Computing the Gerber–Shiu function by frame duality projection

Abstract

Inspired by some works of Kirkby, J. L. [2015. Efficient option pricing by frame duality with the fast Fourier transform. SIAM Journal on Financial Mathematics 6(1), 713–747; 2016. An efficient transform method for Asian option pricing. SIAM Journal on Financial Mathematics 7(1), 845–892], we present a systematic study on effectively computing the Gerber–Shiu function in the Lévy risk model, where the frame duality projection is used for approximation. By introducing an auxiliary function, we provide a smooth extension of the Gerber–Shiu function, which has closed-form Fourier transform and is differentiable over the whole real line under some conditions. The objective function is approximated by its frame duality projection onto a Riesz basis, and the projection coefficients are readily computed by the fast Fourier transform algorithm. Error analysis is made and the effectiveness of our results will be further illustrated in the numerical experiments.

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