Joint distribution of Brownian motion and its maximum, with a generalization to correlated BM and applications to barrier options
type: Item
instance of: scholarly article
title: Joint distribution of Brownian motion and its maximum, with a generalization to correlated BM and applications to barrier options
zbMATH Open document ID: 0854.60084
DOI: 10.1016/0167-7152(95)00092-5
author: Chin-Shan Chuang
published in: Statistics \& Probability Letters
publication date: 1997-01-13T00:00:00Z
review text: Let \(W_t\) be a standard one-dimensional Brownian motion, \(B_t = \sigma W_t + \mu t\) a Brownian motion with drift, \(M_s = \max_{0 \leq u \leq s} B_u\). The author derives the joint distribution of \((M_s, B_t)\) for all \(s\). The explicit formula for \(P (M_s \geq a\), \(B_t \leq z)\), \(s \leq t\), is used to study the joint distribution of \((M_s^{(1)}, B_t^{(2)})\) for the so-called ``correlated Brownian motion \((B_t^{(1)}, B_t^{(2)})\) in the plane'' defined as \[ B_t^{(1)} = \sigma_1 W_t^{(1)} + \mu_1 t, \quad B_t^{(2)} = \sigma_2 (\rho W_t^{(1)} + \sqrt {1 - \rho^2} W_t^{(2)}) + \mu_2 t, \] where \(W_t^{(1)}, W_t^{(2)}\) are independent standard Brownian motions, \(M_s^{(1)} = \sup_{0\leq u\leq s}B_u^{(1)}\). These distributions are applied to study the price barrier options. Some numerical examples are given.
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G70
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 91B06
zbMATH DE Number: 896113
zbMATH Keywords: Brownian motion
zbMATH Keywords: maximum process
zbMATH Keywords: Markov property
zbMATH Keywords: joint distribution
zbMATH Keywords: barrier options
MaRDI profile type: Publication
full work available at URL: 0167-7152(95)00092-5
OpenAlex ID: W1997868088
cites work: Q3796553
cites work: Q5558293
cites work: Q3808989
cites work: Martingales and arbitrage in multiperiod securities markets
cites work: Q5661032
cites work: Q4039796
cites work: Q3994364
cites work: Sequential analysis. Tests and confidence intervals
description: scientific article; zbMATH DE number 896113