Textbook in PDF format This book provides an introduction to probabilistic methods in finance, based on stochastic models in discrete time. It is aimed primarily at graduate students in mathematics but may also benefit mathematicians in academia and the financial industry. In this fifth edition, the entire text has been thoroughly revised to enhance clarity and completeness. This includes new sections on: stop-loss insurance contracts.　 the Fatou property of law-invariant risk measures.　 relative entropy.　 cover’s universal portfolios, and numerous new exercises. Preface to the fifth edition. Preface to the fourth edition. Preface to the third edition. Preface to the second edition. Preface to the first edition. Arbitrage theory. Preferences. Optimality and equilibrium. Monetary measures of risk. Dynamic arbitrage theory. American contingent claims. Superhedging. Efficient hedging. Hedging under constraints. Minimizing the hedging error. Dynamic risk measures. Long-run performance of rebalanced portfolios. A Convexity. B Absolutely continuous probability measures. C Relative entropy. D Quantile functions and their applications. E The Neyman–Pearson lemma. F The essential supremum of a family of random variables. G Spaces of measures. H Some functional analysis. Bibliographical notes. Bibliography. List of symbols. Index