Quasi-Monte Carlo methods, applications in finance and actuarial science

Abstract

In recent years, there had been a growing interest in the application of quasi-Monte Carlo methods in finance and actuarial science. A common application of the Monte Carlo method is in the evaluation of multi-dimensional integrals. Quasi-Monte Carlo uses specially selected deterministic sequences rather than random sequences as in Monte Carlo. These special sequences are known as low discrepancy sequences and have the property that they tend to be evenly dispersed throughout the unit cube. For many applications in finance, the use of low discrepancy sequences seem to provide more accurate answer than random sequences. Nevertheless there are several drawbacks of this method. First, there is no simple criterion to assess the accuracy of the estimates in applications of this technique. Second, the integrand should have certain smoothness properties. This can be a restrictive condition in some situations. Third, ay additional smoothness of the integrands is not reflected in the the error bound of the quasi-Monte Carlo method. In this thesis, we address these issues and examine ways of overcoming these problems so that the efficiency of the quasi-Monte Carlo method can be enhanced.