  In this paper, we develop a Multilayer (ML) method for solving one-factor
parabolic equations. Our approach provides a powerful alternative to the
well-known finite difference and Monte Carlo methods. We discuss various
advantages of this approach, which judiciously combines semi-analytical and
numerical techniques and provides a fast and accurate way of finding solutions
to the corresponding equations. To introduce the core of the method, we
consider multilayer heat equations, known in physics for a relatively long time
but never used when solving financial problems. Thus, we expand the analytic
machinery of quantitative finance by augmenting it with the ML method. We
demonstrate how one can solve various problems of mathematical finance by using
our approach. Specifically, we develop efficient algorithms for pricing barrier
options for time-dependent one-factor short-rate models, such as
Black-Karasinski and Verhulst. Besides, we show how to solve the well-known
Dupire equation quickly and accurately. Numerical examples confirm that our
approach is considerably more efficient for solving the corresponding partial
differential equations than the conventional finite difference method by being
much faster and more accurate than the known alternatives.
