VIX options in the SABR model

Abstract

We study the pricing of VIX options in the SABR model $dS_t = σ_t S_t^βdB_t, dσ_t = ωσ_t dZ_t$ where $B_t,Z_t$ are standard Brownian motions correlated with correlation $ρ<0$ and $0 \leq β< 1$. VIX is expressed as a risk-neutral conditional expectation of an integral over the volatility process $v_t = S_t^{β-1} σ_t$. 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.