A multi-level dimension reduction Monte-Carlo method for jump–diffusion models

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. (2017) 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. Using the first-order strong convergence Lamperti–Backward-Euler scheme, we develop a multi-level estimator with variance convergence rate O ( h 2 ) , resulting in an overall complexity O ( ϵ − 2 ) to achieve a root-mean-square error of ϵ . The proposed mlMC can also avoid potential difficulties associated with the standard multi-level approach in effectively handling simultaneously both multi-dimensionality and jumps, especially in computing hedging parameters. Furthermore, it is considerably more effective than existing mlMC methods, thanks to a significant variance reduction associated with the dimension reduction. Numerical results illustrating the convergence properties and efficiency of the method with jump sizes following normal and double-exponential distributions are presented.