Derivative asset analysis in models with level-dependent and stochastic volatility

In this survey we discuss models with level-dependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical Black-Scholes model; they have been developed in an effort to build models that are flexible enough to cope with the known deficits of the classical Black-Scholes model. We start by briefly recalling the standard theory for pricing and hedging derivatives in complete frictionless markets and the classical Black-Scholes model. After a review of the known empirical contradictions to the classical Black-Scholes model we consider models with level-dependent volatility. Most of this survey is devoted to derivative asset analysis in stochastic volatility models. We discuss several recent developments in the theory of derivative pricing under incompleteness in the context of stochastic volatility models and review analytical and numerical approaches to the actual computation of option values.