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

Abstract

We present new numerical schemes for pricing perpetual Bermudan and American options as well as α-quantile options. This includes a new direct calculation of the optimal exercise boundary for earlyexercise options. Our approach is based on the Spitzer identities for general Lévy processes and on the Wiener-Hopf method. Our direct calculation of the price of α-quantile options combines for the first time the Dassios-Port-Wendel identity and the Spitzer identities for the extrema of processes. Our results show that the new pricing methods provide excellent error convergence with respect to computational time when implemented with a range of Lévy processes.

Efficient and Robust Combinatorial Option Pricing Algorithms on the Trinomial Lattice for Polynomial and Barrier Options

“…Nevertheless, Ritchken [19] demonstrated that the trinomial lattice adopted in this study outperforms Bolye and Lau's [24] method in terms of a faster convergence rate. In addition, the exponential error convergence rate of Fourier transform methods has led to their wide adoption in recent derivative pricing literature, such as discretely monitored barrier options [25,26], continuously monitored barrier options [27], α-quantile and perpetual early exercise options [28], and Asian options [29]. Although lattice methods have slow error convergence rates as in Talponen and Turunen [30], their flexibility has led to their wide adoption in recent literature, as mentioned in footnote 1.).…”

Solution of Wiener-Hopf and Fredholm integral equations by fast Hilbert and Fourier transforms

“…The connection of the Wiener-Hopf equation with probabilistic problems was noticed by Spitzer (1957) and is discussed by Feller (1971), together with the application of Fourier transform methods to stochastic processes. More recently these equations have become of interest in finance to price discretely monitored path-dependent options like barrier, first-touch, lookback (or hindsight), quantile and Bermudan options (Fusai et al, 2006;Green et al, 2010;Fusai et al, 2012;Marazzina et al, 2012;Fusai et al, 2016;Phelan et al, 2018Phelan et al, , 2019Phelan et al, , 2020. The Wiener-Hopf method is employed also to solve a large collection of mixed boundary value problems (Duffy, 2008).…”