Replicating VVIX Exposure with VIX Options

Summary

The document asks whether the VVIX, which measures volatility of the VIX, can be tracked through tradable instruments despite having no directly tradable index, futures, or options. It presents two replication perspectives: variance exposure can be synthesized from options, and a general option replication formula can value a payoff using a continuum of calls and puts across strikes. Applying the log-payoff relationship to VIX as the underlying suggests using VIX options to approximate VVIX exposure; another answer suggests focusing on the vegas of the VIX options most heavily weighted in the index calculation.

These are conceptual proposals rather than a tested trading recipe. The discussion gives no hedge ratios, implementation details, empirical tracking results, or treatment of liquidity and transaction costs. It also does not establish that a small set of options will reproduce the index accurately. The applicability of the general replication argument depends on the available strikes and market prices, so practical tracking error and instrument availability would need separate assessment.

Key ideas

• VVIX measures the volatility of the VIX and lacks a directly tradable contract in the discussion.
• Option prices across strikes can be used to represent certain nonlinear payoffs, including variance-related exposure.
• Treating VIX as the underlying suggests constructing a VVIX proxy from VIX options.
• A simpler approximation would emphasize the vegas of options with larger weights in the VVIX calculation.
• The document provides no empirical evidence on tracking accuracy or implementation costs.

Tags

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


# 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)?

## Answer by Strange (score 3, accepted)

https://quant.stackexchange.com/a/4296

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

## Answer by Andrew (score 4)

https://quant.stackexchange.com/a/4832

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.

## Answer by Rock (score 1)

https://quant.stackexchange.com/a/4803

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.