A highly parallel Black–Scholes solver based on adaptive sparse grids

Abstract

In this paper, we present a highly efficient approach for numerically solving the Black–Scholes equation in order to price European and American basket options. Therefore, hardware features of contemporary high performance computer architectures such as non-uniform memory access and hardware-threading are exploited by a hybrid parallelization using MPI and OpenMP which is able to drastically reduce the computing time. In this way, we achieve very good speed-ups and are able to price baskets with up to six underlyings. Our approach is based on a sparse grid discretization with finite elements and makes use of a sophisticated adaption. The resulting linear system is solved by a conjugate gradient method that uses a parallel operator for applying the system matrix implicitly. Since we exploit all levels of the operator's parallelism, we are able to benefit from the compute power of more than 100 cores. Several numerical examples as well as an analysis of the performance for different computer architectures are provided.

Keywords:

• Black–Scholes PDE
• adaptive sparse grids
• parallelization
• distributed/shared memory systems
• OpenMP/MPI

2010 AMS Subject Classifications::

• 35Q68
• 65N30
• 65M50
• 68W10
• 91G80
• G.1.8
• C.1.4
• C.4

Acknowledgements

We thank Michael Klemm, Andrey Semin, Christopher Dahnken, and Jamie Wilcox at Intel for giving us access to a Xeon E7 (Westmere-EX) machine. Furthermore, we want to thank Reinhold Bader at Leibniz Supercomputing Centre for his support on using the AMD Magny-Cours infiniband-cluster.