Risk-neutral valuation. Pricing and hedging of financial derivatives.
This book is for students of economics, finance, financial engineering, mathematics and statistics with a preknowledge in statistics and probability theory. It has nine chapters. It begins with the background of financial derivatives in chapter 1, followed by the mathematical background in chapter 2. Chapter 3 describes stochastic processes in discrete time. An application to mathematical finance in discrete time follows in chapter 4. The corresponding treatment in continuous time is treated in chapter 5 and 6. The remaining chapters treat incomplete markets, interest rate models, and credit risks. Chapter 1 (Derivative Background) describes the financial markets and instruments, arbitrage, arbitrage relationships, and single period market models. Chapter 2 (Probability Background) begins with measure, integral, probability, equivalent measures and Radon-Nikodym derivatives. Conditional expectations, modes of convergence, convolution and characteristic functions follow. It closes with the central limit theorem, asset return distributions, infinite divisibility and the Lévy-Klintchine formula, elliptically contoured and hyperbolic distributions. Chapter 3 (Stochastic Processes in Discrete Time) begins with information and filtration, discrete-parameter stochastic processes, definition and basic properties of martingales, and martingale transforms. Further, it regards stopping times and optimal stopping, space of martingales and Markov chains. Chapter 4 (Mathematical Finance in Discrete Time) describes the model, the existence of equivalent martingale measures, complete markets, the fundamental theorem of asset pricing, the Cox-Rubinstein model, binominal approximations, american options, further contingent claim valuation in discrete time, and multifactor models. Chapter 5 (Stochastic Processes in Continuous Time) regards filtration, classes of processes, the Brownian motion, point processes, Lévy processes, stochastic integrals (Itô calculus), stochastic calculus for Black-Scholes models. It ends with stochastic differential equations, likelihood estimation for diffusions, martingales, and weak convergence of stochastic processes. Chapter 6 (Mathematical Finance in Continuous Time) treats continuous-time financial market models, the generalized Black-Scholes model, different options, discrete-versus continuous-time models, future and currency markets. Chapter 7 (Incomplete Markets) regards pricing and hedging in incomplete markets, stochastic volatility models, and models driven by Lévy processes. Chapter 8 (Interest Rate Theory) treats the bond market, short rate models, the Heath-Jarrow-Morton methodology, pricing and hedging contingent claims, market models of LIBOR- and Swap-rates, potential models and the Flesaker-Hughston framework. Chapter 9 (Credit Risks) begins with aspects of credit risks, basic credit risk modelling, structural models, reduced form models. It ends with credit derivatives, portfolio credit risk models, and collateralized debt obligations.
scientific article; zbMATH DE number 2107174