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Computing Dupire Local Volatility from Implied Variance Surfaces

Article Quant Q&A · Author: quezac
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Summary

The document asks how to compute Dupire local volatility from an implied variance surface, focusing on the required derivatives with respect to maturity and log strike. The questioner reports that finite differences at small step sizes produce very large second derivatives, and the answer also flags a possible discrepancy between the stated expression and the cited derivation.

The recommended approach is to fit an arbitrage-free parameterization of implied variance, such as SSVI or an extended SSVI surface. This can provide analytic log-strike derivatives; parameters fitted by maturity may be interpolated over time, with a finite difference used for the time derivative. If the surface is represented by splines or another form without convenient analytic derivatives, finite differences may be needed for both dimensions. The answer gives practical guidance but no market data, numerical comparison, or universal rule for choosing a fit.

Key ideas

  • Dupire local volatility requires maturity and log-strike derivatives of implied variance.
  • An arbitrage-free implied variance parameterization can provide analytic log-strike derivatives.
  • Interpolating fitted surface parameters across maturities can support a finite-difference time derivative.
  • Spline-based surfaces may require finite differences for both time and log-strike derivatives.
  • The stated formula may differ from the cited derivation, so the expression should be checked.

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# How to derive Dupire's local volatility?


# 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.

## Answer by Andrew Jacobs (score 1)

https://quant.stackexchange.com/a/71783

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.