VIX options in the SABR model

2025/01/11 by Pirjol, Dan, Zhu, Lingjiong · 2 citations
#FOS: Economics and business #Pricing of Securities (q-fin.PR)

paper · doi:10.48550/arxiv.2501.06398

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 ≤ 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.

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• VIX options in Bergomi models
• VIX and European options with jumps in the short-maturity regime

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