General Prerequisites: B8.1 Probability, Measure and Martingales (previously named Martingales Through Measure Theory) would be good background. Part A Probability is a prerequisite. Part A Integration is also good background, though not a prerequisite.

Course Overview: The course aims to introduce students to derivative security valuation in financial markets. At the end of the course the student should be able to formulate a model for an asset price and then determine the prices of a range of derivatives based on the underlying asset using arbitrage free pricing ideas.

Learning Outcomes: Students will have a familiarity with the mathematics behind the models and analytical tools used in Mathematical Finance. This includes being able to formulate a model for an asset price and then determining the prices of a range of derivatives based on the underlying asset using arbitrage free pricing ideas.

Course Synopsis: Introduction to markets, assets, interest rates and present value; arbitrage and the law of one price: European call and put options, payoff diagrams. Probability spaces, random variables, conditional expectation, discrete-time martingales. The binomial model; European and American claim pricing.

Introduction to Brownian motion and its quadratic variation , continuous-time martingales, informal treatment of ItÃ´'s formula and stochastic differential equations. Discussion of the connection with PDEs through the Feynman-Kac formula.

The Black-Scholes analysis via delta hedging and replication, leading to the Black-Scholes partial differential equation for a derivative price. General solution via Feynman-Kac and risk neutral pricing, explicit solution for call and put options.

American options, formulation as a free-boundary problem. Simple exotic options. Weakly path-dependent options including barriers, lookbacks and Asians. Implied volatility. Introduction to stochastic volatility. Robustness of Black-Scholes formula.