Reducing time-dependent multifactor Black and Scholes equation with knock-out features to equivalent time-constant coefficient equation, and applications

TL;DR

This paper demonstrates that a multifactor Black and Scholes equation with time-dependent coefficients and knock-out boundary conditions can be transformed into an equivalent equation with constant coefficients, simplifying analysis and computation.

Contribution

It introduces a novel reduction technique for time-dependent PDEs with boundary conditions to time-constant coefficient equations using functional analysis methods.

Findings

Reduction of complex PDEs to simpler forms with constant coefficients

Application of the Hille-Yosida approximation technique in this context

Establishment of operator identities for unbounded, commuting operators

Abstract

We consider the multifactor Black and Scholes equation with time-dependent coefficients, and a knock-out feature contingent on the underlying asset values reaching a limit (reflected by a Dirichlet condition on the boundary). We prove that this equation, which has important applications in finance and insurance, can be reduced to an equivalent time-constant coefficient equation, with coefficients defined as averages of the original ones. Equivalent results are also valid for general second order parabolic equations, with applications in other fields in the natural sciences. The result established in this article has not been documented so far in the presence of boundary conditions. The proof is provided in a general framework, as it invokes techniques from the Functional Analysis theory, namely the Hille-Yosida approximation technique. A main ingredient in the proof is establishing the Exp(-A1 - A2)= Exp(-A1) Exp(- A2) identity, proved here in a general setting for unbounded, commuting, monotone and maximal operators A1, A2. This would in particular allow generalization of the result to more general classes of evolution problems with time dependent generators.