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

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(h2), 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.