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

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.