Path Generation for Quasi-Monte Carlo Simulation of Mortgage-Backed Securities

Abstract

Monte Carlo simulation is playing an increasingly important role in the pricing and hedging of complex, path dependent financial instruments. Low discrepancy simulation methods offer the potential to provide faster rates of convergence than those of standard Monte Carlo methods; however, in high dimensional problems special methods are required to ensure that the faster convergence rates hold. Indeed, Ninomiya and Tezuka (1996) have shown highdimensional examples, in which low discrepancy methods perform worse than Monte Carlo methods. The principal component construction introduced by Acworth et al. (1998) provides one solution to this problem. However, the computational effort required to generate each path grows quadratically with the dimension of the problem. This article presents two new methods that offer accuracy equivalent, in terms of explained variability, to the principal components construction with computational requirements that are linearly related to the problem dimension. One method is to take into account knowledge about the payoff function, which makes it more flexible than the Brownian Bridge construction. Numerical results are presented that show the benefits of such adjustments. The different methods are compared for the case of pricing mortgage backed securities using the Hull-White term structure model.Monte Carlo simulation, quasi-Monte Carlo, mortgage backed securities, Brownian bridge, principal component decomposition, option pricing

On the Use of Quasi-Monte Carlo Methods in Computational Finance

“…In any case, generating the point set P n is faster than when MC is used, and this holds for most QMC methods. Two nice properties of the point set (2) are that it is dimension-stationary and fully projection-regular [26,47]. The first property means that if two subsets I = {i 1 , .…”

“…Techniques for reducing the effective dimension are useless to improve MC simulations, but they can greatly enhance the performance of QMC methods [9,32,1,10,27,2]. We discuss two of them: the Brownian bridge (BB) and the principal components (PC) techniques.…”

“…An advantage of this approach over BB is that it can be used to generate prices of correlated assets whereas BB can only be applied for generating prices coming from a single path. However, PC requires more computation time than BB for the generation of the prices, but see [2] for a way of speeding up PC.…”

Efficient simulation of gamma and variance-gamma processes

“…For numerical integration via randomized quasi-Monte Carlo (QMC) techniques, there have been recent publications on the subject of structuring the sampling algorithm so as to concentrate the variance of the integrand to a few coordinates (Caflisch and Moskowitz 1995, Moskowitz and Caflisch 1996, Acworth, Broadie, and Glasserman 1997, Åkesson and Lehoczy 2000, Owen 1998, Liu and Owen 2003. The book of Fox (1999) is centered on such ideas and their synergy with QMC.…”

“…Several variants of the structuring approach have been proposed, with Acworth, Broadie, and Glasserman (1997) suggesting an approach based on the principal components of the covariance matrix of a discretely sampled Brownian motion, and Åkesson and Lehoczy (2000) extending the ideas to more general Gaussian processes. Caflisch, Morokoff, and Owen (1997) and Åkesson and Lehoczy (2000) report computational experience with integrals arising in pricing mortgage-backed securities, and Acworth, Broadie, and Glasserman (1997) also report experience with high-dimensional integrals arising in option pricing. The empirical consensus is that the above path generation schemes, when combined with quasiMonte Carlo, outperform ordinary Monte Carlo (MC) in many situations, sometimes by orders of magnitude.…”