The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options
type: Item
instance of: scholarly article
title: The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options
zbMATH Open document ID: 0837.60076
author: Angelos Dassios
published in: The Annals of Applied Probability
publication date: 1995-12-11T00:00:00Z
review text: 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.
reviewed by: Martin Schweizer
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 811742
zbMATH Keywords: quantiles of Brownian motion with drift
zbMATH Keywords: path-dependent options
zbMATH Keywords: Brownian motion
zbMATH Keywords: Feynman-Kac formula
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1177004770
OpenAlex ID: W2160763071
DOI: 10.1214/AOAP/1177004770
description: scientific article; zbMATH DE number 811742