  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.
