Applications of randomized low discrepancy sequences to the valuation of complex securities

Abstract

Abstract This paper deals with a recent modification of the Monte Carlo method known as quasi-random Monte Carlo. Under this approach, one 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 seems to provide more accurate answers than random sequences. One of the main drawbacks of the use of low discrepancy sequences is that there is no obvious method of computing the standard error of the estimate. This means that in performing the calculations, there is no clear termination criterion for the number of points to use. We address this issue here and consider a partial randomization of Owen's technique for overcoming this problem. The proposed method can be applied to much higher dimensions where it would be computationally infeasible for Owen's technique. The efficiency of these procedures is compared using a particular derivative security. The exact price of this security can be calculated very simply and so we have a benchmark against which to test our calculations. We find that our procedures give promising results even for very high dimensions. Statistical tests are also conducted to support the confidence statement drawn from these procedures.