Risk-Neutral Valuation

Transcription

1 N.H. Bingham and Rüdiger Kiesel Risk-Neutral Valuation Pricing and Hedging of Financial Derivatives W) Springer

2 Contents 1. Derivative Background Financial Markets and Instruments Derivative Instruments Underlying securities Markets Types of Traders Modelling Assumptions Arbitrage Arbitrage Relationships Fundamental Determinants of Option Values The Put-Call Parity Arbitrage Bounds Single-Period Market Models 20 Exercises Probability Background Measure Integral Probability Equivalent Measures and Radon-Nikodym Derivatives Conditional Expectations Properties of Conditional Expectation Modes of Convergence Convolution and Characteristic Functions The Central Limit Theorem 60 Exercises Stochastic Processes in Discrete Time Information and Filtrations Discrete-Parameter Stochastic Processes Discrete-Parameter Martingales Definition and Simple Properties Martingale Convergence Doob Decomposition 73

3 xii Contents 3.4 Martingale Transforms Stopping Times and Optional Stopping The Snell Envelope 78 Exercises Mathematical Finance in Discrete Time The Model Existence of Equivalent Martingale Measures The No-Arbitrage Condition Risk-Neutral Pricing Complete Markets Risk-Neutral Valuation The Cox-Ross-Rubinstein Model Model Structure Risk-Neutral Pricing Hedging Comparison With the General Arbitrage Bounds Binomial Approximations Ill Model Structure The Black-Scholes Option Pricing Formula Further Limiting Models Multifactor Models Extended Binomial Model Multinomial Models Further Contingent Claim Valuation in Discrete Time American Options Barrier Options Lookback Options A Three-Period Example 128 Exercises Stochastic Processes in Continuous Time Filtrations; Finite-Dimensional Distributions Classes of Processes Brownian Motion Quadratic Variation of Brownian Motion Stochastic Integrals; Ito Calculus Itö's Lemma Geometric Brownian Motion Stochastic Differential Equations Stochastic Calculus for Black-Scholes Models Weak Convergence of Stochastic Processes The Spaces C d and D d Definition and Motivation Basic Theorems of Weak Convergence 164

4 Contents xiii Weak Convergence Results for Stochastic Integrals Exercises Mathematical Finance in Continuous Time Continuous-time Financial Market Models The Financial Market Model Equivalent Martingale Measures Risk-neutral Pricing Changes of Numeraire The Generalised Black-Scholes Model The Model Pricing and Hedging Contingent Claims The Greeks Volatility Further contingent claim valuation American Options Asian Options Barrier Options Lookback Options Binary Options Discrete- vs. Continuous-Time Models Convergence Reconsidered Finite Market Approximations Examples of Finite Market Approximations Further Applications Futures Markets Currency Markets 224 Exercises Incomplete Markets Pricing in Incomplete Markets A General Option Pricing Formula The Esscher Measure Hedging in Incomplete Markets Variance Minimising Hedging Risk-Minimising Hedging Stochastic Volatility Models Interest Rate Theory The Bond Market The Term Structure of Interest Rates Mathematical Modelling Bond Pricing, Short Rate Models The Term Structure Equation 255

5 xiv Contents Martingale Modelling Parameter Estimation Heath-Jarrow-Morton Methodology The Heath-Jarrow-Morton Model Class Forward Risk-Neutral Martingale Measures Completeness Pricing and Hedging Contingent Claims Short Rate Models Gaussian HJM Framework Swaps Caps 272 Exercises 274 A. Hubert Space 277 B. Projections and Conditional Expectations 279 C. The Separating Hyperplane Theorem 281 Bibliography 283 Index 293