文档介绍：Proceedings of the 2004 Winter Simulation Conference
R. G. Ingalls, M. D. Rossetti, J. S. Smith, and B. A. Peters, eds.
QUASI-MONTE CARLO METHODS IN FINANCE
Pierre L’Ecuyer
Département d’Informatique et de Recherche Opérationnelle
Université de Montréal, . 6128, . Centre-Ville
Montréal (Québec), H3C 3J7, CANADA
ABSTRACT random number generator, u = (u0,u1,u2,...) is the se-
quence of essive numbers returned by this generator, and
We review the basic principles of Quasi-Monte Carlo (QMC) f represent the (usually plicated) transformation
methods, the randomizations that turn them into variance- of these numbers into the simulation output f(u), which is
reduction techniques, and the main classes of constructions assumed here to be an unbiased estimator of some constant
underlying their implementations: lattice rules, s, µ. In the case where the required number of uniforms is
and permutations in different bases. QMC methods are random and unbounded, we can just take s =∞.
designed to estimate integrals over the s-dimensional unit The estimator of µ considered here has the form
hypercube, for moderate or large (perhaps infinite) values
n−
of s. In principle, any stochastic simulation whose purpose 1 1
Q = f(u ),
is to estimate an integral fits this framework, but the meth- n n i
ods work better for certain types of integrals than others i=0
(., if the integrand can be well approximated by a sum of
={ }⊂[ s
low-dimensional smooth functions). Such QMC-friendly in- where Pn u0,...,un−1 0, 1) is the point set over
tegrals are encountered frequently putational finance which the average is taken. The number of points n corre-
and risk analysis. We give examples and pu- sponds to the number of simulation runs.
tational results that illustrate the efficiency improvement In standard Monte Carlo (MC), the ui are indepen-
[ s
achieved. dent and uniformly distributed over 0, 1) . Then, Qn is
obviously unbiased and has variance σ 2/n, where