A gradient-based calibration method for the Heston model

The Heston model is a well-known two-dimensional financial model. Because the Heston model contains implicit parameters that cannot be determined directly from real market data, calibrating the parameters to real market data is challenging. 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. In this paper, we present a first step towards a novel space mapping approach for parameter calibration of the Heston model. Since the space mapping approach requires an optimization algorithm, we focus on deriving a gradient descent algorithm. To this end, we determine the formal adjoint of the Heston PDE, which is then used to update the Heston parameters. Since the methods are similar, we consider a variation of constant and time-dependent parameter sets. Numerical results show that our calibration of the Heston PDE works well for the various challenges in the calibration process and meets the requirements for later incorporation into the space mapping approach. Since the model and the algorithm are well known, this work is formulated as a proof of concept.