# A Multi-Level Dimension Reduction Monte-Carlo Method for Jump-Diffusion Models

31 Pages Posted: 1 Mar 2015 Last revised: 6 Sep 2017

See all articles by Duy-Minh Dang

## Duy-Minh Dang

University of Queensland - School of Mathematics and Physics

Date Written: March 1, 2015

### Abstract

This paper develops and analyses convergence properties of a novel multi-level Monte-Carlo (mlMC) method for computing prices and hedging parameters of plain-vanilla European options under a very general $b$-dimensional jump-diffusion model, where $b$ is arbitrary. The model includes stochastic variance and multi-factor Gaussian interest short rate(s). The proposed mlMC method is built upon (i) the powerful dimension and variance reduction approach developed in Dang et. al 2015 for jump-diffusion models, which, for certain jump distributions, reduces the dimensions of the problem from $b$ to $1$, namely the variance factor, and (ii) the highly effective multi-level MC approach of Giles 2008 applied to that factor. As a result, the proposed multi-level approach avoids potential difficulties associated with the standard multi-level approach in effectively handling simultaneously both multi-dimensionality and jumps. Numerical analysis shows that, when a time discretization scheme having first-order strong convergence, such as the Lamperti-Backward-Euler or Milstein schemes, is used in Step (ii), the proposed MC method requires only an overall complexity $\mathcal{O}(\epsilon^{-2})$ to achieve a root-mean-square error of $\epsilon$. Numerical results illustrating the convergence properties and efficiency of the method with jump sizes following normal and double-exponential distributions are presented.

Keywords: Monte Carlo, variance reduction, dimension reduction, multi-level, jump-diffusions, Lamperti-Backward-Euler, Milstein

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

Suggested Citation

Dang, Duy-Minh, A Multi-Level Dimension Reduction Monte-Carlo Method for Jump-Diffusion Models (March 1, 2015). Available at SSRN: https://ssrn.com/abstract=2571525 or http://dx.doi.org/10.2139/ssrn.2571525

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