Yet another differentiation trap

In my recent posts on identifying an implied flat volatility, it turned out that numerical differentiation leads to stability problems when noise is comparably high and vega is low. But, to be quite honest, tt was a toy example, because typically it should not be too difficult to identify a single volatality number (maybe it is just reading from your Bloomberg screen).

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.
We will discuss this more detailed next week.