The recent period of extreme volatility in financial markets has once more drawn the attention of academics and practitioners to the insufficiency of Gaussian modeling and the importance of taking into account the extreme market moves. The aim of this course is to show that Lévy processes now offer an easy to use toolkit for pricing and hedging the jump risk in financial markets. After an introduction to the mathematical aspects of Lévy processes, and a general overview of exponential Lévy models, we discuss their uses in risk management, exploring the financial applications where using jump processes really makes a difference.

I. Mathematical introduction to Lévy processes.

Compound Poisson processes and jump-diffusions. Poisson random
measures and the path structure of a Lévy process. Characteristic
functions and the Lévy-Khintchine formula. Basic examples of Lévy
processes used in financial modelling. Simulation of Lévy processes.
Stochastic integrals with respect to Poisson random measures. Itô
formula for semimartingales with jumps.

II. Exponential Lévy models.

Stochastic vs. ordinary exponentials. Measure changes, Esscher
transform and the absence of arbitrage. Market incompleteness. Option
pricing by Fourier methods and the behaviour of implied volatility in
exponential Lévy models. Option pricing using partial
integro-differential equations (PIDE).
Constructing multidimensional exponential Lévy models via Lévy
copulas; application to multi-name gap options.

III. Risk management with exponential Lévy models.

Hedging of options in the presence of jumps: quadratic hedging,
hedging with options, from continuous to discrete rebalancing.
Computing risk measures for portfolio strategies application to the
CPPI strategy.