Numerical Smoothing with Multilevel Monte Carlo for Efficient Option Pricing and Density Estimation

Abstract

When approximating the expectation of a functional of a certain stochastic process, the robustness and performance of multilevel Monte Carlo (MLMC) method, may be highly deteriorated by the low regularity of the integrand with respect to the input parameters. To overcome this issue, a smoothing procedure is needed to uncover the available regularity and improve the performance of the MLMC estimator. In this work, we consider cases where we cannot perform an analytic smoothing. Thus, we introduce a novel numerical smoothing technique based on root-finding combined with a one dimensional integration with respect to a single well-chosen variable. Our study is motivated by option pricing problems and our main focus is on dynamics where a discretization of the asset price is needed. Through our analysis and numerical experiments, we demonstrate how numerical smoothing significantly reduces the kurtosis at the deep levels of MLMC, and also improves the strong convergence rate, when using Euler scheme. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, $\text{TOL}$, this results in an improvement of the complexity from $\mathcal{O}\left(\text{TOL}^{-2.5}\right)$ in the standard case to $\mathcal{O}\left(\text{TOL}^{-2} \log(\text{TOL})^2\right)$. Moreover, we show how our numerical smoothing combined with MLMC enables us also to estimate density functions, which standard MLMC (without smoothing) fails to achieve.