Model Order Reduction in Option Pricing
My name is José Pedro Silva and I am a PhD-Student at the Applied Mathematics and Numerical Analysis Group at the University of Wuppertal together with an Early-Stage Researcher position in the ITN-Strike project. My PhD topic orbits around the use of Model Order Reduction in financial markets, with focus in option pricing.
The goal of MOR is to generate smaller models, faster to solve and, if not with similar, with high enough precision with respect to the original Full Order Model (FOM). The Reduced Order Model (ROM) is then a cheaper and faster proxy of the FOM, making it ideal for multi-query problems: parametric studies, parameter optimization, inverse problems, control problems, all of each are increasingly present in finance due to the recent tighter regulations, stress tests, risk evaluation procedures, etc. In our group we investigate the linear setting using two different classes of models: the N-dimensional Black-Scholes and the Heston Model. In the Proper Orthogonal Decomposition setting, we solve the FOM, obtain a basis for the best subspace, project the system to obtain the ROM and, finally, proceed to solve it.
The Heston model comes as a result of GBM being a very restrictive model regarding the paths of the underlying and the necessity of capturing the non-constant volatility existing in the markets and non-existent in the classic Black-Scholes model.
We applied an ADI-type scheme, modified Craig-Sneyd (MCS), to solve the ODE system resulting from the spatial discretization of the Heston Model and, unlike in the N-dimensional case, we used a non-uniform spatial grid based on the hyperbolic sine with focus around zero volatility and the strike price. This approach allows us to reduce even more the size of our numerical problem.
We evaluate and compared the computational cost of using an ADI MCS scheme to solve a full model and the same cost for of solving a reduced one. In what follows, we took d as the number of dimensions in our problem, n_t the number of time-steps in our time-stepping scheme and n_d the number of discretization points in each direction. We will assume n_d is the same in all dimensions just to simplify the exposition as the extension to the general case is straightforward. The speed-up possibilities look very promising.
More details are available at the ECMI2014 Proceedings: ’Proper Orthogonal Decomposition in Option Pricing: Basket Options and Heston Model, J.P.Silva et al.’ In: Progress in industrial Mathematics at ECMI 2014. G. Russo et al. (Eds)., Springer, Berlin, to appear.
This research was supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE – Novel Methods in Computational Finance).