Is there anywhere I can read the paper, "The Gamma-Vanna-Volga Cost Framework for Constructing Implied Volatility Curves"

Hello there was a paper published by: M. Arslan, G. Eid, J. El Khoury and J. Roth titled "The Gamma-Vanna-Volga Cost Framework for Constructing Implied Volatility Curves" (2009).

I believe they devised a new approach to construct IV curves in it. It has also been cited many times especially by Peter Carr et al in multiple papers that they've written.

But after searching online, I could not find anywhere where this paper was available. I'm wondering if anyone knows where I can read it. Thank you.

2 Answers

Partly because it's hard to get a hold of, the Arslan et. al. paper is starting to assume mythical proportions.

As said by Dimitri Vulis, the general idea of the paper is set out in (one or two of) Peter Carr's papers.

For the benefit of the OP and others I will try to summarize the most salient points of the paper below and also point out the assumptions underlying it. The notes below should be sufficient for anyone in order to implement the GVV model:

Assume the following (local) stochastic volatility model for an asset $$S$$, $$$$dS(t) = \sigma(t) S(t) dW(t)$$$$ There is no need to specify the dynamics of $$\sigma(t)$$. Although I have assumed zero interest rate and dividend yield it is easy to repeat the arguments below with deterministic interest rate and dividend yield.

Suppose at least 3 vanilla options on $$S$$ are traded. Let $$C^{BS}(S,K,\Sigma(S,K))$$ denote the market price of such an option. The change in the market price of any option of strike $$K$$ is $$dC^{BS} = \Delta^{BS} dS + \nu^{BS} d\Sigma + \frac{1}{2} \Gamma^{BS} S^2 ( \sigma^2 - \Sigma^2) dt + \frac{1}{2} vo^{BS} (d\Sigma)^2 + va^{BS} dS d\Sigma$$

Now follow three crucial assumptions of the GVV model:

1. $$E[d\Sigma] = 0$$ for all strikes $$K$$: i.e. all implied volatilities are local martingales
2. $$\frac{dS}{S} \frac{d\Sigma}{\Sigma} = \eta \sigma \rho \, dt$$ for all strikes $$K$$: i.e. all implied volatilities have the same correlation with $$S$$
3. $$(\frac{d\Sigma}{\Sigma})^2 = \eta^2 \, dt$$ for all strikes $$K$$: i.e. all implied volatilities have the same volatility

These are clearly very strong assumptions (and I will show below that assumption 1. in particular cannot be true). However, let's go along with them for the moment.

Since options are tradables, they are local martingales. In other words, $$E [ dC^{BS} ] = 0$$ Using this fact, and the expression for the change in the market price of the option, and the GVV assumptions, we arrive at the following expression: $$$$\boxed{ \frac{1}{2} \Gamma^{BS} S^2 ( \sigma^2 - \Sigma^2) + \frac{1}{2} vo^{BS} \Sigma^2 \eta^2 + va^{BS} S \Sigma \eta\sigma \rho = 0 }$$$$ This is in essence the Gamma-Vanna-Volga model. It basically says that the theta of an option is balanced by its dollar gamma, dollar volga and dollar vanna costs.

So how to use this model? First of all we need to find the three parameters the instantaneous volatility $$\sigma$$, the volatility of implied volatilities $$\eta$$, and the correlation $$\rho$$. It is clear that given three quoted options (preferably two at the wings, and one near ATM) it is possible to back out these three parameters/variables. Once these three quantities have been calibrated, then all other implied volatilities can be solved for by solving the non-linear GVV equation, e.g. using the bi-section method.

Now, returning to what I said about the assumptions, in particular the assumption that all implied volatilities are local martingales. Take specifically the zero vanna implied volatility $$\Sigma_{d_2}$$. That is the strike and implied vol where the vanna and the volga of an option are zero. Under assumption 1 this would lead to $$\Sigma_{d_2} = \sigma$$ regardless of maturity. This cannot be the case. What Peter Carr did was to "generalise" the GVV framework to not assume driftless implied volatilities.

In any case, I personally think the GVV model is a nice model, and if you are willing to overlook its inconsistencies and/or limitations by all means use it. That said, a bit of self-promotion:

Take a look also at my paper It Takes Three to Smile. Although it wasn't my main purpose, in that paper I give an alternative smile interpolation and extrapolation method that also only requires three options. The difference between my method and GVV is that I make minimal assumptions (actually no assumptions) on the dynamics of implied volatilities. I do, however, assume that the smile is generated by a pure stochastic volatility model whereas GVV allows also local stochastic volatility models.

Hope the above helps!

• Thanks for this nice overview. So, basically you say that impl vol is not tradable and hence not a local martingale, i.e., it has a drift. In this case, vega is missing. However, I would suspect that vega should have a less pronounced impact than the other factors, right? Do you have specific examples analyzing the missing impact? As for the other two assumptions, I suppose that it is hard to find dynamics being always consistent with market behaviour and contradicting the GVV assumptions, right?
– SI7
Apr 19 '20 at 10:19
• @SI7 I suppose for short maturities vega has limited impact, but for longer dated smiles I am not sure the approximation remains good. I do not have specific examples, but I would be interested to see some analysis from anyone that looks at this, an also comparison of the accuracy between GVV and the "It takes 3 to smile" approximation. On the other two assumptions of GVV: it basically assumes a lognormal dynamics for the IV and strike independent correlation. Apr 19 '20 at 18:06
• may I come back with a question regarding the missing vega component in GVV. Suppose we go through the math and implement your approach described above. Can't we incorporate the vega effect by just adding a fourth option into this framework? If we then follow the math, at first glance, I would suspect that the matrices/vectors in this algebraic task just have one additional row/column, right? Or am I missing something here?
– SI7
May 8 '20 at 19:05
• @SI7 Feel free to drop me an email to discuss further. My email should be in one of the papers that is available at SSRN or arXiv. May 10 '20 at 9:35

It was a Deutsche Bank Working Paper: http://refhub.elsevier.com/S0304-405X(16)00005-2/sbref0001 Unfortunately, it is very hard to find internal research published by banks. I have not seen this one myself, but as far as I know, Peter Carr's published paper has everything that this paper had.

• Which Peter Carr paper we are talking about please? Apr 21 '20 at 0:11
• If you click on the link in the answer by Dimitri Vulis, you'll see paper statistics. One of them is "cited by", if you click on that you'll see the papers that have cited the Arslan et al paper. Apr 21 '20 at 6:28