Stochastic finance

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This book serves as an introduction to financial mathematics, aimed at graduate students and researchers in both academia and industry. It emphasizes stochastic models in discrete time, offering two key advantages: simpler probabilistic tools and an early exploration of market incompleteness, where perfect hedging of derivatives is not always possible. The first part introduces a one-period model, laying the groundwork for later topics such as arbitrage-free markets, asset profile preferences, equilibrium analysis, and monetary risk measures. The second part expands into dynamic hedging of contingent claims within a multiperiod framework, covering martingale measures, derivative pricing formulas, American options, superhedging, and strategies to minimize shortfall risk. This revised and extended edition features over one hundred exercises and includes new content on risk measures and model uncertainty. Notably, it introduces a chapter on dynamic risk measures and sections on robust utility maximization and efficient hedging with convex risk measures, enhancing its relevance for contemporary financial mathematics.