Fluctuation identities with continuous monitoring and their application to the pricing of barrier options

Pricing methods for α-quantile and perpetual early exercise options based on Spitzer identities

“…The purpose of damping is explained below. As shown by Lewis (2001) and comprehensively used in Fusai et al (2016) and Phelan et al (2018Phelan et al ( , 2019, the expectation can be computed in Fourier space using the Plancherel theorem,…”

“…A damping factor e α d x with a negative damping parameter α d ensures the integrability of the payoff function of a call. In a numerical algorithm the integrability results anyway from the unavoidable truncation of the x range to a finite grid where −∞ < x min ≤ l ≤ x ≤ u ≤ x max < +∞, but a damping factor with α d < 0 for a call and α d > 0 for a put also improves the numerical accuracy because it smooths the payoff and thus mitigates the Gibbs phenomenon (Phelan et al 2018(Phelan et al , 2019. In equation 3the shift of the characteristic function p(ξ + iα d , T) from ξ to ξ + iα d compensates the damping of the payoff, and one must make sure that α d is chosen such that the characteristic function remains within its analiticity strip.…”

“…The selection of the damping parameter was described in detail by Feng and Linetsky (2008) and analysed in depth by Kahl and Lord (2010). The robustness of Spitzer-based numerical pricing methods with respect to the variation of the damping parameter is shown in the supplementary material of Phelan et al (2018Phelan et al ( , 2019. The characteristic function (ξ , t) of a stochastic process in continuous time X (t) is the Fourier transform of its PDF…”

“…Similarly to the discrete-time case, the inverse transform is not generally available in closed form, so we use the numerical inverse Laplace transform by Abate and Whitt (1995), used for option pricing by Phelan et al (2018). The Laplace transform can be obtained from the z-transform setting q = e −s t , t n = n t and taking the limit for t → 0,…”

“…This relationship can be exploited to derive versions of the fluctuation identities with continuous monitoring (Baxter and Donsker 1957, Green et al 2010, Fusai et al 2016, Phelan et al 2018.…”