On the dependence between a Wiener process and its running maxima and running minima processes

Karol DÄ browski; Piotr Jaworski

paper

On the dependence between a Wiener process and its running maxima and running minima processes

arXiv:2109.02024

Abstract

We study a triple of stochastic processes: a Wiener process $W_{t}$ , $t \geq 0$ , its running maxima process $M_{t} = sup {W_{s} : s \in [0, t]}$ and its running minima process $m_{t} = in f {W_{s} : s \in [0, t]}$ . We derive the analytical formulas for the joint distribution function and the corresponding copula. As an application we draw out an analytical formula for pricing double barrier options.