An Analysis of Sobol Sequence and the Brownian Bridge
39 Pages Posted: 1 Nov 2011
Date Written: October 31, 2011
We investigate the loss of efficiency of Sobol low-discrepancy sequence at high dimensions and the apparent improvement provided by the use of the Brownian bridge construction of Brownian motion paths. We show numerically that some often cited potential causes for these phenomena such as low quality of Sobol coordinates at high dimensions are not to blame and instead isolate a bias in Sobol sequence which we conjecture to be the main cause of the problem. We motivate this conjecture by an analysis of the equations defining both the Incremental and Brownian bridge constructions in a simplified setting, showing how the identified bias is removed by the use of the Brownian bridge. We further give numerical evidence that randomizing Sobol sequence can remove most of this bias and achieve a good convergence at high dimensions. We explain why this is particularly relevant for efficient inline implementations in massively parallel environments such as GPUs under the programming language CUDA. Tested products and models include vanilla and barrier options as well as TARN PRDCs in Local Volatility. We also provide proofs of the homogeneity properties of the Gaussian deviates derived from Sobol sequence for particular numbers of iterations.
Keywords: Sobol sequence, Brownian bridge, Quasi Monte Carlo, parallel programming, CUDA
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