# How to price a volatility-index option?

There exist several volatility indices, such as the CBOE Volatility Index (VIX). There are also options on such indicies.

What is the best way to price a volatility-index option? Is there a simple model that works well in terms of performance and precision?

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You can price a VIX option by looking at the volatility of the volatility. This is called vomma: en.wikipedia.org/wiki/Greeks_%28finance%29#Vomma (this doesn't answer your question, but might be trivially helpful) –  barrycarter Oct 10 '11 at 2:47

There is a replicating portfolio for the VIX contract, involving one option and the underlying S&Ps.

Unlike for variance swaps on jump-free underlyings, though, the replicating portfolio requires a dynamic option hedge. In practice, one uses more than one option to do the hedge because a given option's sensitivity to volatility (vega) and bid-offer spread will vary crazily over time.

You ask about a simple model...one thing you can do is start with the variance swap formula and then do a convexity correction by integrating the VIX-related square root over the terminal probability distribution. Beyond that, you're getting into stochastic volatility models, which are not super-simple but do enjoy reasonably efficient pricing schemes via fast fourier transforms. See Jim Gatheral's book for more on that.

Finally, its worth noting that many VIX options traders just hedge against VIX futures, treating the whole thing as a Black-Scholes market with unusual skew, and making local linear approximations where necessary. That takes balls but the spreads are so wide that it works.

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