Valuing Guaranteed Minimum Death Benefits by Complex Fourier Series Expansion

In this paper, we use the complex fourier series expansion method (CFS) to price guaranteed minimum death benefits (GMDB). The main idea is to expand the Fourier series of the auxiliary function. The density function of remaining lifetime has two forms in this paper, namely combination-of-exponentials density and piecewise constant forces of mortality assumption, and the coefficients of series are estimated by using the known characteristic function of the general L\'evy model. We mainly consider the value of GMDB products under call options and put options. In the numerical experiment section, we also demonstrate the advantages of CFS in calculation accuracy and running time by comparing with cosine series expansion method (COS) and Monte Carlo method (MC).