Calibration of a Jump-Diffusion Process Using Optimal Control

A method for calibrating a jump-diffusion model to observed option prices is presented. The calibration problem is formulated as an optimal control problem, with the model parameters as the control variable. It is well known that such problems are ill-posed and need to be regularized. A Hamiltonian system, with non-differentiable Hamiltonian, is obtained from the characteristics of the corresponding Hamilton-Jacobi-Bellman equation. An explicit regularization of the Hamiltonian is suggested, and the regularized Hamiltonian system is solved with a symplectic Euler method. The paper is concluded with some numerical experiments on real and artificial data.