The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options
Let \(X\) be Brownian motion with a constant drift and \(M(\alpha, t)\) the \(\alpha\)-quantile of \(X\) up to time \(t\), i.e., the smallest value \(x\) for which the time spent by \(X\) below \(x\) before \(t\) exceeds \(\alpha t\). The main result of the paper is that \(M(\alpha, t)\) has the same distribution as the sum of the maximum of \(X\) up to time \(\alpha t\) and the minimum of an independent copy of \(X\) up to time \((1- \alpha) t\). This is proved analytically by computing the Laplace transform of \(M(\alpha, t)\) with the help of the Feynman-Kac formula. The same identity in law has recently been proved directly by \textit{P. Embrechts}, \textit{L. C. G. Rogers} and \textit{M. Yor} [ibid. 5, No. 3, 757-767 (1995)], using more probabilistic arguments. As an application of the basic result, the author recovers a formula by Miura and Akahori for the price of a path- dependent financial option.
scientific article; zbMATH DE number 811742