# 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)?

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

• 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.

You could simply trade the vega off VIX options with the heaviest weighting according to the VVIX calculation.

• Rocko, why are your answers usually 99% gibberish? Also, its not just a var swap, its a perpetual var swap with a questionable weighting recipe. You can't replicate it with a single swap, as you'll be out of synch in a day. – Kevin Schmit Dec 19 '12 at 5:44
• I forgot to mention that is is not actually a var swap, it is the square root of an imperfect 30 day synthetic var swap. So there is no static replication with vanillas regardless. – Kevin Schmit Dec 19 '12 at 6:15
• Hey Victor, since your question's around "trading..." a closely replicated product. If you have an edge estimating vix vol, message me privately and I'll walk you through ways to exploit it with vix options. GL – Rock Dec 19 '12 at 20:56