# How to get the local volatility from IV surface?

I have to work on Dupire's model.

If I understand Fengler's paper well enough we can get the local volatility from implied volatility smoothed surface because if not it would look all bumpy like the graphic on the right page 35, and that's not what I have when I use the formulas detailed and approximated in this paper (Kotze et al: Implied and Local Volatility Surfaces for South African Index and Foreign Exchange Options).

So I smoothed it using a non-parametric regression, it's decent according to this paper (Wu, Liu: Curve-Fitting Method for Implied Volatility), and then I don't know what to do.

First thing I absolutely don't understand is why we could use $$\sigma_{1}(t)\sigma_{2}(S)$$ to regularize it and get a function then, instead of $$\sigma(S, t)$$ (I saw it really fast on a board so I may be wrong). And then, how do we get the vol surface from the implied volatility surface ? (I'm not asking for the full procedure, but for a few tips, it's quite hard to understand everything as a beginner).

Thanks.

• This other discussion may help as well link – H.L. Mar 5 at 14:46

$$\sigma^2 \left(T,y\right)=\frac{\frac{\partial w}{\partial T}}{1 -\frac{ y}{w} \frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(\frac{ y^2}{w^2}-\frac{1}{w}-\frac{1}{4}\right)\left( \frac{\partial w}{\partial y}\right)^2}$$
Where y is the money-ness, defined as $$y=\ln \left(\frac{ K}{F} \right)$$, and w is the transformation of Black Scholes implied vol $$w=\sigma_{BS}^2\,T$$
• You can further conditions in the regularisation if you like. Also did you try interpolating $\sigma ^2 times T$? – Magic is in the chain Dec 15 '19 at 23:08