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We consider the multifactor Black and Scholes equation with time-dependentcoefficients, and a knock-out feature contingent on the underlying asset valuesreaching a limit (reflected by a Dirichlet condition on the boundary). We provethat this equation, which has important applications in finance and insurance,can be reduced to an equivalent time-constant coefficient equation, withcoefficients defined as averages of the original ones. Equivalent results arealso valid for general second order parabolic equations, with applications inother fields in the natural sciences. The result established in this articlehas not been documented so far in the presence of boundary conditions. Theproof is provided in a general framework, as it invokes techniques from theFunctional Analysis theory, namely the Hille-Yosida approximation technique. Amain 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 allowgeneralization of the result to more general classes of evolution problems withtime dependent generators.

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