# Local volatility Formula and How To use it

I'm new to Volatility Modelling, so the content of this question may be completely wrong and th question naive. I'm reading "The volatility surface" by Gatheral. I'm trying to get a sense of the first chapters about Local Volatility models.

For the moment I'm trying to get the intuition since the math proved to be a bit tough. My general intuition failed when I saw this formula to compute the Local Volatility surface from market prices.

$$\sigma ^{2}\left ( K,T,S_{0}\right ) = \frac{ \frac{\partial C}{\partial T}}{ \frac{1}{2}K^{2}\frac{\partial^2 C}{\partial K^2}}$$

While at the beginning of the model, we were looking for $$\sigma \left ( S_{t},t;S_{0}\right )$$ that had to be used in

$$\frac{\mathrm{d}S }{S} = \mu_{t} dt + \sigma(S_{t},t;S_{0})dZ$$

So now I'm a bit puzzled. Here is my general intuition about Local Volatility and Implied Volatility (surfaces). I hope that this helps in understanding the flaws in my reasoning.

So, ImpliedVolatiltiy Surface is a mapping from $$(K,T)$$ to ImpVol such that ImpVol reflects the current options market price when used in a model $$(BS)$$. Therefore it is nothing else than a way to express option prices in terms of the corresponding volatility implied by their market price in the BS model. Till here, everything is quite clear.

On the other hand, LocalVolatiltySurface is a map from $$(S,T)$$ to Vol. Where Vol is used in a numerical method (Monte Carlo) to simulate at each step the corresponding level of volatility that has to used to simulate the next step (in this sense, when plotted the local volatility surface should be plotted on the space $$(S,T,Vol)$$ and not $$(K,T,Vol)$$.

Or in theoretical terms, it is the instantaneous volatility of the price diffusion process for each level of $$S$$ and $$T$$. In other terms $$\sigma(S_{t},t;S_{0})$$. Moreover, such mapping is consistent with the current market expectation of volatility extrapolated from the Implied Volatiltiy Surface.

I don't see then how $$\sigma ^{2}\left ( K,T,S_{0}\right )$$ can be dependent from $$K$$, since $$dS$$ should not depend on the strike price of the option.

• This previous question may be useful quant.stackexchange.com/questions/16343/… especially the answer by Quantuple – Alex C Sep 22 '19 at 22:00
• So essentially we see $K$ in $\sigma ^{2}\left ( K,T,S_{0}\right )$ since we use these $Ks$ as a reference point to compute the local volatility for $S_t = K_{i,t}\; for \; i=1...C\; and\; t=0...T$ where C is the number of available strike prices in our volatility surface data, while T the available maturities? If that makes sense – Davide L. Sep 23 '19 at 8:54