Valuing Guaranteed Minimum Death Benefits by Complex Fourier Series Expansion

In this paper, a two-dimensional compound Fourier series expansion (2D-CFS) is used to price the guaranteed minimum death benefits (2D-GMDB) under the influence of two logarithmic assets. The main idea is to model the dynamic price process of risk asset (can be equity funds, or mutual funds) as the index Levy process, which is combined with the density function of the remaining life to construct the auxiliary function, and the constructed auxiliary function that satisfies specific conditions is expanded by a two-dimensional complex Fourier series. In the calculation of the series coefficients, this paper mainly considers two forms of remaining lifetime density functions, namely combination-of-exponentials density and piecewise constant forces of mortality assumption, and uses the characteristic index of the known logarithmic asset model to calculate the progression coefficient. For the choice of logarithmic asset model, we selected geometric Brownian motion (GBM) and jump diffusion process (Jump) to simulate the logarithmic asset process. In the numerical experiment section, we consider the pricing problem of the 2D-CFS method under the exchange option, the maximum option, the minimum option and the geometric option, and the results are compared with the two-dimensional cosine series expansion (2D-COS) method, and the results show that the 2D-CFS method is significantly better than the 2D-COS method in terms of convergence speed and calculation accuracy.