Abstract

The need to numerically simulate stochastic processes arises in many fields. Frequently this is done by discretizing the process into small time steps and applying pseudo-random sequences to simulate the randomness. This paper address the question of how to use quasi-Monte Carlo methods to improve this simulation. Special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps are required. Two such techniques, the generalized Brownian bridge and particle reordering, are described here. These methods are applied to a problem from finance, the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate. When expressed as an integral, this problem is nominally 360 dimensional. The analysis of the integrand presented here explains the effectiveness of the quasi-random sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunc...

Citations

638	Random Number Generation and Quasi-Monte Carlo Methods - Niederreiter - 1992
450	Numerical solutions of stochastic differential equations - Kloeden, Platen - 1992
386	An equilibrium characterization of the term structure - Vasicek - 1977
226	Uniform Distributions of Sequences - Kuipers, Niederreiter - 1974
130	Options, Futures and Other Derivative Securities - Hull - 1989
106	On the distribution of points in a cube and the approximate evaluation of integrals - Sobol - 1967
70	Faster Valuation of financial Derivatives - Paskov, Traub - 1995
62	Orthogonal arrays for computer experiments, integration and visualization - Owen - 1992
59	Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension - Caflisch, Morokoff, et al. - 1997
57	Quasi-sandom sequences and their discrepancies - Morokoff, Caflisch - 1994
37	A quasi�Monte Carlo algo� rithm for the global illumination in the radiosity setting - Keller - 1995
36	Quasi-Monte Carlo integration - Morokoff, Caflisch - 1995
30	Smoothness and dimension reduction in quasi-Monte Carlo methods - Moskowitz, Caflisch - 1996
21	Toward real-time pricing of complex financial derivatives, Applied Mathematical Finance 3 - Ninomiya, Tezuka - 1996
16	A quasi-Monte Carlo approach to particle simulation of the heat equ ation - Morokoff, Caflisch - 1993
9	Quasi-Monte Carlo Computation of a Finance Problem - Caflisch, Morokoff - 1996
8	Radiative heat transfer with quasi-Monte Carlo methods, Transport Theory and Statistical Physics 7 - Kersch, Morokoff, et al. - 1994
5	Quasi- (that is, sub-) random numbers - Press, Teukolsky - 1989
4	A quasi-Monte Carlo scheme using nets for a linear Boltzmann equation - Lécot, Coulibaly - 1998
3	Accelerated quasi-Monte Carlo integration of the radiative transfer equation - O'Brien - 1992
2	Markets, Analysis and Strategies - Bond - 1996
2	A quasi-Monte Carlo method for the Boltzmann equation - Lecot - 1991
2	New methodolgies for valuing derivatives - Paskov - 1996
1	Low disecrepancy sequences for the Boltzmann equation - Lecot - 1989
1	Application of quasirandom points for simulation of gamma radiation transfer - Shuhman - 1990