VIX options in the SABR model

2025-01-11

AI Summary

This paper addresses the ill-posedness of VIX option pricing under the SABR model: when the implied volatility process (v_t) explodes with positive probability in finite time, VIX futures and call option prices diverge, while put option prices collapse to zero, rendering the model unstable. We rigorously establish— for the first time—that this explosion arises from the violation of the Feller condition in the SABR volatility dynamics. To resolve this, we propose a modified volatility process incorporating a truncation mechanism that preserves the original model’s structural features while ensuring global existence and non-explosion. Leveraging risk-neutral pricing and short-maturity asymptotics, we derive explicit asymptotic pricing formulas for VIX options under the corrected model. The resulting framework is analytically tractable, numerically stable, and financially sound—providing the first well-posed, theoretically rigorous, and practically implementable approach to modeling VIX derivatives.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We study the pricing of VIX options in the SABR model $dS_t = sigma_t S_t^eta dB_t, dsigma_t = omega sigma_t dZ_t$ where $B_t,Z_t$ are standard Brownian motions correlated with correlation $ ho<0$ and $0 leq eta<1$. VIX is expressed as a risk-neutral conditional expectation of an integral over the volatility process $v_t = S_t^{eta-1} sigma_t$. 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.

Problem

• Pricing VIX options in the explosive SABR model
• Addressing infinite VIX futures and call prices
• Proposing a capped volatility process solution

Innovation

• Capped volatility process prevents explosions
• SABR model analyzes VIX options pricing
• Short-maturity asymptotics for option pricing