Short Note: VIX Derivatives Pricing

- Lest it has not been emphasized enough in the previous post, note that there is no straightforward no-arbitrage relationship to get VIX future price from sopt VIX.

- Perhaps one day, the market would be so well-developed and liquidly-traded that we could have a "Volatility Market Model." Until then, we will have to rely on more modest approaches.

- Why do people say that VIX options are priced off VIX future instead of spot? Well, for options on, say, FX or stock, there is no need to distinguish between pricing options off futures or spots because spot and future are strictly connected by no-arbitrage; for VIX, however, such connection is absent. If you start of with VIX spot, you first have to price the future contract before you can price the option. So it is better to directly use VIX future price as input when pricing option on VIX, so as to avoid "an extra layer of error."

-  Currently the different approaches of pricing VIX derivatives are:
1. Modeling the instantaneous spot variance as a stochastic process
This is similar to modeling the short rate in interest rate derivatives pricing. Instantaneous spot variance, much like short rate, is neither observed nor tradable. Yet, they are rather well-studied and people are quite familiar with them, at least in the context of vanilla and exotic stock options. The idea is to pick your favorite model that describes the evolution of the instantaneous variance (say, Heston or 3/2 model). Calibrate it to the current volatility surface (considering VIX is a 30-day measure, you may want to calibrate only to the front and forthcoming month). Then use the calibrated parameters to price the VIX derivatives with numerical integration.

2. Modeling the instantaneous forward variance as a stochastic process
This is similar to the HJM framework in interest rate derivatives pricing. One example is the model proposed by Bergomi (see, for example, Bergomi's Smile Dynamics series).

3. Modeling VIX itself as a stochastic process
This differs from the previous two cases in the sense that VIX itself, which is observed, is modeled as a stochastic process.

Bottom line:
In addition to calibration efficiency, one important desirable feature is for the model to fit to both the SPX option prices and the VIX option prices.

Reference
Mencía and Sentana, 2012, Valuation of VIX Derivatives
http://www.bde.es/f/webbde/SES/Secciones/Publicaciones/PublicacionesSeriadas/DocumentosTrabajo/12/Fich/dt1232e.pdf
Wang and Daigler, 2008, The Performance of VIX Options Pricing Models: Empirical Evidence Beyond Simulation
http://www2.fiu.edu/~zwang001/Research/papers/VIX_Option_Pricing_FMA.pdf