  We discuss in this note applications of the Multidimensional Positive
Definite Advection Transport Algorithm (MPDATA) to numerical solutions of
partial differential equations arising from stochastic models in quantitative
finance. In particular, we develop a framework for solving Black-Scholes-type
equations by first transforming them into advection-diffusion problems, and
numerically integrating using an iterative explicit finite-difference approach,
in which the Fickian term is represented as an additional advective term. We
discuss the correspondence between transport phenomena and financial models,
uncovering the possibility of expressing the no-arbitrage principle as a
conservation law. We depict second-order accuracy in time and space of the
embraced numerical scheme. This is done in a convergence analysis comparing
MPDATA numerical solutions with classic Black-Scholes analytical formulae for
the valuation of European options. We demonstrate in addition a way of applying
MPDATA to solve the free boundary problem (leading to a linear complementarity
problem) for the valuation of American options. We finally comment on the
potential the MPDATA framework has with respect to being applied in tandem with
more complex models typically used in quantitive finance.
