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Stochastic finance

An Introduction in Discrete Time

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Hans Föllmer

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This book serves as an introduction to financial mathematics, targeting graduate students and researchers in both academia and industry. It emphasizes stochastic models in discrete time, which simplifies the probabilistic framework and allows for immediate exploration of key issues in pricing and hedging financial derivatives. The discussion reveals that a complete financial market, where all derivatives can be perfectly hedged, is more of an exception, highlighting the intrinsic risks associated with market incompleteness early on. The first part covers a simple one-period model, laying the groundwork for later concepts. Key topics include arbitrage-free markets, asset profile preferences, equilibrium analysis, and monetary measures of financial risk. The second part expands into dynamic hedging of contingent claims within a multiperiod context, addressing martingale measures, derivative pricing formulas, American options, superhedging, and strategies aimed at minimizing shortfall risk. This newly revised fourth edition features over one hundred exercises and introduces material on risk measures and model uncertainty, including dynamic risk measures, robust utility maximization, and efficient hedging with convex risk measures. The contents are structured into two main parts: mathematical finance in one period and dynamic hedging.