Multilevel Monte Carlo combined with numerical smoothing for robust and efficient option pricing and density estimation

The multilevel Monte Carlo (MLMC) is a highly efficient approach to estimate expectations of a functional of a solution to a stochastic differential equation. However, MLMC estimators may be unstable and have a nonoptimal complexity in case of low regularity of the observable. To overcome this issue, we extend our idea of numerical smoothing, introduced in our previous work (ArXiv abs/2111.01874 (2021)) in the context of deterministic quadrature methods, to the MLMC setting. The numerical smoothing technique is based on root finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. Motivated by option pricing and density estimation problems, our analysis and numerical experiments show that the employed numerical smoothing significantly improves the complexity and robustness (making the kurtosis at deep levels bounded) of the MLMC method. In particular, the smoothness theorem presented in our previous work (ArXiv abs/2111.01874 (2021)) enables us to recover the MLMC complexities obtained for Lipschitz functionals. Moreover, our approach efficiently estimates density functions, a task that previous methods based on Monte Carlo or MLMC fail to achieve at least in moderate to high dimensions. Finally, our approach is generic and can be applied to solve a broad class of problems, particularly for approximating distribution functions, financial Greeks computation, and risk estimation