Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives

ContentsPreface to the Second EditionPreface to the First Edition1. Derivative Background1.1 Financial Markets and Instruments1.1.1 Derivative Instruments1.1.2 Underlying Securities1.1.3 Markets1.1.4 Types of Traders1.1.5 Modeling Assumptions1.2 Arbitrage1.3 Arbitrage Relationships1.3.1 Fundamental Determinants of Option Values1.3.2 Arbitrage Bounds1.4 Single-period Market Models1.4.1 A Fundamental Example1.4.2 A Single-period Model1.4.3 A Few Financial-economic ConsiderationsExercises 2. Probability Background2.1 Measure2.2 Integral2.3 Probability2.4 Equivalent Measures and Radon-Nikodym Derivatives2.5 Conditional Expectation2.6 Modes of Convergence2.7 Convolution and Characteristic Functions2.8 The Central Limit Theorem2.9 Asset Return Distributions2.10 In.nite Divisibility and the L´evy-Khintchine Formula2.11 Elliptically Contoured Distributions2.12 Hyberbolic DistributionsExercises 3. Stochastic Processes in Discrete Time3.1 Information and Filtrations3.2 Discrete-parameter Stochastic Processes3.3 De.nition and Basic Properties of Martingales3.4 Martingale Transforms3.5 Stopping Times and Optional Stopping3.6 The Snell Envelope and Optimal Stopping3.7 Spaces of Martingales3.8 Markov ChainsExercises 4. Mathematical Finance in Discrete Time4.1 The Model4.2 Existence of Equivalent Martingale Measures4.2.1 The No-arbitrage Condition4.2.2 Risk-Neutral Pricing4.3 Complete Markets: Uniqueness of EMMs4.4 The Fundamental Theorem of Asset Pricing: Risk-Neutral Valuation4.5 The Cox-Ross-Rubinstein Model4.5.1 Model Structure4.5.2 Risk-neutral Pricing4.5.3 Hedging4.6 Binomial Approximations4.6.1 Model Structure4.6.2 The Black-Scholes Option Pricing Formula4.6.3 Further Limiting Models4.7 American Options4.7.1 Theory4.7.2 American Options in the CRR Model4.8 Further Contingent Claim Valuation in Discrete Time4.8.1 Barrier Options4.8.2 Lookback Options4.8.3 A Three-period Example4.9 Multifactor Models4.9.1 Extended Binomial Model4.9.2 Multinomial ModelsExercises 5. Stochastic Processes in Continuous Time5.1 Filtrations; Finite-dimensional Distributions5.2 Classes of Processes5.2.1 Martingales5.2.2 Gaussian Processes5.2.3 Markov Processes5.2.4 Diffusions5.3 Brownian Motion5.3.1 Definition and Existence5.3.2 Quadratic Variation of Brownian Motion5.3.3 Properties of Brownian Motion5.3.4 Brownian Motion in Stochastic Modeling5.4 Point Processes5.4.1 Exponential Distribution5.4.2 The Poisson Process5.4.3 Compound Poisson Processes5.4.4 Renewal Processes5.5 Levy Processes5.5.1 Distributions5.5.2 Levy Processes5.5.3 Levy Processes and the Levy-Khintchine Formula5.6 Stochastic Integrals; Ito Calculus5.6.1 Stochastic Integration5.6.2 Ito's Lemma5.6.3 Geometric Brownian Motion5.7 Stochastic Calculus for Black-Scholes Models5.8 Stochastic Differential Equations5.9 Likelihood Estimation for Diffusions5.10 Martingales, Local Martingales and Semi-martingales5.10.1 Definitions5.10.2 Semi-martingale Calculus5.10.3 Stochastic Exponentials5.10.4 Semi-martingale Characteristics5.11 Weak Convergence of Stochastic Processes5.11.1 The Spaces Cd and Dd5.11.2 Definition and Motivation5.11.3 Basic Theorems of Weak Convergence5.11.4 Weak Convergence Results for Stochastic IntegralsExercises 6. Mathematical Finance in Continuous Time6.1 Continuous-time Financial Market Models6.1.1 The Financial Market Model6.1.2 Equivalent Martingale Measures6.1.3 Risk-neutral Pricing6.1.4 Changes of Numeraire