VIX options in the SABR model

Abstract

We study the pricing of VIX options in the SABR model dS_t = _t S_t^ dB_t, d_t = _t dZ_t where B_t,Z_t are standard Brownian motions correlated with correlation <0 and 0 < 1. VIX is expressed as a risk-neutral conditional expectation of an integral over the volatility process v_t = S_t^-1 _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.

Tasks

Reproductions