I suspect Gatheral is correct and you have made an error. Here is my suggestion. Since Gatheral has given a very detailed derivation in his book on page 12, if you are correct, you should be able to point out an error on one of the lines on pg. 12. If you can't find a mistake there, then that strongly suggests an error in your derivation.
Alan, thank you for your response. I totally agree that it is the most reasonable deduction to make. I also feel that it's logical that Gatheral's formula (tried and tested through time) is correct. What gives me a slight, very slight feeling that there's a chance our derivation is correct, are the following points:
- The difference between our result and the original derivation in Gatheral's paper is very tiny, perhaps hardly noticeable in a Monte-Carlo computation: half of a second order derivative in the denominator
- Every paper we have found online seems to follow exactly the derivation in Gatheral's original paper, whereas we have followed a different, more mechanical approach, which (arguably) should be easier to verify: the level of maths involved in successively applying multivariate chain-rule is relatively easy. Jim Gatheral uses a more involved approach in his derivation (which however, makes his much more elegant).
Comparing our result with Jim Gatheral's derivation, the discrepancy seems to occur in the second order derivative of the call price with respect to strike, respectively transforming this to derivatives with respect to the variables "y" and "v" (which are denoted "y" and "w" in Gatheral's original paper and his book). That's where the extra term arises. I have tried to pin down exactly the point in Gatheral's derivation where the two approaches diverge, and will try some more.
If anyone spots an error in the mechanical approach we have chosen, please do let us know: we haven't been able to find an error there so far.