Efficient Weighted Lattice Rules with Applications to Finance

Good lattice rules are an important type of quasi-Monte Carlo algorithm. They are known to have good theoretical properties, in the sense that they can achieve an error bound (or optimal error bound) that is independent of the dimension for weighted function spaces with suitably decaying weights. To use the theory of weighted function spaces for practical applications, one has to determine what weights should be used. After explaining that lattice rules based on figures of merit with classical weights (classical lattice rules) may not give good results for high-dimensional problems, we propose a "matching strategy," which chooses the weights in such a way that the typical functions of a weighted Korobov space are similar (in the sense of similar relative sensitivity indices) to those of the given function. The matching strategy relates the weights of the spaces to the sensitivity indices of the given function. We apply the Korobov construction and the component-by-component construction of lattice rules with the suitably chosen weights to the valuation of high-dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules; moreover, they are competitive with other types of low discrepancy sequences. We also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly. We show that an acceptance-rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy.

[1]  Henryk Wozniakowski,et al.  Tractability of Multivariate Integration for Weighted Korobov Classes , 2001, J. Complex..

[2]  Fred J. Hickernell,et al.  Integration and approximation in arbitrary dimensions , 2000, Adv. Comput. Math..

[3]  I. H. SLOAN,et al.  Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces , 2002, SIAM J. Numer. Anal..

[4]  Henryk Wozniakowski,et al.  Liberating the weights , 2004, J. Complex..

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

[6]  Xiaoqun Wang,et al.  Constructing Robust Good Lattice Rules for Computational Finance , 2007, SIAM J. Sci. Comput..

[7]  Henryk Wozniakowski,et al.  Tractability of Integration in Non-periodic and Periodic Weighted Tensor Product Hilbert Spaces , 2002, J. Complex..

[8]  A. Owen,et al.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 1997 .

[9]  P. Glasserman,et al.  Monte Carlo methods for security pricing , 1997 .

[10]  Fred J. Hickernell,et al.  The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension , 2002, Math. Comput..

[11]  Frances Y. Kuo,et al.  Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces , 2003, J. Complex..

[12]  R. Caflisch,et al.  Smoothness and dimension reduction in Quasi-Monte Carlo methods , 1996 .

[13]  Henryk Wozniakowski,et al.  Good Lattice Rules in Weighted Korobov Spaces with General Weights , 2006, Numerische Mathematik.

[14]  P. L’Ecuyer,et al.  Variance Reduction via Lattice Rules , 1999 .

[15]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[16]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[17]  Ian H. Sloan,et al.  Why Are High-Dimensional Finance Problems Often of Low Effective Dimension? , 2005, SIAM J. Sci. Comput..

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

[19]  J. Hull.  Options, Futures, and Other Derivatives , 1989 .

[20]  A. Owen,et al.  Control variates for quasi-Monte Carlo , 2005 .

[21]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[22]  P. Glasserman,et al.  A Comparison of Some Monte Carlo and Quasi Monte Carlo Techniques for Option Pricing , 1998 .

[23]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

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

[25]  Fred J. Hickernell,et al.  Randomized Halton sequences , 2000 .

[26]  Ian H. Sloan,et al.  Component-by-component construction of good lattice rules , 2002, Math. Comput..

[27]  Kai-Tai Fang,et al.  The effective dimension and quasi-Monte Carlo integration , 2003, J. Complex..

[28]  F. J. Hickernell.  Quadrature Error Bounds with Applications to Lattice Rules , 1997 .

[29]  I. Sloan.  Lattice Methods for Multiple Integration , 1994 .

[30]  R. Cranley,et al.  Randomization of Number Theoretic Methods for Multiple Integration , 1976 .

[31]  Dirk Nuyens,et al.  Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces , 2006, Math. Comput..

[32]  Xiaoqun Wang,et al.  Strong tractability of multivariate integration using quasi-Monte Carlo algorithms , 2003, Math. Comput..

[33]  Josef Dick,et al.  On Korobov Lattice Rules in Weighted Spaces , 2004, SIAM J. Numer. Anal..