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Strictly speaking, indices such as the VIX are built to approximate the expected variance (of log-returns) that would effectively realise under a pure diffusion setting (i.e. no jumps) $$\frac{dX_t}{X_t} = \mu(t) dt + \sigma(t,.) dW_t^{\mathbb{Q}}$$ Writing out the equations (*) yields the famous static replication formula in terms of strike-weighted OTMF ...

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The motive is indeed to construct a constant gamma portfolio. A position in a VVIX portfolio replicates the volatility of VIX forward prices. VVIX portfolio prices have usually been at a premium relative to future realized volatility. The discount is a volatility risk premium. For nearby expirations, these prices have also tended to surge at the same time ...

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Historically, there has been little correlation between the VVIX and the VIX except at extreme values of the VIX. You are correct it will definitely lead to different greeks, but it does not matter at lot as your Portfolio objective would be fulfilled as methodology for both are same.

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