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 LÚvy 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 LÚvy processes, and a general overview of exponential LÚvy models, we discuss their uses in risk management, exploring the financial applications where using jump processes really makes a difference.

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

II. Exponential LÚvy 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 LÚvy models. Option pricing using partial integro-differential equations (PIDE). Constructing multidimensional exponential LÚvy models via LÚvy copulas; application to multi-name gap options.

III. Risk management with exponential LÚvy 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.