  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.
