A correction note on the first passage time of an Ornstein-Uhlenbeck process to a boundary

Abstract

Abstract. This paper provides the derivation of the hitting time density of an Ornstein-Uhlenbeck process to a flat boundary. The derivation relies on a change of measure approach and delivers an explicit formula. This formula is an amended expression of the result given in Leblanc and Scaillet (1998). It corresponds to the formula given by a time substitution approach when the boundary level coincides with the mean of the invariant measure. It can for example be used to price digital up-and-in credit spread options when the logarithm of the credit spread is assumed to follow an Ornstein-Uhlenbeck process.

Tail Asymptotics of the Supremum of a Regenerative Process

“…we obtain an Ornstein-Uhlenbeck process with parameter η (see [21]). Let We now derive the behavior of the ruin probability ψ(x) as x → ∞.…”

Representations of the First Hitting Time Density of an Ornstein-Uhlenbeck Process1

“…Recently, Leblanc & Scaillet [16] and Leblanc, Renault & Scaillet [15] showed that the density can be expressed as the Laplace transform of a functional of a 3-dimensional Bessel bridge. However, the authors used therein an erroneous spatial homogeneity property for the 3-dimensional Bessel bridge, a mistake that has been noticed by several authors, including Göing-Jaeschke & Yor [6].…”

Hitting time and time change

“…where T I, K is the first passage time of the transformed process I up through K, with I t~It=s 2 {a=b, so Proof: Refer to Leblanc et al (2000). % However, Gö ing-Jaschke and Yor (2003) point out that this formula is actually correct only if K~0.…”

“…Focuses of attention have been the underlying dynamics and the range structure. As for the underlying process, Leblanc et al (2000) derive the hitting time density of an Ornstein-Uhlenbeck process to a flat boundary (a correction is needed though, as pointed out by Gö ing-Jaschke and Yor, 2003), and Kou and Wang (2003) find the first passage times for a jump-diffusion process by specifying double exponential distributions for the jump. As for the range structure, an example is the pricing of double barrier options, such as Geman and Yor (1996) who derive approximation formulas, or Kolkiewicz (2002) who provides a general valuation method for those options by utilizing a generalized version of the Lévy formula proven by Kunitomo and Ikeda (1992).…”