Modelling 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 Wt, t≥0, its running maxima process Mt=sup{Ws:s∈[0,t]}, and its running minima process mt=inf{Ws:s∈[0,t]}. We derive the analytical formula for the corresponding copula and show that it is supported on the hemicube, a convex hexahedron with seven vertices. As an application, we draw out an analytical formula for pricing of a double barrier option.

Keywords

• copulas
• Wiener process
• running maxima and minima
• strong Markov property
• reflection principle
• Spearman rho