Reduced order models in PIDE constrained optimization

Mathematical models for option pricing often result in partial differential equations originally starting with the Black-Scholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one solves the problems mentioned last numerically, this yields large linear systems of equations with dense matrices. However, by using the special structure and an iterative solver the problem can be handled efficiently. To further reduce the computational cost in the calibration phase we implement a reduced order model, like proper orthogonal decomposition (POD), which proves to be very efficient. In this paper we use a special multi-level trust region POD algorithm to calibrate the option pricing model and give numerical results supporting the gain in efficiency.