Abstract

We consider the problem of pricing path-dependent contingent claims. Classically, this problem can be cast into the Black-Scholes valuation framework through inclusion of the path-dependent variables into the state space. This leads to solving a degenerate advection-diffusion Partial Differential Equation (PDE). Standard Finite Difference (FD) methods are known to be inadequate for solving such degenerate PDE. Hence, path-dependent European claims are typically priced through Monte-Carlo simulation. To date, there is no numerical method for pricing path-dependent American claims. We first establish necessary and sufficient conditions amenable to a Lie algebraic characterization, under which degenerate diffusions can be reduced to lower-dimensional non-degenerate diffusions on a sub-manifold of the underlying asset space. We apply these results to pathdependent options. Then, we describe a new numerical technique, called Forward Shooting Grid (FSG) method, that efficiently copes with de...

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