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.
