Pricing VIX Options with Replication and Volatility Models
Summary
The document outlines several ways to think about pricing options on a volatility index such as VIX. One approach starts from the option portfolio used to replicate variance and adds a convexity adjustment that accounts for the square root linking variance to the volatility index. More detailed pricing can use stochastic volatility models, with Fourier transform methods cited as a way to make computation efficient.
The discussion also describes practical hedging: traders may hedge VIX options with VIX futures and approximate the market locally using a Black–Scholes framework with a nonstandard volatility skew. The replies emphasize that replication involves dynamic option hedging and that option sensitivities and bid–offer spreads can vary substantially. These are conceptual suggestions rather than a worked comparison of pricing accuracy; the document gives no calibration, empirical results, or definitive best model. One reply frames volatility derivatives as combinations of options weighted to maintain roughly constant gamma, pointing toward the related problem of pricing options on options.
Key ideas
- A variance replication portfolio can provide a starting point for valuing a volatility-index option.
- A convexity adjustment is needed because the index reflects a square root of variance.
- Stochastic volatility models offer a more involved route, and Fourier methods can make pricing efficient.
- VIX futures are commonly used as hedging instruments, with local approximations used in practice.
- Dynamic hedging and changing option liquidity and sensitivities limit simple replication.
Tags
Full text
# How to price a volatility-index option? # 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? ## Answer by Brian B (score 7) https://quant.stackexchange.com/a/1961 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. ## Answer by William (score 2) https://quant.stackexchange.com/a/1958 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? ## Answer by bill_080 (score 2) https://quant.stackexchange.com/a/2208 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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