# How to derivate Dupire's local volatility?

I want to calculate the expression of local volatility expressed in terms of implied volatility given by Fabrice Douglas Rouah in Derivation of Local Volatility :

$$v_{l} = \frac{ \frac{\partial w}{\partial T} }{\left[1 - \frac{y}{w} \frac{\partial w}{\partial y}+ \frac{1}{2}\frac{\partial^2 w}{\partial y^2}+ \frac{1}{4}\left( - \frac{1}{4} + \frac{1}{w} + \frac{y^2}{w^2} \right) \left(\frac{\partial w}{\partial y}\right)^2 \right]}$$

I don't know how to calculte the EDP, should I use finite difference ? When I try with a low $$h$$ I find very high values for $$\frac{\partial^2 w}{\partial y^2}$$. Thank you for your help.

• what does EDP stand for – develarist Dec 4 '19 at 1:44
• PDE, I forgot about it, sorry. – quezac Dec 4 '19 at 6:39
• You could compute the derivatives using finite difference formulas indeed (not PDE). Except if you have a closed and arb free parametrisation of your total implied variance surface $w(y,T)$. Arb free because in that formula, you want both a positive numerator (no cal arb) and a positive denominator (no fly arb). – Quantuple Dec 4 '19 at 7:51
• Sorry for the stupid question, but I have to calculate this: I have different strikes for the same period, 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 $y$ ? – quezac Dec 6 '19 at 8:31