Double-barrier first-passage times of jump-diffusion processes

Abstract

Abstract. Required in a wide range of applications in, e.g., finance, engineering, and physics, first-passage time problems have attracted considerable interest over the past decades. Since analytical solutions often do not exist, one strand of research focuses on fast and accurate numerical techniques. In this paper, we present an efficient and unbiased Monte-Carlo simulation to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution; extending single-barrier results by [Journal of Derivatives 10 (2002), 43–54]. In mathematical finance, the double-barrier first-passage time is required to price exotic derivatives, for example corridor bonus certificates, (step) double barrier options, or digital first-touch options, that depend on whether or not the underlying asset price exceeds certain threshold levels. Furthermore, it is relevant in structural credit risk models if one considers two exit events, e.g., default and early repayment.

Piecewise linear double barrier options

“…As quickly expected, the higher volatility is, the more terms are needed. The numerical stability with finite terms for approximation is reported in several pieces of research (e.g., Buchen & Konstandatos, 2009;Fernandez et al, 2013;Guillaume, 2010;Konstandatos, 2018;Kunitomo & Ikeda, 1992;Sidenius, 1998;Wang et al, 2021). Specifically, Buchen and Konstandatos (2009) argue that only k = ±1 i terms beyond the dominant k = 0 i terms are necessary to achieve numerical convergence.…”

The double barrier problem for Brownian motion with Poissonian resetting

“…For the study of 'run and tumble mobiles' under resetting see [54], where it is proven that velocity randomization improves position-only resetting. Finally for other related problems-like double-barrier first escape times of jump-diffusion-processes see [13,55] where discontinuous processes with finite jumps are considered.…”

A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

“…It is shown, indeed, that some of these representations reveal to be particularly useful for the determination of the closed form of first-exit-time (FET) from an open set confined between two boundaries. A large number of papers is devoted to investigate first passage problem of diffusion processes restricted between two boundaries, in the past but also recently, (see, for instance, [29][30][31][32][33][34][35][36][37][38]), also for possible applications ( [39,40]). Here, we focus on Gauss-Markov processes between Daniels-type boundaries ( [25]) for which the closed form can be derived also by the proposed symmetry approach.…”