Option Pricing Models with Jumps: Integro-differential Equations and Inverse Problems

Observation of sudden, large movements in the prices of financial assets has led to the use of stochastic processes with discontinuous trajectories – jump processes – as models for financial assets. Exponential Lévy models provide an analytically tractable subclass of models with jumps and the flexibility in choice of the Lévy process allows to calibrate the model to market prices of options and reproduce a wide variety of implied volatility skews/smiles. We discuss the characterization of prices of European and barrier options in exponential Lévy models in terms of solutions of partial integro-differential equations (PIDEs). These equations involve, in addition to a second-order differential operator, a non-local integral term which requires specific treatment both at the theoretical and numerical level. The study of regularity of option prices in such models shows that, unlike the diffusion case, option price can exhibit lack of smoothness. The proper relation between option prices and PIDEs is then expressed using the notion of viscosity solution. Numerical solution of the PIDE allows efficient computation of option prices. The identification of exponential Lévy models from option prices leads to an inverse problem for such PIDEs. We describe a regularization method based on relative entropy and its numerical implementation. This inversion algorithm, which allows to extract an implied Lévy measure from a set of option prices, is illustrated by numerical examples.