Numerical smoothing and hierarchical approximations for efficient option pricing and density estimation

When approximating the expectation of a functional of a certain stochastic process, the efficiency and performance of deterministic quadrature methods, and hierarchical variance reduction methods such as multilevel Monte Carlo (MLMC), is highly deteriorated in different ways 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 aforementioned methods. 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. We prove that under appropriate conditions, the resulting function of the remaining variables is highly smooth, potentially allowing a higher efficiency of adaptive sparse grids quadrature (ASGQ), in particular when combined with hierarchical representations to treat the high dimensionality effectively. Our study is motivated by option pricing problems and our main focus is on dynamics where a discretization of the asset price is needed. Our analysis and numerical experiments illustrate the advantage of combining numerical smoothing with ASGQ compared to the Monte Carlo method. Furthermore, we demonstrate how numerical smoothing significantly reduces the kurtosis at the deep levels of MLMC, and also improves the strong convergence rate. Given a pre-selected tolerance, , this results in an improvement of the complexity from in the standard case to . Finally, we show how our numerical smoothing enables MLMC to estimate density functions, which standard MLMC (without smoothing) fails to achieve.

C. Bayer gratefully acknowledges support from the German Research Foundation (DFG), via the Cluster of Excellence MATH+ (project AA4-2) and the individual grant BA5484/1. This work was supported by the KAUST Office of Sponsored Research (OSR) under Award No. URF/1/2584-01-01 and the Alexander von Humboldt Foundation.



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