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 $S$,
\begin{equation}
dS(t) = \sigma(t) S(t) dW(t)
\end{equation}
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:
- $E[d\Sigma] = 0$ for all strikes $K$: i.e. all implied volatilities are local martingales
- $\frac{dS}{S} \frac{d\Sigma}{\Sigma} = \eta \sigma \rho \, dt$ for all strikes $K$: i.e. all implied volatilities have the same correlation with $S$
- $ (\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:
\begin{equation}
\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
}
\end{equation}
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!
