Fourier-cosine method for Gerber–Shiu functions

Abstract

h i g h l i g h t s• An efficient approximant for the Gerber-Shiu functions is proposed.• The approximation is based on the Fourier-cosine method.• The approximation is of linear computational complexity. • An explicit error bound is provided for functions satisfying some mild technical conditions. a b s t r a c tIn this article, we provide a systematic study on effectively approximating the Gerber-Shiu functions, which is a hardly touched topic in the current literature, by incorporating the recently popular Fouriercosine method. Fourier-cosine method has been a prevailing numerical method in option pricing theory since the work of Fang and Oosterlee (2009). Our approximant of Gerber-Shiu functions under Lévy subordinator model has O(n) computational complexity in comparison with that of O(n log n) via the fast Fourier transform algorithm. Also, for Gerber-Shiu functions within our proposed refined Sobolev space, we introduce an explicit error bound, which seems to be absent from the literature. In contrast with our previous work (Chau et al., 2015), this error bound is more conservative without making heavy assumptions on the Fourier transform of the Gerber-Shiu function. The effectiveness of our result will be further demonstrated in the numerical studies.

Approximating the Density of the Time to Ruin via Fourier-Cosine Series Expansion

“…The COS method can be easily used as long as the corresponding Fourier transform (or characteristic function) is available. Because the Fourier transforms of the ruin probability and Gerber-Shiu function are easily obtained, Chau et al (2015aChau et al ( , 2015b capitalized on the COS method to compute the ultimate ruin probability and Gerber-Shiu function in a class of Lévy risk models. The Fourier transform method has also been used by Zhang andYang (2013, 2014) to estimate the ruin probability in the Lévy risk model, where the FFT algorithm is used for computation.…”

Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion

“…Except for option pricing, this method has been adopted in insurance ruin theory. For example, Chau et al [25,26] used the 1D COS method to compute the ruin probability and the expected discounted penalty function; Zhang [27] approximated the density function of the time to ruin by both 1D and 2D COS methods; Yang et al [28] proposed a nonparametric estimator for the deficit at ruin by the 2D COS method; Wang et al [29] and Huang et al [30] used the 1D COS method to estimate the expected discounted penalty function under some risk models with stochastic premium income. The COS method has also been used by some authors to value variable annuities.…”

Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model

“…Biffis and Morales [5] generalized the Gerber-Shiu function to pathdependent penalties. Chau et al [6] used the Fourier-cosine method to evaluate the Gerber-Shiu function. For more studies on Gerber-Shiu function, the interested readers are referred to Yin and Wang [7,8], Asmussen and Albrecher [9], Chi [10], Wang et al [11], Chi and Lin [12], Zhao and Yin [13,14], Shen et al [15], Yu [16][17][18], Yin and Yuen [19,20], Zhao and Yao [21], Zheng et al [22], Huang et al [23], Li et al [24], Zhang et al [25], Yu et al [26], Zeng et al [27,28], Li et al [29], and Dong et al [30].…”