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

Abstract

This paper proposes to study volatility index (VIX) futures pricing by directly modeling the logarithmic VIX while incorporating observable dynamic jumps of the VIX, which are derived based on VIX high-frequency data. The impacts of several different interday and intraday jump tests for VIX futures prices are investigated. We obtain the analytical expression by deducing the forward iteration relations of the lagged logarithm VIX, as well as the conditional variance and jump intensity, and use the maximum likelihood method to estimate the parameters under the risk-neutral measure. The empirical results prove the superiority of our newly proposed model (especially the model based on the LM jump test), which indicates that considering the heteroscedasticity effect of conditional variance, introducing VIX high-frequency data information and separating realized jump variation from the realized variance are very important to obtain much more accurate VIX futures pricing.

VIX futures pricing based on high‐frequency VIX: A hybrid approach combining SVR with parametric models

“…We can observe that the logarithm of VIX futures price is a linear function of the contemporaneous and lagged items of the logarithmic VIX. This is consistent with previous studies (Jiang et al, 2022; Wang et al, 2022; Yin et al, 2021), but different from continuous‐time models (Mencía & Sentana, 2013; Park, 2016) and the discrete‐time HN‐GARCH model (Wang et al, 2017), where VIX futures price only depends on VIX at the current time. More importantly, the realized semivariances of VIX information at current time t are also incorporated into the VIX futures pricing via the conditional variances $${h}_{u,t}$$ and $${h}_{d,t}$$ in our proposed model.…”

“…Considering that the VIX itself is the underlying asset of VIX futures, some studies propose constructing related models by modeling the VIX directly, more effectively utilizing the information implied in the VIX (Kaeck & Alexander, 2013; Mencía & Sentana, 2013; Park, 2016; Zang et al, 2017). Recently, Yin et al (2021), Wang et al (2022), and Jiang et al (2022) proposed a direct pricing approach for VIX futures by extending the HAR‐type model, which is commonly used in realized volatility prediction (Corsi, 2009). This kind of direct pricing method avoids many problems of traditional GARCH‐type models for VIX futures pricing, such as time‐consuming computational routines and inaccurate numerical integration.…”

“…This kind of direct pricing method avoids many problems of traditional GARCH‐type models for VIX futures pricing, such as time‐consuming computational routines and inaccurate numerical integration. Specifically, Jiang et al (2022) extend the HAR model by modeling the logarithmic VIX while considering jump‐diffusion decomposition based on realized volatility by using high‐frequency VIX data. Motivated by this idea, we employ another decomposition of RV, the realized upside and downside semivariances calculated from the high‐frequency data of VIX itself, to capture the heteroscedasticity of VIX.…”

“…Moreover, since volatility derivatives only contain risk exposures of volatility, this approach is theoretically feasible and flexible enough to consider other factors that might impact VIX futures prices. Compared with the continuous‐jump decomposition in Jiang et al (2022), it is more concise and does not require complex statistical tests (such as jump tests); thus, it is easier to understand and faster in practical operation.…”

“…First, we examine VIX futures pricing (prediction) by considering the realized upside and downside semivariances (or good and bad volatility) of high‐frequency VIX information to characterize the heteroscedasticity of the VIX based on the direct pricing framework; this method is essentially different from the traditional method of modeling the stock index. We are the first to introduce the asymmetric volatility shocks of VIX into the study of VIX futures pricing, which complements the direct modeling method of VIX futures pricing research (Jiang et al, 2022; Wang et al, 2022; Yin et al, 2021). Second, we prove that the logarithm prices of VIX futures can be calculated as a linear function of the contemporaneous logarithmic VIX, the lagged logarithmic VIX, and the dynamic conditional variances related to the realized semivariances of the VIX.…”