VIX options in the SABR model

Abstract

We study the pricing of VIX options in the SABR model $d{S}_t={\sigma }_t{S}_t^{\beta }d{B}_t,d{\sigma }_t=\omega {\sigma }_{t}d{Z}_t$ where $B_t,Z_t$ are standard Brownian motions correlated with correlation $\rho <0$ and $0\le \beta <1$. VIX is expressed as a risk-neutral conditional expectation of an integral over the volatility process $v_t=S_t^{\beta -1}{\sigma }_t$. We show that $v_t$ is the unique solution to a one-dimensional diffusion process. Using the Feller test, we show that $v_t$ explodes in finite time with non-zero probability. As a consequence, VIX futures and VIX call prices are infinite, and VIX put prices are zero for any maturity. As a remedy, we propose a capped volatility process by capping the drift and diffusion terms in the $v_t$ process such that it becomes non-explosive and well-behaved, and study the short-maturity asymptotics for the pricing of VIX options.