Numerical Methods for the Quadratic Hedging Problem in Markov Models with Jumps

40 Pages Posted: 15 Jun 2016

See all articles by Carmine De Franco

Carmine De Franco

OSSIAM

Peter Tankov

ENSAE, Institut Polytechnique de Paris

Xavier Warin

EDF Energy

Date Written: November 04, 2015

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.

Keywords: Quadratic Hedging, Electricity Markets, Markov Jump Processes, Partial Integrodifferential Equation, Hamilton–Jacobi–Bellman Equation, Discretization Schemes for Partial Integrodifferential Equations

Suggested Citation

De Franco, Carmine and Tankov, Peter and Warin, Xavier, Numerical Methods for the Quadratic Hedging Problem in Markov Models with Jumps (November 04, 2015). Journal of Computational Finance, Vol. 19, No. 2, Pages 29–67, 2015, Available at SSRN: https://ssrn.com/abstract=2795670

Carmine De Franco

OSSIAM ( email )

80 Avenue de la Grande Armée
Paris, 75017
France

Peter Tankov (Contact Author)

ENSAE, Institut Polytechnique de Paris ( email )

Palaiseau
France

Xavier Warin

EDF Energy ( email )

France

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