SERVING THE QUANTITATIVE FINANCE COMMUNITY

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JanStuller
Topic Author
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Joined: June 5th, 2017, 2:25 pm

### Local Vol: Jim Gatheral formula

Hello, I have spent a few weeks now re-deriving the transformed Dupire formula and have had my work checked by friends and professionals. We consistently get an extra half term in the denominator of the formula. I am (almost) sure we've made some logical mistake somewhere, but maybe our derivation is correct? I would appreciate any feedback on the attached for those interested.

I have not found any similar detailed derivation online (apologies about the format, it's a summary typed using Word equation editor).

Thank you so much for any hints and feedback, including criticism,

Best regards,

Jan

Ps: I've made small adjustment to the attachment to make it more legible.
Attachments
FullDerivation.pdf
Last edited by JanStuller on June 7th, 2017, 10:52 am, edited 1 time in total.

Alan
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### Re: Local Vol: Jim Gatheral formula

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.

JanStuller
Topic Author
Posts: 7
Joined: June 5th, 2017, 2:25 pm

### Re: Local Vol: Jim Gatheral formula

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.

Alan
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### Re: Local Vol: Jim Gatheral formula

Another check is to google for derivations of $Q(K) \sim C_{KK}$, the risk-neutral density. This is essentially the denominator in question here. Unfortunately, I do not have a link, but I know it exists in some papers, and provides essentially the same chain rule computation when $C = C_{BS}(K,IV(K))$.

BTW, I think I have mentioned elsewhere that this is a routine (but tedious!) computation that all quants should do at least once because of the importance of $Q(K)$ -- maybe this will encourage somebody else on the forum to confirm Gatheral's result by this route.  That is to say, in the absence of cost-of-carry terms:

$\nu_L \equiv \sigma^2(K,T,S_0) = \frac{2 \, C_T}{K^2 Q(K)}$
Last edited by Alan on June 7th, 2017, 10:10 pm, edited 1 time in total.

frolloos
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### Re: Local Vol: Jim Gatheral formula

Btw, Alan, dropped you a pm on a related topic.

JanStuller
Topic Author
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Joined: June 5th, 2017, 2:25 pm

### Re: Local Vol: Jim Gatheral formula

Btw, Alan, dropped you a pm on a related topic.
Don't think I've received any personal messages thus far (just checked: inbox empty . ). But any hints or comments on the topic are certainly most welcome!

JanStuller
Topic Author
Posts: 7
Joined: June 5th, 2017, 2:25 pm

### Re: Local Vol: Jim Gatheral formula

Another check is to google for derivations of $Q(K) \sim C_{KK}$, the risk-neutral density. This is essentially the denominator in question here. Unfortunately, I do not have a link, but I know it exists in some papers, and provides essentially the same chain rule computation when $C = C_{BS}(K,IV(K))$.

BTW, I think I have mentioned elsewhere that this is a routine (but tedious!) computation that all quants should do at least once because of the importance of $Q(K)$ -- maybe this will encourage somebody else on the forum to confirm Gatheral's result by this route.  That is to say, in the absence of cost-of-carry terms:

$\nu_L \equiv \sigma^2(K,T,S_0) = \frac{2 \, C_T}{K^2 Q(K)}$
Alan, I've gone through the derivation of this one too: if I understood you correctly, you mean the second partial derivative of the Call price with respect to strike price, to obtain the risk-neutral density of the underlying stock price at option maturity (with the strike being the argument in the PDF).

The usual approach for this one is to write the call price as the risk-neutral expectation of the PayOff and follow the Fundamental Theorem of Calculus, rather than chain rule per-se. The differentiation eventually gets rid of the integral in the expectation, and the rest follows.

The result is very short, I don't mind posting it here, it's a one page ("X" in the attachment stands for the Stock price at option Maturity: X=S(T) ). What is interesting though (and I love that idea) is to do an empirical test: plug in values for the Gatheral formula and the derived density: and see if they are the same. Is that what you meant?

Jan
Attachments
StockPriceDerivedDensity.pdf

Alan
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### Re: Local Vol: Jim Gatheral formula

No - I just meant that the Gatheral denominator is essentially $\partial_K^2 C_{BS}(K,IV(K))$, which can be found in some papers if googled. (For example, eqn (10) here). So, another way to check is to start from that result (or derive it anew) and then convert from $(IV,K)$ to Gatheral's variables: $(w,y)$.

This shouldn't be so hard!

frolloos
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Location: Netherlands

### Re: Local Vol: Jim Gatheral formula

Btw, Alan, dropped you a pm on a related topic.
Don't think I've received any personal messages thus far (just checked: inbox empty . ). But any hints or comments on the topic are certainly most welcome!
Although not directly related to your question, I've attached a derivation of Gatheral's formula for variance strike which I did for myself long time ago. It probably contains expressions relevant to your question as well, in particular how to express the probability density in terms of implied volatility as Alan mentioned.
Attachments
vsfromiv.pdf

Alan
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My suggestion for a quick resolution. The OP should offer a reward of $100, payable by Paypal, for the first person to find the error in his derivation. JanStuller Topic Author Posts: 7 Joined: June 5th, 2017, 2:25 pm ### Re: Local Vol: Jim Gatheral formula My suggestion for a quick resolution. The OP should offer a reward of$100, payable by Paypal, for the first person to find the error in his derivation.
Lol. Loving it!
Ok: I am offering \$50!
J.

outrun
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### Re: Local Vol: Jim Gatheral formula

or 0.02 btc

Alan
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### Re: Local Vol: Jim Gatheral formula

Excellent. I think all internet Q&A sites should work like this. Finally, a legit use case for bitcoin!

outrun
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### Re: Local Vol: Jim Gatheral formula

Our only hope is list1! He's very into the technical aspect and definitions.

JanStuller
Topic Author
Posts: 7
Joined: June 5th, 2017, 2:25 pm

### Re: Local Vol: Jim Gatheral formula

If our derivation is correct, and Gatheral's contains a small mistake: wouldn't that be just incredible?