This text is designed for 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 time framework, allowing an accessible account of Brownian motion and stochastic calculus. The Black-Scholes pricing formula is first derived in the simplest financial context. Subsequent chapters are 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 large number of exercises and examples illustrate how the methods and concepts can be applied to realistic financial questions Contents Preface 1. Single period models 2. Binomial trees and discrete parameter martingales 3. Brownian motion 4. Stochastic calculus 5. The Black-Scholes model 6. Different payoffs 7. Bigger models Bibliography and further reading Notation Index