Parallelizing Computation of Expected Values in Recombinant Binomial Trees

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 finance when pricing an option. For example, valuation of a European option can be carried out by evaluating the expected value of asset payoffs with respect to random paths in the tree. In many variants of the option valuation problem, a closed form solution cannot be obtained and computational methods are needed. 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 propose a parallel Monte Carlo method which takes advantage of the mapping to achieve a reduced variance over the basic Monte Carlo estimator. Performance results from R and Julia implementations of the parallelization method on a distributed computing cluster indicate that both the implementations are scalable, but Julia is significantly faster than a similarly written R code. A simulation study is carried out to verify the convergence and the variance reduction behavior in the proposed Monte Carlo method.

Sai K. Popuri, Andrew M. Raim, Nagaraj K. Neerchal, Matthias K. Gobbert · 2017