A short cut: Directly pricing VIX futures with discrete‐time long memory model and asymmetric jumps

Abstract

This paper proposes a simple but rich framework to directly price volatility index (VIX) futures by applying the heterogeneous autoregressive structure and asymmetric jumps to the logarithm of the VIX. Compared with other discrete‐time models, our model imposes fewer parameter constraints. The analytical solution is also free from time‐consuming and sometimes unstable numerical integration. Empirical results suggest that our model can significantly reduce pricing errors compared with existing models using realized variance, both in‐ and out‐of‐sample. The improvement indicates that besides looking for a better measure of current volatility, it is also important to utilize information embedded in the VIX itself.

Directly pricing VIX futures with observable dynamic jumps based on high‐frequency VIX

“…In addition, the expression of the VIX futures price under the HAR‐DJI‐GARCH model is a linear function of the lagged logarithm VIX and the contemporaneous logarithm VIX, as well as the conditional variance and jump intensity; however, VIX futures prices under the HN‐GARCH model employ only the contemporaneous VIX via the conditional variance. Furthermore, compared with Yin et al (2021), they assume the logarithm VIX has constant variance, while we introduce the heteroscedasticity effect of conditional variance in our newly proposed model. On the other hand, they use the parametric model to characterize the jumping of logarithm VIX.…”

“…Huang et al (2018) studied the VIX term structure and VIX futures pricing with RV based on the LHARG model. Different from the above studies on the pricing of VIX futures by modeling the underlying asset stock price index, Yin et al (2021) proposed directly pricing VIX futures by modeling the logarithm of VIX based on the HAR model and pointed out the importance of using the information contained in VIX itself. Wang et al (2022) extended the direct pricing framework and proposed a new model for VIX futures by adding dynamic volatility with long‐ and short‐run components and dynamic jump intensity.…”

“…At present, there are two main categories of literature about VIX derivatives pricing. One is to model the S&P 500 index under continuous‐time stochastic volatility models (Zhang et al, 2010; Zhang & Zhu, 2006; S. P. Zhu & Lian, 2012; Zhu & Zhang, 2007) or based on discrete‐time Generalized AutoRegressive Conditional Heteroskedasticity (GARCH)‐type models (Guo & Liu, 2020; Wang et al, 2017; Xie et al, 2020; Yang & Wang, 2018; Yang et al, 2019), and the other way is to directly model VIX (the underlying asset of VIX futures; Kaeck & Alexander, 2013; Mencía & Sentana, 2013; Park, 2016; Wang et al, 2022; Yin et al, 2021; Zang et al, 2017). Compared with the former modeling method, the latter, also called the direct pricing method, does not require integration, which can effectively overcome the time‐consuming and sometimes inaccurate integration problems.…”

“…In this way, our new model, which is named the HAR‐DJI‐GARCH model in our study, can accommodate the observable high‐frequency data information of the VIX and is more intuitive. It is worth emphasizing that this is also the typical difference between our model specification and the existing methods to characterize jumps (Yang et al, 2019; Yin et al, 2021).…”

“…In contrast, our newly proposed method avoids the assumption of certain dynamics for the underlying stock index. Compared with Yin et al (2021) and Wang et al (2022), we assume that the conditional variance of the VIX logarithm depends on the observable dynamic jumps, which can make full use of the high-frequency information of the VIX itself. To the best of our knowledge, this is the first study to consider the high-frequency data of VIX in terms of VIX futures pricing.…”