In the case of local volatility, the situation gets more interesting. You would have implied volatility information derpending on the strike K and on the expiry T and would like to recover a local volatility surface depending on the spot price S and on time t.
Bruno Dupire developed his famous inversion formula in 1994. It reads as
C(K,T) is the call price function obtained from the implied volatilities. The ingredients to obtain the local volatility are:
- (first order) differentiation by T
- (first order) differentiation by K
- (second order) differentaition by K
- divide by second derivative.