On the Scrambled Sobol Sequence

The Soboĺ sequence is the most popular quasirandom sequence because of its simplicity and efficiency in implementation. We summarize aspects of the scrambling technique applied to Soboĺ sequences and propose a new simpler modified scrambling algorithm, called the multi-digit scrambling scheme. Most proposed scrambling methods randomize a single digit at each iteration. In contrast, our multi-digit scrambling scheme randomizes one point at each iteration, and therefore is more efficient. After the scrambled Soboĺ sequence is produced, we use this sequence to evaluate a particular derivative security, and found that when this sequence is numerically tested, it is shown empirically to be far superior to the original unscrambled sequence.

[1]  Jirí Matousek,et al.  On the L2-Discrepancy for Anchored Boxes , 1998, J. Complex..

[2]  Hongmei Chi,et al.  Parallel linear congruential generators with Sophie-Germain moduli , 2004, Parallel Comput..

[3]  Joseph F. Traub,et al.  Faster Valuation of Financial Derivatives , 1995 .

[4]  Russel E. Caflisch,et al.  Quasi-Random Sequences and Their Discrepancies , 1994, SIAM J. Sci. Comput..

[5]  Frances Y. Kuo,et al.  Remark on algorithm 659: Implementing Sobol's quasirandom sequence generator , 2003, TOMS.

[6]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[7]  Eric Fournié,et al.  Monte Carlo Methods in Finance , 2002 .

[8]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[9]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[10]  Shu Tezuka,et al.  Uniform Random Numbers , 1995 .

[11]  I. Sobol.  Uniformly distributed sequences with an additional uniform property , 1976 .

[12]  Julian V. Noble,et al.  The full Monte , 2002, Comput. Sci. Eng..

[13]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[14]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[15]  Hongmei Chi,et al.  On the optimal Halton sequence , 2005, Math. Comput. Simul..

[16]  G. Marchuk,et al.  Numerical methods and applications , 1995 .

[17]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[18]  Jian Cheng,et al.  Computational Investigation of Low-Discrepancy Sequences in Simulation Algorithms for Bayesian Networks , 2000, UAI.

[19]  A. Owen.  Randomly Permuted (t,m,s)-Nets and (t, s)-Sequences , 1995 .

[20]  J. M. Sek,et al.  On the L2-discrepancy for anchored boxes , 1998 .

[21]  S. Tezuka.  Uniform Random Numbers: Theory and Practice , 1995 .

[22]  Emanouil I. Atanassov.  A New Efficient Algorithm for Generating the Scrambled Sobol' Sequence , 2002, Numerical Methods and Application.

[23]  L. R. Moore,et al.  An Exhaustive Analysis of Multiplicative Congruential Random Number Generators with Modulus $2^{31} - 1$ , 1986 .

[24]  Peter Jaeckel,et al.  Monte Carlo methods in finance , 2002 .

[25]  Paul Bratley,et al.  Algorithm 659: Implementing Sobol's quasirandom sequence generator , 1988, TOMS.

[26]  Michael Mascagni.  Parallel Linear Congruential Generators with Prime Moduli , 1998, Parallel Comput..

[27]  Fred J. Hickernell,et al.  Algorithm 823: Implementing scrambled digital sequences , 2003, TOMS.

[28]  MatoušekJiří.  On the L2-discrepancy for anchored boxes , 1998 .

[29]  H. Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing , 1995 .