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I am trying to implement a local volatility pricer using Monte Carlo and Dupire's equation in function of implied volatilities and I was told that first of all I have to check Dupire is well implemented so I was reccommended to prove it like this:

-The first step, to check Dupire's function, was to get the local volatility surface from a flat implied volatility surface. As it is flat and we know that local vol skew is double of the implied vol skew I expect also a flat local vol surface with the same value of the implied vol surface (since ATM value is the same for both), am I wrong?

-The second step would be to chek Dupire with an implied volatility surface with some slope in the expirys term but without skew (flat in the strikes). The question is: what I should expect from this? I understand that it should be a local volatility surface without skew in the moneyness term (as I said its skew is double of implied volatility and it is flat) but don't know about the expiry term.

-Finally I can't prove this case because I get negative local variances (complex local vols) so I can't plot the surface. I am using a bivariable cubic spline over moneyness and linear in expiry with python functions bisplrep and bisplev but I would like if anyone can tell me which interpolation scheme I should use.

This is the implied volatility surface I artificially created and which I am using to check step 2, but as I said I obtain complex local vols with that kind of interpolation. enter image description here

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    $\begingroup$ Negative local variance may come from calendar arbitrage, can check by plotting total variance $\sigma^2T$ and see whether it is increasing. Since the skew is flat, negative local variance is less likely to come from butterfly arbitrage. $\endgroup$
    – ryc
    Mar 20, 2021 at 7:45
  • $\begingroup$ Thanks! I checked it by plotting total variance and it was pretty weird due to the way I generated the temporal slope. Then I fixed it and now it is increasing so I dont get negative local variance. My final question is: what should I expect about de local vol surface, to be the same as the implied volatility surface? or maybe to have some more/less temporal slope than IVS? $\endgroup$
    – Nico Blanco
    Mar 23, 2021 at 16:43
  • $\begingroup$ this is well described in Gatheral book "The volatility surface". The slope of the local variance in log-moneyness is around twice that of the implied variance. In time, it all depends on your interpolation. Linear interpolation will lead to piecewise flat looking local variance. $\endgroup$
    – jherek
    Sep 25 at 7:48

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I would suggest to use linear interpolation. The second derivative might cause problems with cubic spline.

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  • $\begingroup$ Very odd recommendation. What the author of the post does is relatively standard. Not sure how linear interpolation can work with Dupire since it involves second derivatives. $\endgroup$
    – jherek
    Sep 25 at 7:45

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