Time-Varying Skew in VIX Derivatives Pricing

This paper proposes a new reduced-form model for the pricing of VIX derivatives that includes an independent stochastic jump intensity factor and cojumps in the level and variance of VIX, while allowing the mean of VIX variance to be time varying. I fit the model to daily prices of futures and European options from April 2007 through December 2017. The empirical results indicate that the model significantly outperforms all other nested models and improves on benchmark by 21.6% in sample and 31.2% out of sample. The model more accurately portrays the tail behavior of VIX risk-neutral distribution for both short and long maturities, as it better captures the time-varying skew found to be largely independent of the level of the VIX smile. This paper was accepted by Kay Giesecke, finance.

Willow tree algorithms for pricing VIX derivatives under stochastic volatility models

VIX futures and options are the most popular contracts traded in the Chicago Board Options Exchange. The bid-ask spreads of traded VIX derivatives remain to be wide, possibly due to the lack of reliable pricing models. In this paper, we consider pricing VIX derivatives under the consistent model approach, which considers joint modeling of the dynamics of the S&P index and its instantaneous variance. Under the affine jump-diffusion formulation with stochastic volatility, analytic integral formulas can be derived to price VIX futures and options. However, these integral formulas invariably involve Fourier inversion integrals with cumbersome hyper-geometric functions, thus posing various challenges in numerical evaluation. We propose a unified numerical approach based on the willow tree algorithms to price VIX derivatives under various common types of joint process of the S&P index and its instantaneous variance. Given the analytic form of the characteristic function of the instantaneous variance of the S&P index process in the Fourier domain, we apply the fast Fourier transform algorithm to obtain the transition density function numerically in the real domain. We then construct the willow tree that approximates the dynamics of the instantaneous variance process up to the fourth order moment. Our comprehensive numerical tests performed on the willow tree algorithms demonstrate high level of numerical accuracy, runtime efficiency and reliability for pricing VIX futures and both European and American options under the affine model and 3/2-model. We also examine the implied volatility smirks and the term structures of the implied skewness of VIX options.