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

TL;DR

This paper analyzes the dependence structure between a Wiener process and its running extrema, deriving joint distributions and copulas, with applications to double barrier option pricing.

Contribution

It provides analytical formulas for the joint distribution and copula of a Wiener process with its maxima and minima, and applies these to option pricing.

Findings

Derived joint distribution functions for Wiener process and extrema.

Formulated copulas capturing dependence structure.

Applied results to double barrier option pricing.

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 $m_t=\inf \{W_s: s \in [0,t]\}$. 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.