Some Applications of First-Passage Ideas to Finance

Abstract

Capital Fund Management, 23-25, rue de l'Université, 75 007 Paris Some applications of first-passage ideas to finance 5

Bounds on Mean Exit Time for Purely Time-Dependent Drift and Diffusion

“…In structural dynamics, failure of a mechanical system due to random vibrations is a first passage problem [16]. The optimum time to sell an asset is an example of a first exit time problem in finance [8]. In biology, the persistence time, or extinction time, is a first exit time problem [1,5,6].…”

Fast calculation of boundary crossing probabilities for Poisson processes

“…Finally, since Brownian motion can be described as a limit of a Poisson process, one may apply our method to approximate the boundary crossing probability and first passage time of a Brownian motion, see for example Khmaladze and Shinjikashvili [8]. The latter quantity has multiple applications in finance and statistics (Siegmund [16], Chicheportiche and Bouchaud [17]). In this case an accurate approximation may require a fine discretization of the continuous boundaries, or equivalently a large value of n. Hence, fast algorithms are needed.…”

First detection of threshold crossing events under intermittent sensing

“…The first occurrence of breakdown of an engineered structure, triggering of bio-chemical reactions, or a stock market index reaching a specific value are all examples that can be posed as first passage problems (FPP). This class of problems has been extensively studied in statistical physics for many decades [1][2][3], and has a large number of applications in many areas of physical sciences [1,[4][5][6], engineering [3,[7][8][9], finance [10][11][12] and biology [13][14][15]. In its simplest formulation, the FPP is solved under an implicit assumption of perfect detection conditions -the notional sensor, which monitors the first occurrence of a threshold crossing event, is active at all the times.…”

“…These rates were previously used to model threshold crossing processes [30] in the context of triggering of biochemical reactions [29]. Figure 2 shows the FDTD for this process with parameters n 0 = 0, N = 20, λ = 0.1 and k = 1, for threshold values m = 3, 5 and 7, and (α, β) = (1, 1), (1,10) and (10,1). This figure shows analytical results (solid lines) for which inverse Laplace transform of Eq.…”