How do path generation methods affect the accuracy of quasi-Monte Carlo methods for problems in finance?

Abstract

Quasi-Monte Carlo (QMC) methods are important numerical tools in computational finance. Path generation methods (PGMs), such as Brownian bridge and principal component analysis, play a crucial role in QMC methods. Their effectiveness, however, is problem-dependent. This paper attempts to understand how a PGM interacts with the underlying function and affects the accuracy of QMC methods. To achieve this objective, we develop efficient methods to assess the impact of PGMs. The first method is to exploit a quadratic approximation of the underlying function and to analyze the effective dimension and dimension distribution (which can be done analytically). The second method is to carry out a QMC error analysis on the quadratic approximation, establishing an explicit relationship between the QMC error and the PGM. Equalities and bounds on the QMC errors are established, in which the effect of the PGM is separated from the effect of the point set (in a similar way to the Koksma–Hlawka inequality). New measures for quantifying the accuracy of QMC methods combining with PGMs are introduced. The usefulness of the proposed methods is demonstrated on two typical high-dimensional finance problems, namely, the pricing of mortgage-backed securities and Asian options (with zero strike price). It is found that the success or failure of PGMs that do not take into account the underlying functions (such as the standard method, Brownian bridge and principal component analysis) strongly depends on the problem and the model parameters. On the other hand, the PGMs that take into account the underlying function are robust and powerful. The investigation presents new insight on PGMs and provides constructive guidance on the implementation and the design of new PGMs and new QMC rules.