Correct Theta for VIX options, Part 1

There's been many discussions on the internet regarding the way different software platforms calculate hedge ratios, or Greeks for VIX options. The main problem is in the underlying. Because VIX is not tradable, the underlying for VIX options is the VIX futures contract of the same month. If your software platform calculates delta and gamma using VIX index as the underlying, your greeks will be significantly off.

If futures are not available, one can calculate a synthetic future. For example on can take future to be equal deep ITM call price + strike. Today front month 10-strike call closed at 18.20-18.60. Taking mid-mark 18.40 + 10 = 28.40, which is exactly where futures closed. Another approach - probably more general is to calculate future value from least squares regression on risk-reversals. I believe that approach is used by LiveVol.

However while using the correct underlying (future or synthetic) will give you correct value for delta and gamma, theta (time decay) will not be correct. I am not aware of any commercial platform that calculates VIX theta correctly!

Why is theta different? Because of the term structure of VIX volatility. Since VIX prices are mean-reverting, its volatility is a curve that increases as option gets closer to expiration. That means that every day as option moves closer to expiration, it also gains some portion of Vega proportional to increase in implied volatility.