The authors propose the first closed-form price formulas for VIX futures under the widely used discrete-time symmetric GARCH(1, 1) and asymmetric Glosten–Jagannathan–Runkle (GJR) GARCH(1, 1) models. For VIX futures expired before July 21, 2017, the proposed methods, which are truly simple, perform reasonably well in out-of-sample pricing. In regard to pricing errors and efficiency, the new methods significantly outperform a continuous-time benchmark based on the Heston volatility model and a discrete-time benchmark based on the Heston–Nandi GARCH(1, 1). Empirically, GJR is the most “potent”—a term the authors apply to the ability of the model to successfully price VIX futures in the data set. The GJR potency in this study is as high as 96.6%. The novel GARCH approaches are unique with the implication of applicability in real time. Finally, an insight is gained into the research of pricing, namely, that potency is an important gauge of a pricing method. TOPICS: Futures and forward contracts, derivatives, factor-based models Key Findings • Closed-form price formulas for VIX futures under GARCH(1,1) and GJR GARCH(1,1) models are proposed. • The novel approaches are shown to be really competitive for out-of-sample, and more importantly imply applicability in real time, pricing of VIX futures. • Potency, as a gauge of the success rate of pricing, is proposed.

中文翻译：

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VIX期货的有效样品外定价

作者在广泛使用的离散时间对称GARCH（1，1）和非对称Glosten-Jagannathan-Runkle（GJR）GARCH（1，1）模型下提出了VIX期货的第一个封闭式价格公式。对于2017年7月21日之前到期的VIX期货，建议的方法非常简单，在非样本定价中表现良好。关于定价误差和效率，新方法明显优于基于Heston波动率模型的连续时间基准和基于Heston–Nandi GARCH（1，1）的离散时间基准。从经验上讲，GJR是最“有力”的术语—作者将其应用到该模型中的能力，即该模型可以成功地为数据集中的VIX期货定价。这项研究中的GJR效能高达96.6％。新颖的GARCH方法在实时性方面具有独特性。最后，可以从定价研究中获得见识，即效力是定价方法的重要指标。主题：期货和远期合同，衍生品，基于因子的模型主要发现•提出了在GARCH（1,1）和GJR GARCH（1,1）模型下的VIX期货的封闭式价格公式。•新颖的方法被证明对样品外样品具有真正的竞争力，更重要的是，它暗示了VIX期货实时定价的适用性。•建议使用效能作为衡量定价成功率的标准。•新颖的方法被证明对样品外样品具有真正的竞争力，更重要的是，它暗示了VIX期货实时定价的适用性。•建议使用效能作为衡量定价成功率的标准。•新颖的方法被证明对样品外样品具有真正的竞争力，更重要的是，它暗示了VIX期货实时定价的适用性。•建议使用效能作为衡量定价成功率的标准。