Variance swap volatility - ATMF vol, Skew and Curvature

Question

In a pure diffusion setting, it is a well known result that the volatility $\sigma_T$ of a fresh-start variance swap of maturity $T$ as seen of $t=0$ verifies \begin{align} \sigma_T^2 &= \Bbb{E}_0^\Bbb{Q} \left[ \frac{1}{T}\int_0^T d\langle \ln S \rangle_t \right] \\ &= \int_{-\infty}^{+\infty} \tilde{\sigma}^2(z,T) \phi(z) dz \\ &= \int_{-\infty}^{+\infty} \sigma^2(f^{-1}(z),T) \phi(z) dz\\ &= \int_{-\infty}^{+\infty} \sigma^2(k,T) \phi(f(k)) f'(k) dk \end{align} where $\phi(\cdot)$ is the pdf of a standard Gaussian, $\sigma(k,T)$ is the implied volatility smile for log-forward moneyness $k$ and time to expiry $T$ and $$ f : k \to \frac{k}{\sigma(k,T)\sqrt{T}} + \frac{1}{2} \sigma(k,T) \sqrt{T} $$

As a result, if the smile in the modified moneyness space $z$ admits the following parameterisation $$ \tilde{\sigma}^2 = \tilde{\sigma}_0^2 + \alpha z + \beta z^2 $$ then $$ \sigma_T^2 = \tilde{\sigma}_0^2 + \beta $$ by the properties of the standard Gaussian and it is often said that only the "ATM" level and "ATM" convexity contribute to the price of the VS. Of course "ATM" is here to be understood as at the money in the modified moneyness space, i.e. $z=0$ not $k=0$.

I've been looking to obtain a similar expression relating the VS variance, ATMF variance, ATMF skew and curvature under the standard moneyness space $k$ but am surprised by how difficult this is. Certainly missing something, if you have pointers, I'd gladly take them.

I am aware that such relationships exist for generic stochastic volatility models as given by the Guyon-Bergomi expansion, but I was (secretly) hoping for a model-free result.

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[Edits]

Since this question seems harder than it seems. I'm also willing to take answers to a lighter version of it.

Indeed,a model-free result can be obtained when considering a smile linear in log-forward moneyness $$ \sigma(k, T) = \sigma_{F_T T} + \mathcal{S}_T k $$ It is not an exact result but rather an approximation based on a perturbation analysis. More specifically, $\sigma(k,T)$ is perturbed around $\sigma_0 = \sigma_{F_T T}$ at order 1 in the ATMF skew $\mathcal{S}_T \ll 1$ and the resulting order 1 expansion of VS volatility then reads $$ \sigma_T = \sigma_{F_T T} - \frac{1}{2} \sigma_{F_T T}^2 T \mathcal{S}_T $$ which is already one step towards the relationship I'm looking for.

Starting from the integral formulation of VS volatility above, I have worked out a (very sloppy) demonstration of this result as I'm not an expert of perturbation theory.

So if anyone is willing to provide a rigourous demonstration of how perturbation theory can be used here, I would be a happy man. Using a similar approach I find that at order 1 in convexity $\mathcal{C}_T$ $$ \sigma_T = \sigma_{F_T T} + \mathcal{C}_T \frac{\sigma_{F_T T} T }{2} \left( 1 + \sigma_{F_T T}^2 T \right) $$ hence both positive convexity and negative skew mean that VS volatility will be higher than ATMF volatility.

Answer 1

I think you can consider the following as a starting point.

Let's start from the daily realized variance of a stock price $S_t$ approximated by the forward prices $F_t$: $$\sigma^2_{t,T}=\frac{1}{T}\sum_{t=1}^{T}\left(\log\frac{S_t}{S_{t-1}}\right)^2\approx \frac{1}{T}\sum_{t=1}^{T}\left(\log\frac{F_t}{F_{t-1}}\right)^2$$

By the Taylor expansion,

\begin{align} \log\frac{F_t}{F_{t-1}}&=\log\bigg(1+\underbrace{\frac{F_t-F_{t-1}}{F_{t-1}}}_{=:R_{\Delta t}}\bigg)=R_{\Delta t}-\frac{1}{2}R_{\Delta t}^2+\mathcal{O}[(R_{\Delta t})^3] \\ \Rightarrow & \left(\log\frac{F_t}{F_{t-1}}\right)^2 = 2R_{\Delta t}-\log\frac{F_t}{F_{t-1}}+\mathcal{O}[(R_{\Delta t})^3] \end{align}

since $\left(\log\frac{F_t}{F_{t-1}}\right)^2\approx R_{\Delta t}^2$

\begin{align} \Rightarrow \sigma^2_{t,T}&\approx\frac{1}{T}\sum_{t=1}^{T}\left(\log\frac{F_t}{F_{t-1}}\right)^2 \\ &=\frac{1}{T}\sum_{t=1}^{T}\left[2\frac{F_t-F_{t-1}}{F_{t-1}}+\log\frac{F_t}{F_{t-1}}+\mathcal{O}[(R_{\Delta t})^3]\right] \\ &=\frac{2}{T}\sum_{t=1}^{T}\frac{F_t-F_{t-1}}{F_{t-1}}-\frac{2}{T}\log\frac{F_T}{F_0}+\frac{1}{T}\sum_{t=1}^{T}\mathcal{O}[(R_{\Delta t})^3] \end{align}

With this formula, one has a relation between the forward price density and the spot variance via the second term in the RHSof the last equation. One can use a model for the implied volatility to capture the skew, which is represented by the error term $\mathcal{O}[(R_{\Delta t})^3]$. This will of course differ when using a LV and SV.