Complete specification of some partial differential equations that arise in financial mathematics

Abstract

We consider some well-known partial differential equations that arise in Financial Mathematics, namely the BlackScholesMerton, Longstaff, Vasicek, CoxIngersollRoss and Heath equations. Our central aim is to discover any underlying connections taking into account the Lie remarkability property of the heat equation. For a few of these equations there is a known connection with the heat equation through a coordinate transformation. We investigate further that connection with the help of modern group analysis. This is realized by obtaining the Lie point symmetries of these equations and comparing their algebras with that of the heat equation. For those with an algebra identical to that of the heat equation a systematic way is shown to obtain the coordinate transformation that links them: the Lie remarkability property is a direct consequence. For the rest this is achieved only in certain subcases. © 2009 The Author(s).