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.