# How to derive 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 PDE, 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 Dec 4, 2019 at 1:44
• PDE, I forgot about it, sorry. Dec 4, 2019 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). Dec 4, 2019 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 $y$ ? Dec 6, 2019 at 8:31

As for calculating the derivatives, the best case would be if your underlying market was such that you could first calibrate a paramterisation of the implied variance, $$w(y,t)$$, such as one of Gatheral's SSVIs, perhaps with time-dependent parameters as discussed in the work on extended SSVI surfaces. In this case your surface is guaranteed to be arbitrage free in both the log-strike ($$y$$) and time dimensions and the derivatives you require for the local volatility function can be obtained analytically. In practice you can often fit such parameterisations per maturity and find that the parameters are well-behaved enough through time to allow linearly interpolating them to obtain the time derivative by finite difference - log-strike derivatives are still given analytically. Otherwise, for example if using some form of spline interpolation, you will need to resort to finite difference calculations for both the time and log-strike derivatives.