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

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.

Figures (4)

• Figure 1: Polyhedron $P$---the support of the $C_{W,M,m}$ copula.
• Figure 2: Copula $C_{W,M}$ (scatterplot).
• Figure 3: Copula $C_{W,m}$ (scatterplot).
• Figure 4: Copula $C_{M,m}$ (scatterplot).

Theorems & Definitions (27)

• Theorem 2.1
• Proposition 2.2
• Definition 3.1
• Theorem 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof