Large deviations in mathematical finance ∗

The area of large deviations is a set of asymptotic results on rare events probabilities and a set of methods to derive such results. Large deviations theory is a very active field in applied probability, and finds important applications in finance, where questions related to extremal events play an increasingly major role. Financial applications are various, and range from Monte-Carlo methods and importance sampling in option pricing to estimates of large portfolio losses subject to credit risk, or long term portfolio investment The purpose of these lectures is to explain some essential techniques in large deviations theory, and to illustrate how they are applied recently for example in stochastic volatility models to compute implied volatilities near maturities.