VIX options in the SABR model

TL;DR

This paper analyzes the pricing of VIX options within the SABR model, revealing issues with volatility explosion and proposing a capped volatility process to ensure well-behaved option prices.

Contribution

It demonstrates the explosion of the volatility process in the SABR model and introduces a capped process to address this, enabling practical VIX option pricing.

Findings

VIX futures and calls are infinite due to volatility explosion.

VIX puts are zero for any maturity in the original model.

A capped volatility process ensures finite, well-behaved option prices.

Abstract

We study the pricing of VIX options in the SABR model $dS_t = \sigma_t S_t^\beta dB_t, d\sigma_t = \omega \sigma_t dZ_t$ where $B_t,Z_t$ are standard Brownian motions correlated with correlation $\rho<0$ and $0 \leq \beta < 1$. 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.