Numerical methods for the quadratic hedging problem in Markov models with jumps

Carmine De Franco, Peter Tankov and Xavier Warin

Abstract

ABSTRACT

We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by a pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integrodifferential equations (PIDEs), which can be shown to possess unique smooth solutions in our setting. The first equation is nonlinear, but does not depend on the payoff of the option to hedge (the pure investment problem), while the other two equations are linear. We propose convergent finite-difference schemes for the numerical solution of these PIDEs and illustrate our results with an application to electricity markets, where time-inhomogeneous pure jump Markov processes naturally occur.