Stochastic Finance

Introduction Assets, Portfolios and Arbitrage Definitions and Formalism Portfolio Allocation and Short-Selling Arbitrage Risk-Neutral Measures Hedging of Contingent Claims Market Completeness Example Exercises Discrete-Time Model Stochastic Processes Portfolio Strategies Arbitrage Contingent Claims Martingales and Conditional Expectation Risk-Neutral Probability Measures Market Completeness Cox{Ross{Rubinstein (CRR) Market Model Exercises Pricing and Hedging in Discrete Time Pricing of Contingent Claims Hedging of Contingent Claims - Backward Induction Pricing of Vanilla Options in the CRR Model Hedging of Vanilla Options in the CRR model Hedging of Exotic Options in the CRR Model Convergence of the CRR Model Exercises Brownian Motion and Stochastic Calculus Brownian Motion Wiener Stochastic Integral Ito Stochastic Integral Deterministic Calculus Stochastic Calculus Geometric Brownian Motion Stochastic Differential Equations Exercises The Black-Scholes PDE Continuous-Time Market Model Self-Financing Portfolio Strategies Arbitrage and Risk-Neutral Measures Market Completeness Black-Scholes PDE The Heat Equation Solution of the Black-Scholes PDE Exercises Martingale Approach to Pricing and Hedging Martingale Property of the Ito Integral Risk-Neutral Measures Girsanov Theorem and Change of Measure Pricing by the Martingale Method Hedging Strategies Exercises Estimation of Volatility Historical Volatility Implied Volatility Black-Scholes Formula vs. Market Data Local Volatility Exotic Options Generalities Reexion Principle Barrier Options Lookback Options Asian Options Exercises Contents vii American Options Filtrations and Information Flow Martingales, Submartingales, and Supermartingales Stopping Times Perpetual American Options Finite Expiration American Options Exercises Change of Numeraire and Forward Measures Notion of Numeraire Change of Numeraire Foreign Exchange Pricing of Exchange Options Self-Financing Hedging by Change of Numeraire Exercises Forward Rate Modeling Short-Term Models Zero-Coupon Bonds Forward Rates HJM Model Forward Vasicek Rates Modeling Issues BGM Model Exercises Pricing of Interest Rate Derivatives Forward Measures and Tenor Structure Bond Options Caplet Pricing Forward Swap Measures Swaption Pricing on the LIBOR Exercises Default Risk in Bond Markets Survival Probabilities and Failure Rate Stochastic Default Defaultable Bonds Credit Default Swaps Exercises Stochastic Calculus for Jump Processes Poisson Process Compound Poisson Processes Stochastic Integrals with Jumps Ito Formula with Jumps Stochastic Differential Equations with Jumps Girsanov Theorem for Jump Processes Exercises Pricing and Hedging in Jump Models Risk-Neutral Measures Pricing in Jump Models Black-Scholes PDE with Jumps Exponential Models Self-Financing Hedging with Jumps Exercises Basic Numerical Methods The Heat Equation Black-Scholes PDE Euler Discretization Milshtein Discretization Appendix: Background on Probability Theory Probability Spaces and Events Probability Measures Conditional Probabilities and Independence Random Variables Probability Distributions Expectation of a Random Variable Conditional Expectation Moment Generating Functions Exercises Bibliography Index