Fast Risk Estimation through Fourier Transform Based Multilevel Monte Carlo Simulation

Abstract

In this paper, we consider a problem of estimating a large loss probability of financial derivatives portfolio, which are commonly modeled as nested expectations. However, the cost of nested simulation may be too expensive and thus multilevel Monte Carlo (MLMC) method is recently used to reduce the nested simulation complexity. When using antithetic MLMC to solve the indicator function, we get the complexity of O(e-5^2). To decrease the computational burden, we use a Fourier transform method to modify the form of indicator function. The new estimator is sufficiently smooth and enables the antithetic MLMC method to achieve a better complexity. In addtion, we combine quasi-Monte Carlo (QMC) with MLMC to reduce the variance of inner estimator. Numerical results show that using the Fourier transform method in both MLMC and MLQMC can attain the optimal complexity О (e-2).