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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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  • $\begingroup$ 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) $\endgroup$ – user59 Oct 10 '11 at 2:47
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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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Volatility and variance derivatives, such as the VIX, are priced by creating a replicating portfolio of options, weighted so as to have constant gamma for a wide range price levels. So an option on such a structure would necessarily be the sum of options on each of the individual options in the replicating portfolio. Thus, the question of their pricing reduces to the question of pricing options on options, which is easier to research. Or is the question how you price an option on an option?

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    $\begingroup$ Hi Jason, welcome to quant.SE and thanks for contributing your answer. However, I think you are incorrect. The VIX methodology actually takes the square root of the price of a replicating portfolio of options. Hence a VIX option is an option on the square root of prices of a portfolio of a bunch of other options. This is a different and presumably more difficult problem than pricing options on options. $\endgroup$ – Tal Fishman Sep 19 '11 at 14:55
  • $\begingroup$ Oh the VIX is a volatility swap? I thought it was a variance swap. So I guess any option-option model would have to be adapted to consider the square root of the assumed process -- that does sound pretty nontrivial. That said, it seems like you could do it pretty easily by adopting Monte Carlo integration as your solution approach. $\endgroup$ – William Sep 19 '11 at 15:40
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Here are some papers:

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1524193

http://www.rmi.nus.edu.sg/conferences/RMC2008/submission/papers/33.pdf

http://www.iaeng.org/publication/WCE2011/WCE2011_pp433-438.pdf

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This is an old post, but here is a recent paper on a market model for pricing and hedging volatility derivatives:

A multi-factor shifted lognormal model for forward starting variance swaps

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