  In a multi-dimensional diffusion framework, the price of a financial
derivative can be expressed as an iterated conditional expectation, where the
inner conditional expectation conditions on the future of an auxiliary process
that enters into the dynamics for the spot. Inspired by results from non-linear
filtering theory, we show that this inner conditional expectation solves a
backward SPDE (a so-called 'conditional Feynman-Kac formula'), thereby
establishing a connection between SPDE and derivative pricing theory. Unlike
situations considered previously in the literature, the problem at hand
requires conditioning on a backward filtration generated by the noise of the
auxiliary process and enlarged by its terminal value, leading us to search for
a backward Brownian motion in this filtration. This adds an additional source
of irregularity to the associated SPDE which must be tackled with new
techniques. Moreover, through the conditional Feynman-Kac formula, we establish
an alternative class of so-called mixed Monte-Carlo PDE numerical methods for
pricing financial derivatives. Finally, we provide a simple demonstration of
this method by pricing a European put option.
