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

TL;DR

This paper introduces finite difference algorithms for solving the quadratic hedging problem in markets modeled by pure jump Markov processes, with applications demonstrated in electricity markets.

Contribution

It develops convergent numerical schemes for a system of PIDEs related to quadratic hedging in jump Markov models, including a practical electricity market application.

Findings

Successfully solves the PIDE system with finite difference schemes.

Demonstrates applicability in electricity markets with jump processes.

Ensures convergence and uniqueness of solutions.

Abstract

We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by 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 integro-differential equations (PIDE), which can be shown to possess unique smooth solutions in our setting. The first equation is non-linear, but does not depend on the pay-off 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 appear in a natural manner.