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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.


[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.

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    $\begingroup$ How do you define the VS variance ? the variance of a VS fair strike in the future ? $\endgroup$ – Antoine Conze Feb 10 at 11:54
  • $\begingroup$ Hi @AntoineConze, I've edited the question to make that clear. My final objective would be to obtain a relationship that would allow me to compare future ATMF volatility in a LV vs. SV framework calibrated to the same smile (hence implied forward start vols), knowing that I can easily compare future ATMF skew and curvature under both models (and since they are both calibrated to the vanilla market, forward variances will coincide). $\endgroup$ – Quantuple Feb 10 at 12:14
  • $\begingroup$ @Quantuple have you tried to expand the realized variance of the forward price?When you say "ATMF volatility in a LV" you mean the Dupire formula? $\endgroup$ – FunnyBuzer Feb 11 at 12:31
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    $\begingroup$ No I didn't, expand it around what? What I meant is that for an ATM forward start option $$ V_0 = \Bbb{E}_0^\Bbb{Q}\left[ \left(\frac{S_{T_2}}{S_{T_1}}-1 \right)^+ \right]$$ $$\iff BS(1,\Delta,\sigma_{1T_1T_2}) = \Bbb{E}_0^\Bbb{Q} \left[ BS(1,\Delta,\sigma_{1\Delta}) \right] $$ where $\sigma_{1T_1T_2}$ represents the IV of an ATM forward start option of forward starting date $T_1$ and expiry $T_2$ as of today and $\sigma_{1\Delta}$ a random variable representing the future IV (prevailing at $T_1$) of an ATM vanilla of residual maturity $\Delta$ under the model at hand: be it LV or SV. $\endgroup$ – Quantuple Feb 11 at 12:44
  • $\begingroup$ If you have a relationship such as the one I am looking for, then you can compare the distributions of $\sigma_{1\Delta}$ under the two models knowing that the forward variances (calendar spread of VS) will be the same, just by comparing their future skew and curvature. I'm sure most of you have heard the: future skew and convexity are underestimated under LV hence the price of forward starts is higher, I'm trying to show that. But this was only to provide context to my question, which is independent of what I'm trying to prove. $\endgroup$ – Quantuple Feb 11 at 12:48
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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.

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  • $\begingroup$ I appreciate the effort but down-voted nevertheless since it really doesn't answer the question. In particular, you don't establish the link to the derivatives of the implied volatility as a function of the log-forward moneyness at all, which is the main point of the question. $\endgroup$ – LocalVolatility Feb 12 at 12:54
  • $\begingroup$ @LocalVolatility probably there is still something that I miss, however the term $\log\frac{F_T}{F_0}$ can be expressed in terms of forward moneyness, though probably no ATM $\endgroup$ – FunnyBuzer Feb 13 at 11:09
  • $\begingroup$ Hi Funny buzzer, I'm not sure how this really answers the question sorry. $\endgroup$ – Quantuple Feb 13 at 13:42

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