The double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

Gatheral [(2008) 引入的双均值回归模型。SPX 和 VIX 期权的一致建模。在7 月 18 日伦敦 Bachelier Finance Society 第五届世界大会上，众所周知，它是一个成功的三因素模型，可以校准到 CBOE 波动率指数 (VIX) 和标准普尔 500 指数 (SPX) 期权。但是，该模型的校准可能会很慢，因为欧式期权没有封闭形式的解决方案。在本文中，我们使用由 Huh等人开发的模型的重新缩放版本。[（2018）。VIX 衍生品的双均值回归模型的缩放版本。数学和金融经济学12: 495–515] 并明确获得欧式期权价格的封闭式定价公式。我们的一阶和二阶近似公式不需要任何复杂的积分计算。我们证明了双均值反转模型的更快校准结果是可用的，并且可以产生实际的 SPX 期权隐含波动率表面。特别是，我们的公式不仅可以捕捉到隐含波动率表面的通常凸行为，而且可以捕捉到 COVID-19 市场中显示的不寻常的下凹行为。