This accessible introduction to the mathematical underpinnings of finance concentrates on the probabilistic theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory, and Merton's fund separation theory. It includes a solved example for every new technique presented, numerous exercises, and a Further Reading list in each chapter. This new edition includes new chapters on measure theory, probability theory, Girsanov transformations, the LIBOR and Swap Market Models, and martingale representations INDICE 1 Introduction 2 The Binomial Model 3 A More General One Period Model 4 Stochastic Integrals 5 Differential Equations 6 Portfolio Dynamics 7 Arbitrage Pricing 8 Completeness and Hedging 9 Parity Relations and Delta Hedging 10 The Martingale Approach to Arbitrage Theory (For advanced readers) 11 The Mathematics of the Martingale Approach (For advanced readers) 12 Black-Scholes from a Martingale Point of View (For advanced readers) 13 Multidimensional Models: Classical Approach 14 Multidimensional Approach: Martingale Approach (For advanced readers) 15 Incomplete Markets 16 Dividends 17 Currency Derivatives 18 Barrier Options 19 Stochastic Optimal Control 20 Bonds and Interest Rates 21 Short Rate Models 22 Martingale Models for the Short Rate 23 Forward Rate Models 24 Change of Numeraire (For advanced readers) 25 LIBOR and Swap Market Models 26 Forwards and Futures