Quasi-Monte Carlo in finance: extending for problems of high effective dimension

Abstract

RESUMONeste artigo mostramos que é possível usar métodos de simulação quase-Monte Carlo em problemas de alta dimensão efetiva. Isto é feito por meio de uma combinação de uma cuidadosa construção das seqüên-cias de Sobol e de uma decomposição apropriada da matriz de covariância dos fatores de risco. A eficácia do método é ilustrada por meio da precificação de opções que envolve a solução de problemas com dimensão nominal da ordem de 550 (e dimensão efetiva da ordem de 300). Acreditamos que o método apresentado seja de fácil implementação e de grande interesse para os participantes do mercado financeiro.Palavras-chave: quase-Monte Carlo, baixa discrepância, Sobol, dimensão efetiva, seqüências deterministas. ABSTRACTIn this paper we show that it is possible to extend the use of quasi-Monte Carlo for applications of high effective dimension. This is achieved through a combination of a careful construction of the Sobol sequence and an appropriately chosen decomposition of a covariance matrix. The effectiveness of this procedure is demonstrated as we price average options with nominal dimensions ranging up to 550 (effective dimension around 300). We believe the method we present is easy to implement and should be of great interest to practitioners.

On improving the least squares Monte Carlo option valuation method

“…The use of control variates reduces the RMSRE, but these reductions are only statistically significant, at a 5% level, for the LSMC with Halton (1960), Niederreiter (1988, Bratley and Fox (1988), and Silva and Barbe (2003) LDS sequences. However these improvements come with an increase of the computational cost.…”

“…CV stands for control variates, BS for Black-Scholes valuation at the step before maturity, AV for antithetic variates, Nied. for Niederreiter, BF for Sobol with Bratley and Fox (1988) initialization, SB for Sobol with Silva and Barbe (2003) initialization, MM1 for one moment matching and MM2 for two moment matching *p-value < 10%, **p-value < 5%,***p-value < 1% (2000) MRG31k3p uniform random variates generator was used with the seed 12345. All the low discrepancy sequences (LDS) were used with Brownian bridges (BB).…”

“…Nied. stands for Niederreiter, BF for Sobol with Bratley and Fox (1988) initialization, SB for Solbol with Silva and Barbe (2003) initialization, MM1 for one moment matching and MM2 for two moment matching. The bechmark value of the American option was obtained using the binomial method with Cox et al (1979) parameters and 10,000 steps.…”

“…Moro (1995) normal variates were used. The regression was performed using SVD algorithm with all the paths and 6 weighted Laguerre polynomial functions *p-value < 10%, **p-value < 5%, ***p-value < 1% BF stands for Bratley and Fox (1988), SB for Silva and Barbe (2003), MM1 for one moment matching and MM2 for two moment matching. The bechmark value of the American option was obtained using the binomial method with Cox et al (1979) parameters and 10,000 steps.…”

“…Therefore, we consider Halton (1960), Sobol (1967), Faure (1982) and Niederreiter (1988) sequences, along with the use of BB, to select the one which produces the best results for the American option valuation problem. 11 To implement Sobol sequences we use two different algorithms: one suggested by Bratley and Fox (1988), and the other the implementation proposed by Press et al (1992) along with initialization numbers suggested by Silva and Barbe (2003) which, according to the authors, produces better results for high dimensional problems than the original algorithm or even the initialisation numbers suggested by Jackel (2002). 12…”