Project CEMAPRE internal
| Title | Option pricing in Lévy models and integro-differential equations |
| Participants | Nicola Cantarutti, Manuel Guerra, João Guerra (Principal Investigator), Raúl Narciso |
| Summary | Partial integro-differential equations (PIDE's) appear in option pricing with discontinuous models. These equations generalize the Black-Scholes PDE when the continuous diffusion dynamics for the underlying price is replaced by a Lévy process dynamics (including jumps). The integral operator in the PIDE propagates a possible irregularity of the solution. For many exponential Lévy models (such as Variance Gamma) or for Barrier options, the option price may not be sufficiently regular. This drawback led several authors to consider option prices as weak solutions (viscosity solutions) of the PIDEs. Our main goals are: (i) develop a Markov chain approximation for the optimization problem related to the option pricing problem in a exponential Lévy model market with transaction costs, prove the convergence of this Markov chain approximation and, if possible, prove the existence of a viscosity solution for the PIDE associated to this problem. (ii) develop numerical schemes based on finite differences, finite element methods and Radial basis functions for pricing PIDE's in general exponential Lévy models. |