Path-dependent volatility and the preservation of PDEs

Abstract

The classical theory of risk neutral derivative pricing relies on the underlying market model being Markovian and complete. We present the theory of stochastic di erential equations relevant to risk neutral pricing, with a particular focus on the Markov property and its links to partial di erential equations. We demonstrate when this classical theory can still be applied to derivative pricing in models with path dependent volatility. A link between these models and the local volatility framework is derived via the representation of local volatility as the conditional expectation of some, more complicated, process. Julien Guyon used this link as a tool in tting a large class of models to the market. We will propose a tted, complete and Markovian market model, which incorporates past asset levels in future volatility levels. The numerical implementation of such a model is addressed through a Monte Carlo scheme incorporating Guyon's particle method, as well as a nite difference scheme.