Local Volatility - Formulation

Formulation

In mathematical finance, the assets S_t which underlie financial derivatives, are typically assumed to follow stochastic differential equations of the type

where are Brownian motions correlated as follows: . The correlation coefficients are considered constant.

where is the volatility of the FX rate and is the correlation

where r is the instantaneous risk free rate, giving an average local direction to the dynamics, and W 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 (naive) model, this instant volatility is assumed to be constant, but in reality realized volatility of an underlier actually rises and falls over time.

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 asset level S_t 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. This model is used to calculate exotic option valuations which are consistent with observed prices of vanilla options.