Research projects

Project CEMAPRE internal

TitleOption pricing in Lévy models and integro-differential equations
ParticipantsNicola Cantarutti, João Guerra (Principal Investigator), Manuel Guerra, Raúl Narciso
SummaryPartial 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.