Finance provides a dramatic example of the successful application of advanced mathematical techniques to a practical problem : the pricing of financial derivatives. This self-contained text is designed fer first courses in financial Calculus aimed at students with a good background in mathematics. Key concepts such as martingales and change of measure are introduced in the discrete Lime framework, allowing an accessible account of Brownian motion and stochastic calculus : proofs in the continuons time world follow naturally. The Black-Scholes pricing formula is first derived in the simplest financial context. The second half of the book is then devoted to increasing the financial sophistication of the models and instruments. The final chapter introduces more advanced topics including stock price models with jumps, and stochastic volatility. A valuable feature is the large number of exercises and examples, designed to test technique and illustrate how the methods and concepts can be applied to realistic financial questions.