The local volatility first developed by Bruno Dupire in 1994 was an attempt to model the volatility smile skew using actual prices observed in the market. It has been widely adopted because of the simplicity of use. However the dynamics of the moves in the smiles are very poorly reproduced by a Dupire Local volatility model, which required the development of a more sophisticated volatility model. Stochastic modelling of the volatility explains much better the dynamic of the volatility smile in a self-consistent way, but is self-burdened by its complexity.

## The case for stochastic modelling of volatility

Stochastic volatility modelling starts with the observation that volatility is mean-reverting, somewhat auto-correlated (we observe clustering in historical volatility), follows a distribution that would be close in shape to a normal distribution.

In a Stochastic volatility model, we will model both the underlying price and its volatility, as they are assumed to take each other as input and to be linked to correlated Brownian variables.

A typical formulation of a stochastic model is made of three stochastic differential equations

**The underlying : $dS(t)= μ(t)×S(t)dt + √{v(t)} ×S(t) dW1$**
**The volatility : $dv(t)= ∝(S,v,t)dt+ n × β(S,v,t)× √(v(t) ) dW2 $**
**The correlated Wiener processes: $dW1 dW2= ρdt$**

Where:

- μ(t) is the drift parameter of the underlying price
- The parameter ρ is the correlation between the stock price returns and the changes in volatility.
- The parameter n is the volatility of the volatility
- W1 and W2 are Wiener processes

## The case for local volatility

A local volatility model can be seen as a useful simplification of the stochastic volatility model, where the volatility is simply a function of the stock price and time horizon. This function is set-up to be consistent with all observed current vanilla option market prices on a specific underlying. This volatility function has to be calibrated on the available options on the market.