Modelling the Dependence between a Wiener Process and Its Running Maxima and Running Minima Processes

Karol Da̧browski and Piotr Jaworski ()
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Karol Da̧browski: Institute of Mathematics, University of Warsaw, 02-097 Warszawa, Poland
Piotr Jaworski: Institute of Mathematics, University of Warsaw, 02-097 Warszawa, Poland

Mathematics, 2024, vol. 12, issue 17, 1-27

Abstract: We study a triple of stochastic processes: a Wiener process W t , t ≥ 0 , its running maxima process M t = sup { W s : s ∈ [ 0 , t ] } , and its running minima process m t = inf { W s : 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; double-barrier options (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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