The effect of randomized low discrepancy sequences in option pricing

New derivative products usually have complex payoff structures depending on multiple risk factors. In such situation numerical computation methods, such as Monte Carlo and quasi-Monte Carlo methods, become very powerful tools because of difficulty in evaluating their pricing model analytically. Low discrepancy sequences installed in quasi-Monte Carlo methods make it possible to produce the uniformity of distribution over the domain of integration, i.e. one or more dimensional unit cube, even for a small number of sample points, which makes numerical integrations to be efficient. Classical low discrepancy sequences, e.g. Faure sequences, are not always satisfactory for multi-dimensional integrations. However, some of generalized Faure sequences can attain quite high performance to compute high-dimensional integrations practically required in financial derivatives pricing, which have been reported in some papers. Unfortunately, none of detailed techniques for the practical construction of such high performance generalized Faure sequences is shown in them. Incidentally, we can confirm that applying a kind of randomization to the classical sequences leads to realize better convergence performance than the original sequence, as we have reported. And recently, some error estimation methods for quasi-Monte Carlo simulation were proposed and experiments taking up such error estimations in numerical evaluations were reported. These methods require certain probabilistic structures of their internal sequences and the most generalized class of them demands huge computational quantity, which is a major problem to be solved in analyzing the quasi-Monte Carlo errors. In this paper, we try to modify low discrepancy sequences with randomized structures originally based on generalized Niederreiter sequences, to keep their performance of convergence and apply the quasi-Monte Carlo error estimation method to it. The structure of this sequence is simple, which can reduce computational complexity. These simplified sequences are applied to the numerical evaluation of path-dependent options in order to compare other low discrepancy and pseudo-random number sequences. We demonstrate that the sequences we proposed attain comparable performance of convergence, error estimation and evaluation time to sequences with more generalized and complicated probabilistic structures.