Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation

TLDR: In this paper, the authors presented reduced models for pricing basket options with the Black-Scholes and the Heston model, which achieved speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options.

Abstract: This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models.