Local volatility

A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level and of time . As such, it is a generalisation of the Black–Scholes model, where the volatility is a constant (i.e. a trivial function of and ). Local volatility models are often compared with stochastic volatility models, where the instantaneous volatility is not just a function of the asset level but depends also on a new "global" randomness coming from an additional random component. In mathematical finance, the asset St that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form under the risk neutral measure, where is the instantaneous risk free rate, giving an average local direction to the dynamics, and is a Wiener process, representing the inflow of randomness into the dynamics. The amplitude of this randomness is measured by the instant volatility . In the simplest model i.e. the Black–Scholes model, is assumed to be constant, or at most a deterministic function of time; in reality, the realised volatility of an underlying actually varies with time and with the underlying itself. When such volatility has a randomness of its own—often described by a different equation driven by a different W—the model above is called a stochastic volatility model. And when such volatility is merely a function of the current underlying asset level St and of time t, we have a local volatility model. The local volatility model is a useful simplification of the stochastic volatility model. "Local volatility" is thus a term used in quantitative finance to denote the set of diffusion coefficients, , that are consistent with market prices for all options on a given underlying, yielding an asset price model of the type This model is used to calculate exotic option valuations which are consistent with observed prices of vanilla options.