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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.

quezac
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  • what does EDP stand for – develarist Dec 04 '19 at 01:44
  • PDE, I forgot about it, sorry. – quezac Dec 04 '19 at 06:39
  • 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 04 '19 at 07:51
  • 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 06 '19 at 08:31

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First a comment, as discussed here the (correct) expression you have stated here is not the one stated and derived in Rouah.

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.

Andrew Jacobs
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  • Thx but as it’s currently written, your answer is unclear. Please edit it to add additional details that will help others understand how this addresses the question asked. Otherwise, if you wished to simply comment, pls wait until you have enough reputation points and don't put in a comment as an answer. – Alper Aug 01 '22 at 11:51
  • Thanks, edited. – Andrew Jacobs Aug 01 '22 at 21:10