Three One-Factor Processes for Option Pricing with a Mean-Reverting Underlying: The Case of VIX

We challenge the two most prominent one‐factor mean‐reverting models for variance/volatility indices and propose the inhomogeneous geometric Brownian motion process to price volatility index (VIX) options. We study the roles of the equilibrium level, speed of reversion, volatility and expiry date in the pricing of VIX options and obtain analytic solutions for perpetual American options as well as some Greeks. We price the finite‐lived American options using their transformed process to build a binomial tree. We also derive the closed‐form mean first‐passage time for perpetual American options and find a long optimal exercising time. Other things being equal, the European ones are cheaper than their American counterparts, which reflect an early exercise premium.