VIX options in the SABR model

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.

• The paper demonstrates that within the SABR model for 0≤β<1 and ρ<0, the volatility process explodes, leading to infinite call and future prices and zero put prices.
• The authors propose a capped volatility process to remedy the explosion and derive explicit short-maturity asymptotics for VIX options.
• Numerical simulations confirm the analytic asymptotics while underlining the model’s limitations in matching the observed concave VIX smile.

VIX Option Pricing in the SABR Model: Explosions, Remedies, and Asymptotics

Overview and Motivation

This paper rigorously investigates the pricing of VIX options within the SABR stochastic volatility framework, focusing on the regime $0 \leq \beta < 1$ and negative correlation $\rho < 0$. The VIX, defined as a risk-neutral expectation of future integrated variance, is modeled via the SABR dynamics: $$dS_t = \sigma_t S_t^\beta dB_t, \quad d\sigma_t = \omega \sigma_t dZ_t$$ with $B_t$ and $Z_t$ correlated Brownian motions. The volatility process relevant for VIX pricing is $v_t = S_t^{\beta-1} \sigma_t$, which is shown to satisfy a one-dimensional SDE with non-linear drift and diffusion coefficients.

The study is motivated by both theoretical and practical considerations. Theoretically, the SABR model is a limiting case of local-stochastic volatility (LSV) models, but for $0 \leq \beta < 1$ it violates technical conditions required for standard LSV results. Practically, SABR is widely used in financial engineering, making its behavior for VIX derivatives of significant interest.

Volatility Process Dynamics and Explosion

Applying Itô's lemma, the authors derive the SDE for $v_t$: $$\frac{dv_t}{v_t} = \sigma_V(v_t) dW_t + \mu_V(v_t) dt$$ where

$$\sigma_V(v) = \sqrt{\omega^2 + (\beta-1)^2 v^2 + 2\rho (\beta-1) \omega v}, \quad \mu_V(v) = v (\beta-1)\left[ \frac12 (\beta - 2) v + \rho \omega \right]$$

and $W_t$ is a Brownian motion constructed from $B_t$ and $Z_t$.

Under the parameter regime $0 \leq \beta < 1$, $\rho < 0$, the drift and diffusion coefficients are strictly positive and super-linear in $v$. The authors employ the Feller test for explosion, showing that the process $v_t$ explodes in finite time with non-zero probability. This result is robust, as the scale function $p(x)$ associated with the SDE is finite as $x \to \infty$, and the Feller criterion confirms the presence of explosions.

Implications for VIX Derivatives

The explosion of $v_t$ has immediate and severe consequences for VIX derivatives:

• VIX futures and call option prices are infinite for any maturity.
• VIX put option prices are identically zero.

This is a strong and counterintuitive result, contradicting the practical use of the SABR model for VIX option pricing in the $\beta < 1$ regime. The asset price $S_t$ itself remains a true martingale, but the volatility process relevant for VIX is not integrable.

Remedy: Capped Volatility Process

To address the pathological behavior, the authors propose a capped volatility process: $$\frac{dv_t}{v_t} = \hat{\sigma}_V(v_t) dW_t + \hat{\mu}_V(v_t) dt$$ where $\hat{\sigma}_V(v)$ and $\hat{\mu}_V(v)$ are capped versions of the original coefficients, with caps $a, b > 0$: $$\hat{\sigma}_V(v) = \min(a, \sigma_V(v)), \quad \hat{\mu}_V(v) = \min(b, \mu_V(v)) \cdot 1_{\mu_V(v)>0} + \max(-b, \mu_V(v)) \cdot 1_{\mu_V(v)\leq 0}$$ This modification ensures non-explosive behavior and finite moments for $v_t$, making VIX option pricing tractable.

Short-Maturity Asymptotics

The paper derives sharp short-maturity asymptotics for VIX options under the capped process. As $\tau, T \to 0$, the VIX index converges to the spot volatility $v_T$, and the option prices admit large deviation rate function representations: $$\lim_{T \to 0} T \log C_V(K,T) = -J_V(K), \quad K > v_0$$

$$J_V(K) = \frac12 \left( \int_{v_0}^K \frac{dz}{z \hat{\sigma}_V(z)} \right)^2$$

Similar expressions hold for put options with $K < v_0$. The implied volatility smile for VIX options is then given by: $$\sigma_{\mathrm{VIX}}(x) = \frac{\log(K/v_0)}{\int_{v_0}^K \frac{dz}{z \hat{\sigma}_V(z)}}$$ where $x = \log(K/v_0)$ is the log-strike.

ATM, Skew, and Convexity

The paper provides explicit formulas for the ATM level, skew, and convexity of the VIX implied volatility:

• ATM: $\sigma_{\mathrm{VIX}}(0) = \sigma_V(v_0)$
• Skew: $$s_{\mathrm{VIX}} = v_0 (\beta-1) \frac{\rho\omega + (\beta-1) v_0}{2\sigma_{V}(v_0)}$$
• Convexity: $\kappa_{\mathrm{VIX}}$ given by a cubic polynomial in $v_0$ and $\omega$

For $\beta < 1$ and $\rho < 0$, the VIX skew is positive, matching empirical observations of an up-sloping VIX smile. However, the convexity is also positive, which contradicts the observed concave VIX smile in market data. This limits the realism of the SABR model for VIX option calibration.

Numerical Results

Monte Carlo simulations of the capped volatility process confirm the theoretical short-maturity asymptotics. The VIX forward price $F_V(T,a)$ converges to $v_0$ for small $T$, and the implied volatility smile matches the analytic asymptotics. The cap parameter $a$ is chosen sufficiently large so that the capped region is not reached for typical strikes, ensuring the model's practical applicability.

Theoretical and Practical Implications

The analysis demonstrates that the SABR model with $\beta < 1$ is fundamentally unsuitable for VIX option pricing without modification, due to the explosion of the volatility process. The capped process provides a practical workaround, but the model's inability to reproduce the observed concave VIX smile remains a limitation. The results highlight the importance of verifying integrability and explosion properties when applying stochastic volatility models to derivative pricing.

From a theoretical perspective, the paper extends the understanding of SDE explosion criteria in financial modeling and provides explicit short-maturity asymptotics for VIX options in non-standard volatility regimes. Practically, the capped SABR model may serve as a simple approximation for VIX option pricing, but more sophisticated models (e.g., rough volatility, multi-factor LSV) are required for accurate market calibration.

Future Directions

The findings suggest several avenues for further research:

• Development of alternative stochastic volatility models that avoid explosion and reproduce the observed VIX smile convexity.
• Empirical calibration studies comparing the capped SABR model to market data.
• Extension of the explosion analysis to other derivative products sensitive to integrated variance.

Conclusion

This paper provides a comprehensive analysis of VIX option pricing in the SABR model, revealing that for $0 \leq \beta < 1$ and $\rho < 0$, the volatility process explodes, rendering VIX futures and call prices infinite and put prices zero. The proposed capped volatility process remedies this issue and enables tractable short-maturity asymptotics, but the model's inability to match the observed VIX smile convexity limits its practical utility. The work underscores the necessity of rigorous SDE analysis in financial engineering and motivates the search for more robust volatility models for VIX derivatives.