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I want to compute Dupire Local volatility using the identity that links Dupire local variance to BS implied total variance. I calibrated an SVI on options data to get the implied total variance surface and to compute the derivative of the implied total variance wrt time, i do a finite difference. The problem is i always stumble on a negative nominator or denominator, hence I cannot apply a square root to get dupire local volatility.

Is there a solution to the problem ? some kind of regularization to do ? I am 99% sure that my surface is free of arbitrage

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What data are you using? Local volatility requires an arbitrage-free implied volatility surface. In general, equities rarely satisfy these conditions on their options because their bid-ask spreads are too large. I have had more success calibrating local volatility surfaces with FX options because of this very reason.

Additionally, SVI is generally used for each tenor slice and so you will satisfy strike arbitrage, but will generally fail calendar arbitrage. There is another method called quasi-SVI, which has worked for me, or you can calibrate each slice and then attempt to adjust the slices - but I often find this will lead to unrealistic results for the surface.

But in general, there isn't a robust method to calibrate equity options across a vast amount of strikes and maturities.

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  • $\begingroup$ Hello, thank you for your response. I am using data of European options of cac40 stocks like LVMH, AXA .. ( here is a sample live.euronext.com/fr/product/stock-options/CS4-DPAR). For the SVI, i am calibrating it for each tenor, and I added a condition that W(t1,k) < W(t2,k) to enforce no calendar spread arbitrage, and to construct the surface I am interpolating linearly the time and the implied local variances. $\endgroup$
    – M2000
    Commented May 21 at 8:17

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