Enhancing Quasi-Monte Carlo Methods by Exploiting Additive Approximation for Problems in Finance
The valuation of many financial securities can be formulated as high-dimensional integrals for which quasi-Monte Carlo (QMC) methods are becoming indispensable numerical tools. It is known that some typical high-dimensional problems in finance and in many other fields have a strong additive nature. This paper investigates the enhancement of QMC methods by exploiting the additive approximation in combination with suitable transformation methods. To this end, we study (a) the approximation of a multivariate function by a sum of one-dimensional functions; and (b) the effective use of the possible strong additive property inherent in finance problems to enhance QMC. For problem (a), we establish a relationship between the approximation quality resulting from anchored decomposition and the global sensitivity indices. For problem (b), we propose new methods for efficiency improvement—the additive weighted uniform sampling and the additive control variate, which exactly exploit the strong additive structure. The required importance distribution and control variate are found constructively based on the anchored additive approximation with a suitably chosen anchor. Numerical examples on bond valuation and option pricing illustrate that the proposed methods reduce the variance by huge factors in QMC. Transformation methods play a crucial role, since they may enhance the degree of additivity of a function. Traditional methods, such as Brownian bridge and principal component analysis, do not necessarily work well, whereas methods that take into account the underlying functions are robust and powerful.