New Brownian bridge construction in quasi-Monte Carlo methods for computational finance
Quasi-Monte Carlo (QMC) methods have been playing an important role for high-dimensional problems
in computational finance. Several techniques, such as the Brownian bridge (BB) and the principal component
analysis, are often used in QMC as possible ways to improve the performance of QMC. This paper proposes
a new BB construction, which enjoys some interesting properties that appear useful in QMC methods. The
basic idea is to choose the new step of a Brownian path in a certain criterion such that it maximizes the
variance explained by the new variable while holding all previously chosen steps fixed. It turns out that
using this new construction, the first few variables are more “important” (in the sense of explained variance)
than those in the ordinary BB construction, while the cost of the generation is still linear in dimension.
We present empirical studies of the proposed algorithm for pricing high-dimensional Asian options and
American options, and demonstrate the usefulness of the new BB.