A Space Mapping approach for the calibration of financial models with the application to the Heston model

We present a novel approach for parameter calibration of the Heston model for pricing an Asian put option, namely space mapping. Since few parameters of the Heston model can be directly extracted from real market data, calibration to real market data is implicit and therefore a challenging task. In addition, some of the parameters in the model are non-linear, which makes it difficult to find the global minimum of the optimization problem within the calibration. Our approach is based on the idea of space mapping, exploiting the residuum of a coarse surrogate model that allows optimization and a fine model that needs to be calibrated. In our case, the pricing of an Asian option using the Heston model SDE is the fine model, and the surrogate is chosen to be the Heston model PDE pricing a European option. We formally derive a gradient descent algorithm for the PDE constrained calibration model using well-known techniques from optimization with PDEs. Our main goal is to provide evidence that the space mapping approach can be useful in financial calibration tasks. Numerical results underline the feasibility of our approach.