# Vol swaps Vol of vol

I an quite unfamiliar with those products and would like to understand why they require a vol of vol model for pricing. The variance swap is replicated (assuming no cash dividends) with a delta hedged log contract and vol swaps cannot be replicated (volatility is not time linear) but where does the vol of vol come into play? And what is the convexity adjustment applied from between the implied variance of the variance swap and the implied vol of the vol swap?

Is there any paper that explains how do we model this vol of vol?

I am not entirely sure what you are asking. I cannot answer from a valuation of contract perspective but I can help on an asset pricing perspective. Vol-of-vol is important to model a volatility swap because absent vol-of-vol (or jumps in the underlying) the variance risk premium on the contract is zero.

There are two relevant references:

1. First Carr and Wu (2009) show that there is a large variance risk premium. Take a look at table 3 from their paper:

1. Second vol-of-vol is important in generating this premium. Take a look at Bollerslev, Tauchen and Zhou (2009). They basically embed vol-of-vol in a standard asset pricing framework:

\begin{equation*} \Delta c_{t+1} \equiv g_{t+1} = \mu_c +x_t + \sigma_t \eta_{t+1} \end{equation*} \begin{equation*} \sigma_{t+1}^2 = a_\sigma + \rho_\sigma \sigma_t^2 + \sqrt{q_t} z_{\sigma,t+1} \end{equation*} \begin{equation*} q_{t+1} = a_q + \rho_q q_t + \varphi_q \sqrt{q_t} z_{q,t+1} \end{equation*}

The last equation is the vol-of-vol. They show that in such a framework the variance risk premium is given by:

\begin{eqnarray*} E_t^Q\left(\sigma_{r, t+1}^2\right) - E_t\left(\sigma_{r, t+1}^2\right) = (\theta - 1) k_1^2 (A_\sigma^2 + A_q^2 \varphi_q^2) \varphi_q^2 ]q_t \end{eqnarray*}

So you can immediatly see that if there is no stochastic vol-of-vol the variance risk premium is zero. Also if $q_t$ is constant the variance risk premium is not time-varying.

There is quite a recent paper by Rolloos and Arslan (Wilmott, January 2017) that shows how to obtain a good approximation of the volswap strike from the implied volatility smile without having to specify any model, i.e. a model-free approximation that is immune to correlation to first order. Their method is much easier to follow and implement than Carr and Lee's but just as accurate.

Once you have the volswap strike and the varswap strike then you can back out the vol of vol.