VIX options in the SABR model

We study the pricing of VIX options in the SABR model dS_t=σ_tS_t^βdB_t,dσ_t=ωσ_tdZ_t where B_t,Z_t are standard Brownian motions correlated with correlation ρ<0 and 0≤β<1. VIX is expressed as a risk-neutral conditional expectation of an integral over the volatility process v_t=S_t^β−1σ_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.