ForwardContracts ForwardPrice FuturesCallandPutOptions Futures Contracts and Futures Prices One-periodFuturesMarket Martingale Measure for a Futures Market Absence of Arbitrage One-period Spot/Futures Market OptionsofAmericanStyle Universal No-arbitrage Inequalities

3 xii Contents 2 Discrete-time Security Markets The Cox-Ross-Rubinstein Model Binomial Lattice for the Stock Price Recursive Pricing Procedure CRROptionPricingFormula Martingale Properties of the CRR Model MartingaleMeasures Risk-neutral Valuation Formula Change of a Numeraire The Black-Scholes Option Pricing Formula ValuationofAmericanOptions AmericanCallOptions AmericanPutOptions AmericanClaims Options on a Dividend-paying Stock Security Markets in Discrete Time Finite Spot Markets Self-financing Trading Strategies Replication and Arbitrage Opportunities ArbitragePrice Risk-neutral Valuation Formula Existence of a Martingale Measure Completeness of a Finite Market Separating Hyperplane Theorem Change of a Numeraire Discrete-time Models with Infinite State Space FiniteFuturesMarkets Self-financing Futures Strategies Martingale Measures for a Futures Market Risk-neutral Valuation Formula Futures Prices Versus Forward Prices American Contingent Claims Optimal Stopping Problems Valuation and Hedging of American Claims AmericanCallandPut Game Contingent Claims Dynkin Games Valuation and Hedging of Game Contingent Claims Benchmark Models in Continuous Time The Black-Scholes Model Risk-free Bond StockPrice Self-financing Trading Strategies Martingale Measure for the Black-Scholes Model

4 Contents xiii Black-Scholes Option Pricing Formula Case of Time-dependent Coefficients Merton s Model Put-Call Parity for Spot Options Black-Scholes PDE ARisklessPortfolioMethod Black-Scholes Sensitivities MarketImperfections 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 Bachelier Sensitivities Black Model Self-financing Futures Strategies Martingale Measure for the Futures Market Black s Futures Option Formula OptionsonForwardContracts ForwardandFuturesPrices 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 DomesticMartingaleMeasure ForeignMartingaleMeasure Foreign Stock Price Dynamics Currency Forward Contracts and Options Forward Exchange Rate CurrencyOptionValuationFormula ForeignEquityForwardContracts ForwardPriceofaForeignStock Quanto Forward Contracts ForeignMarketFuturesContracts ForeignEquityOptions Options Struck in a Foreign Currency Options Struck in Domestic Currency Quanto Options Equity-linked Foreign Exchange Options

5 xiv Contents 5 American Options ValuationofAmericanClaims AmericanCallandPutOptions Early Exercise Representation of an American Put Analytical Approach Approximations of the American Put Price Option on a Dividend-paying Stock Game Contingent Claims Exotic Options Packages Forward-startOptions Chooser Options Compound Options DigitalOptions BarrierOptions Lookback Options AsianOptions BasketOptions Quantile Options OtherExoticOptions Volatility Risk Implied Volatilities of Traded Options Historical Volatility Implied Volatility Implied Volatility Versus Historical Volatility ApproximateFormulas Implied Volatility Surface Asymptotic Behavior of the Implied Volatility Marked-to-Market Models Vega Hedging CorrelatedBrownianMotions Forward-startOptions 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

6 Contents xv Hull 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 FuturesMarket ChoiceofaNumeraire Existence of a Martingale Measure Fundamental Theorem of Asset Pricing Multidimensional Black-Scholes Model Market Completeness Variance-minimizing Hedging Risk-minimizing Hedging MarketImperfections Part II Fixed-income Markets 9 Interest Rates and Related Contracts Zero-coupon Bonds Term Structure of Interest Rates ForwardInterestRates Short-term Interest Rate Coupon-bearing Bonds Yield-to-Maturity MarketConventions InterestRateFutures TreasuryBondFutures BondOptions TreasuryBillFutures EurodollarFutures InterestRateSwaps ForwardRateAgreements Stochastic Models of Bond Prices Arbitrage-free Family of Bond Prices Expectations Hypotheses CaseofItôProcesses

7 xvi Contents MarketPriceforInterestRateRisk Forward Measure Approach ForwardPrice ForwardMartingaleMeasure ForwardProcesses ChoiceofaNumeraire Short-Term Rate Models Single-factor Models Time-homogeneous Models Time-inhomogeneous Models Model Choice AmericanBondOptions Options on Coupon-bearing Bonds Multi-factor Models StateVariables Affine Models Yield Models Extended CIR Model Squared Bessel Process Model Construction Change of a Probability Measure Zero-coupon Bond CaseofConstantCoefficients Case of Piecewise Constant Coefficients Dynamics of Zero-coupon Bond Transition Densities BondOption 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 MarkovianCase European Spot Options BondOptions StockOptions Option on a Coupon-bearing Bond Pricing of General Contingent Claims ReplicationofOptions Volatilities and Correlations Volatilities

8 Contents xvii Correlations FuturesPrice FuturesOptions PDE Approach to Interest Rate Derivatives PDEs for Spot Derivatives PDEsforFuturesDerivatives Recent Developments Market LIBOR Models ForwardandFuturesLIBORs One-period Swap Settled in Arrears One-period Swap Settled in Advance EurodollarFutures LIBOR in the Gaussian HJM Model InterestRateCapsandFloors Valuation in the Gaussian HJM Model Plain-vanilla Caps and Floors ExoticCaps Captions LIBOR Market Models Black s Formula for Caps Miltersen, Sandmann and Sondermann Approach Brace, G atarek and Musiela Approach Musiela and Rutkowski Approach SDEs for LIBORs under the Forward Measure Jamshidian s Approach Alternative Derivation of Jamshidian s SDE Properties of the Lognormal LIBOR Model Transition Density of the LIBOR Transition Density of the Forward Bond Price Valuation in the Lognormal LIBOR Model PricingofCapsandFloors Hedging of Caps and Floors Valuation of European Claims BondOptions Extensions of the LLM Model Alternative Market Models SwapsandSwaptions ForwardSwapRates Swaptions ExoticSwapDerivatives Valuation in the Gaussian HJM Model Swaptions CMSSpreadOptions

9 xviii Contents YieldCurveSwaps Co-terminalForwardSwapRates Jamshidian s Model Valuation of Co-terminal Swaptions Hedging of Swaptions Bermudan Swaptions Co-initial Forward Swap Rates Valuation of Co-initial Swaptions ValuationofExoticOptions Co-slidingForwardSwapRates Modelling of Co-sliding Swap Rates ValuationofCo-slidingSwaptions 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 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 ForwardPriceofaForeignAsset Valuation of Foreign Contingent Claims Cross-currencyRates Gaussian Model CurrencyOptions ForeignEquityOptions Cross-currencySwaps Cross-currencySwaptions BasketCaps Model of Forward LIBOR Rates Quanto Cap Cross-currencySwap Concluding Remarks Part III APPENDIX A An Overview of Itô Stochastic Calculus A.1 Conditional Expectation A.2 Filtrations and Adapted Processes A.3 Martingales

10 Contents xix A.4 Standard Brownian Motion A.5 Stopping Times and Martingales A.6 Itô Stochastic Integral A.7 Continuous Local Martingales A.8 Continuous Semimartingales A.9 Itô slemma A.10 Lévy s Characterization Theorem A.11 Martingale Representation Property A.12 Stochastic Differential Equations A.13 Stochastic Exponential A.14 Radon-Nikodým Density A.15 Girsanov s Theorem A.16MartingaleMeasures A.17Feynman-KacFormula A.18FirstPassageTimes References Index