Parallelizing computation of expected values in recombinant binomial trees

ABSTRACT

Recombinant binomial trees are binary trees where each non-leaf node has two child nodes, but adjacent parents share a common child node. Such trees arise in option pricing in finance. For example, an option can be valued by evaluating the expected payoffs with respect to random paths in the tree. The cost to exactly compute expected values over random paths grows exponentially in the depth of the tree, rendering a serial computation of one branch at a time impractical. We propose a parallelization method that transforms the calculation of the expected value into an embarrassingly parallel problem by mapping the branches of the binomial tree to the processes in a multiprocessor computing environment. We also discuss a parallel Monte Carlo method and verify the convergence and the variance reduction behavior by simulation study. Performance results from R and Julia implementations are compared on a distributed computing cluster.

KEYWORDS:

• Binomial tree
• Bernoulli paths
• Monte Carlo estimation
• option pricing

Acknowledgments

The first author acknowledges financial support from the UMBC High Performance Computing Facility (HPCF) at the University of Maryland, Baltimore County (UMBC). The hardware used in the computational studies is part of HPCF. The facility is supported by the U.S. National Science Foundation through the MRI program [grant no. CNS–0821258 and CNS–1228778] and the SCREMS program [grant no. DMS–0821311], with additional substantial support from the University of Maryland, Baltimore County (UMBC). See hpcf.umbc.edu for more information on HPCF and the projects using its resources.

Disclosure statement

This article is released to inform interested parties of ongoing research and to encourage discussion of work in progress. The views expressed are those of the authors and not necessarily those of the U.S. Census Bureau.

ORCID

Sai K. Popuri  http://orcid.org/0000-0002-5151-9721