Efficient Parallel Solution Methods for High-Dimensional Option Pricing Problems

Many problem classes in Finance lead to high-dimensional partial differential equations (PDEs), which need to be solved efficiently. Currently, several methods exist to either circumnavigate the curse of dimensionality or use parallel High Performance Computing to calculate solutions despite it. In Schröder et al. (2013b) the authors present a special class of decomposition techniques to decompose a high-dimensional PDE into a linear combination of independent, low-dimensional PDEs, which can be solved in parallel. We combine this decomposition with the combination technique introduced Griebel et al., 1992 to circumnavigate the curse of dimensionality for these low-dimensional PDEs using sparse grids. The combination technique also allows for a straightforward parallelization of the so-called component grids that are used to construct the solution in the sparse grid space. Therefore, we introduce a two-level parallelization technique, which facilitates the solution of the whole set of low-dimensional PDEs in parallel. For each of these PDEs, we employ the combination technique and compute the solution on the component grids again in parallel.The presented parallelization approach significantly reduces the overall runtime of solution routines for decomposed high-dimensional PDEs. We show strong scalability of our approach, even for problems of very high dimensionality, using basket options on the DAX 30 and the S&P 500 as numerical examples