On Stochastic PDEs for the pricing of derivatives in a multi-dimensional diffusion framework

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. The benefits of this representation are potentially significant and of both theoretical and practical interest. In particular, this representation leads to an alternative class of so-called mixed Monte-Carlo / PDE numerical methods.