Dimension and Variance Reduction for Monte Carlo Methods for High-Dimensional Models in Finance

27 Pages Posted: 21 Jan 2015 Last revised: 13 Dec 2015

See all articles by Duy-Minh Dang

Duy-Minh Dang

University of Queensland - School of Mathematics and Physics

Kenneth R. Jackson

University of Toronto - Department of Computer Science

Mohammadreza Mohammadi

University of Toronto - Department of Statistics

Date Written: December 4, 2015

Abstract

One-way coupling often occurs in multi-dimensional models in finance. In this paper, we present a dimension reduction technique for Monte Carlo (MC) methods, referred to as drMC, that exploits this structure for pricing plain-vanilla European options under a N-dimensional one-way coupled model, where N is arbitrary. The dimension reduction also often produces a significant variance reduction.

The drMC method is a dimension reduction technique built upon (i) the conditional MC technique applied to one dimension and (ii) the derivation of a closed-form solution for the conditional Partial Differential Equation (PDE) that arises via Fourier transforms. In the drMC approach, the option price can be computed simply by taking the expectation of this closed-form solution. Hence, the approach results in a powerful dimension reduction from N to one, which often results in a significant variance reduction as well, since the variance associated with the other (N-1) factors in the original model are completely removed from the drMC simulation. Moreover, under the drMC framework, hedging parameters, or Greeks, can be computed in a much more efficient way than in traditional MC techniques.

A variance reduction analysis of the method is presented and numerical results illustrating the method's efficiency are provided.

Keywords: conditional Monte Carlo, variance reduction, dimension reduction, cross-currency, Fourier transform, partial differential equations

JEL Classification: E40, E43, G12, G13, C61, C63

Suggested Citation

Dang, Duy-Minh and Jackson, Kenneth R. and Mohammadi, Mohammadreza, Dimension and Variance Reduction for Monte Carlo Methods for High-Dimensional Models in Finance (December 4, 2015). Available at SSRN: https://ssrn.com/abstract=2553044 or http://dx.doi.org/10.2139/ssrn.2553044

Duy-Minh Dang (Contact Author)

University of Queensland - School of Mathematics and Physics ( email )

Priestly Building
St Lucia
Brisbane, Queesland 4067
Australia

HOME PAGE: http://people.smp.uq.edu.au/Duy-MinhDang/

Kenneth R. Jackson

University of Toronto - Department of Computer Science ( email )

Sandford Fleming Building
10 King's College Road, Room 3302
Toronto, Ontario M5S 3G4
Canada

Mohammadreza Mohammadi

University of Toronto - Department of Statistics ( email )

100 St. George St.
Toronto, Ontario M5S 3G3
Canada

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