Strike and Time Interpolation for Dupire Local Volatility

Summary

The document asks whether implied volatility should be interpolated across both strike and time when deriving a local volatility surface with the Dupire method. It presents a formula in which local volatility depends on the implied volatility surface and its derivatives with respect to maturity and strike, including the second strike derivative.

That dependence means the strike dimension matters to the calculation: the method needs a sufficiently smooth implied volatility representation to estimate those derivatives. The question contrasts this with a cited claim that interpolation should occur only across time, on the grounds that volatility at different strikes is not connected. No answer or empirical evidence is included, so the document does not resolve the practical interpolation choice or explain the cited claim. It also leaves implementation details such as smoothing, arbitrage constraints, and numerical stability open.

Key ideas

• Dupire local volatility depends on implied volatility and its strike and maturity derivatives.
• The presented formula includes both first and second derivatives with respect to strike.
• The document asks whether implied volatility should be interpolated across strike as well as time.
• No answer or evidence is supplied to settle the interpolation question.

Tags

# When getting the local vol surface from the implied vol surface, do we interpolate the strikes?


# When getting the local vol surface from the implied vol surface, do we interpolate the strikes?












Using the dupire method: $$\sigma(T, K)=\sqrt{\frac{\sigma_{\mathrm{imp}}^2+2 \sigma_{\mathrm{imp}} T\left(\frac{\partial \sigma_{\mathrm{imp}}}{\partial T}+(r-q) K \frac{\partial \sigma_{\mathrm{imp}}}{\partial K}\right)}{1+2 d_1 K \sqrt{T} \frac{\partial \sigma_{\mathrm{imp}}}{\partial K}+K^2 T\left(d_1 d_2\left(\frac{\partial \sigma_{\mathrm{imp}}}{\partial K}\right)^2+\sigma_{\mathrm{imp}} \frac{\partial^2 \sigma_{\mathrm{imp}}}{\partial K^2}\right)}}$$

According to Clark (2011) we're only supposed to interpolate across the time domain because, there is no connection with vol across different strikes. For stock options, do I interpolate the implied vol surface across both strike and time?

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