Improving The Monte Carlo Simulation Of Financial Derivatives And Value-at-Risk Estimation

The quasi-Monte Carlo method is a numerical tool becoming increasingly popular in computational finance, especially in pricing financial derivatives and risk measurement such as Value-at-Risk (VAR). Implementations of various quasi-Monte Carlo (low-discrepancy) sequences are now available in general mathematical software, as well as specialized financial software. In addition, many patents have been issued on the application of these algorithms to problems from finance in the last decade.In this paper, first we consider various numerical examples from financial derivatives, VAR, and sensitivity estimation to show the superiority of quasi-Monte Carlo method. Second, we analyze the effects of different methods to transform the low discrepancy sequences to normal distribution in order to obtain best accuracy our estimates. In particular, we will discuss a recent finding which claims that the Box-Muller method is at least as good as the inverse transformation method in the context of low-discrepancy sequences, which contradicts the common folklore among financial engineers and some researchers. We propose an alternative algorithm, based on radial stratification of Box-Muller method to sample the normal random variables, which is effective in VAR estimation. Finally, we derive error bounds for the quasi-Monte Carlo estimation of European and Asian geometric call option pricing problems.