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 ${\displaystyle S_{t}}$ and of time ${\displaystyle t}$. As such, it is a generalisation of the Black–Scholes model, where the volatility is a constant (i.e. a trivial function of ${\displaystyle S_{t}}$ and ${\displaystyle t}$). Local volatility models are often compared with stochastic volatility models, where the instantaneous volatility is not just a function of the asset level ${\displaystyle S_{t}}$ but depends also on a new "global" randomness coming from an additional random component.