I want to compute Dupire's local volatility, but I'm struggling since several days. Here is the formula to get the local variance, with $y=\ln \left(\frac{ K}{F} \right)$ and $w=\sigma_{BS}^2\,T$, and I get $\sigma_{BS}$ from $\tilde{BS}^{-1}$ : $v_{l} = \frac{ \frac{\partial w}{\partial T} }{\left[1 - \frac{y}{w} \frac{\partial w}{\partial y}+ \frac{1}{2}\frac{\partial^2 w}{\partial y^2}+ \frac{1}{4}\left( - \frac{1}{4} + \frac{1}{w} + \frac{y^2}{w^2} \right) \left(\frac{\partial w}{\partial y}\right)^2 \right]}$
Sorry for the stupid question, but I have different strikes for the same maturity, so do I have to calculate $\frac{\partial w}{\partial T}$ by doing $\frac{w_{2}-w_{1}}{T_2-T_1}$ for the first one while knowing that I also have to calculate $\frac{\partial w}{\partial y}$ and so should I take the $y$ from the different period too? Or do I take a more or less arbitrary $h$ and calculate $\frac{w_{T+h}-w_{T}}{h}$ and do the same with $y$ ?
Thanks for the help.
