Financial Mathematics

Finance Mathematics is devoted to financial markets both with discrete and continuous time, exploring how to make the transition from discrete to continuous time in option pricing. This book features a detailed dynamic model of financial markets with discrete time, for application in real-world environments, along with Martingale measures and martingale criterion and the proven absence of arbitrage.

With a focus on portfolio optimization, fair pricing, investment risk, and self-finance, the authors provide numerical methods for solutions and practical financial models, enabling you to solve problems both from mathematical and from financial point of view.

Chapter 1. Financial Markets with Discrete Time 1.1. General description of a market model with discrete time 1.2. Arbitrage opportunities, martingale measures and martingale 1.3. Contingent claims: complete and incomplete markets 1.4. The Cox–Ross–Rubinstein approach to option pricing 1.5. The sequence of the discrete-time markets as an intermediate 1.6. American contingent claims Chapter 2. Financial Markets with Continuous Time 2.1. Transition from discrete to continuous time 2.2. Black–Scholes formula for the arbitrage-free price of the 2.3. Arbitrage theory for the financial markets with continuous time 2.4. American contingent claims in continuous time 2.5. Exotic derivatives in the model with continuous time