# Trading a synthetic replication of the VVIX (volatility of VIX)

In the same spirit as this question: Trading a synthetic replication of the VIX index.

The VVIX tracks the volatility of the VIX.

One cannot directly buy and sell the VVIX index and, as opposed to VIX, there are no futures or options to trade it.

Is it possible to synthetically replicate the VVIX index using VIX options and futures (or other tradable instruments)?

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It is variance swap, so obviously, it's possible, but what's the point?

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Thanks. The point: the mean reversion of the vvix. –  Victor P Oct 9 '12 at 22:50
there's mean reversion with any index vol/var –  Rock Dec 18 '12 at 19:15
vix is mean reverting but vol of vol does not seem to be mean reverting. std is well known to decay at ~$\frac{1}{\sqrt{t}}$ –  Andrew Dec 23 '12 at 6:00

you definitely can track this not even by just using vix options, but even by using spx options.

Let $g(S_T)$ be the exotic payoff that you are trying to replicate, then: $\mathbb{E} [g(S_T)] = g(F) + \int^F_0 dK \tilde{P}_K g''(K) + \int^\infty_F dK \tilde{C}_K g''(K)$ where $C_K, P_K$ are the values of the call options and put options which we can get from the market. Now, the funky payoff that turns out the most interesting is $log\frac{S_T}{S_0}$ since this payoff replicates the total variance. If you are familiar with stochastic calculus, you can perform ito's lemma on this and then you can value exactly the vix spot index. In your case, if you let $S_T$ be the vix index, then you would need a few vix options to replicate vvix.

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