英文文献：The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well-指数期权的形状和期限结构在傻笑:为什么多因素随机波动模型如此有效
英文文献作者：Peter Christoffersen,Steven Heston,Kris Jacobs
英文文献摘要：
State-of-the-art stochastic volatility models generate a "volatility smirk" that explains why out-of-the-money index puts have high prices relative to the Black-Scholes benchmark. These models also adequately explain how the volatility smirk moves up and down in response to changes in risk. However, the data indicate that the slope and the level of the smirk fluctuate largely independently. While single-factor stochastic volatility models can capture the slope of the smirk, they cannot explain such largely independent fluctuations in its level and slope over time. We propose to model these movements using a two-factor stochastic volatility model. Because the factors have distinct correlations with market returns, and because the weights of the factors vary over time, the model generates stochastic correlation between volatility and stock returns. Besides providing more flexible modeling of the time variation in the smirk, the model also provides more flexible modeling of the volatility term structure. Our empirical results indicate that the model improves on the benchmark Heston model by 24% in-sample and 23% out-of-sample. The better fit results from improvements in the modeling of the term structure dimension as well as the moneyness dimension.

最先进的随机波动率模型产生了一种“波动率假笑”，这解释了为什么钱外指数看跌期权相对于布莱克-斯科尔斯基准价格高。这些模型也充分解释了波动率是如何随着风险的变化而上下波动的。然而，数据表明，笑容的斜率和水平很大程度上是独立波动的。虽然单因素随机波动模型可以捕捉假笑的斜率，但它们不能解释随着时间的推移，其水平和斜率在很大程度上独立的波动。我们建议用一个双因素随机波动模型来建模这些运动。由于这些因素与市场回报有明显的相关性，并且由于这些因素的权重随时间而变化，该模型产生了波动率与股票回报之间的随机相关性。该模型除了提供了更灵活的smirk时间变化模型，还提供了更灵活的波动期限结构模型。我们的实证结果表明，该模型比基准的Heston模型在样本内和样本外分别改进了24%和23%。通过对期限结构维度和金钱度维度模型的改进，可以得到较好的拟合结果。