中国股市期权波动率微笑特征的实证分析

波动率微笑现象显示了期权隐含波动率和执行价格之间的关系.在理想的完全符合Black-Scholes期权定价模型假设的情况下,期权隐含波动率关于执行价格应该是一条水平线.然而,在实证分析中,对隐含波动率和执行价格进行拟合并绘制曲线,会产生一个倾斜或微笑形状的曲线,证明Black-Scholes期权定价模型存在一定的缺陷.本文从Black-Scholes期权定价模型和回归分析出发,尝试用不同的函数形式(对数函数、二次函数和三角函数)拟合波动率的解析表达式并绘制图形,最终以调整的可决系数最大为最优.首先拟合截面数据,对一固定的时间期限拟合出波动率关于执行价格的解析表达式以及波动率微笑曲线,然后将不同时间期限的波动率微笑曲线排列成时间序列,拟合面板数据,即波动率微笑曲面.然而由于面板数据的复杂性,该模型的拟合优度相对于截面数据有所降低,但是在考虑了期限与执行价格对隐含波动率的交互影响后,面板数据模型调整的可决系数显著增大,拟合优度得到提高.

An Empirical Analysis of Option Volatility Smile in Chinese Stock Market

Volatility smile demonstrates the relationship between the implied volatility and the strike prize of options.In the case with the full range of assumptions of the Black-Scholes formula,the implied volatility should be a horizontal line to the strike price.However,in empirical analyses,if we fit the implied volatility to the strike price,a smiling shaped curve will be generated,which proves that the Black-Scholes option pricing model has certain flaws.This paper is based on a Black-Scholes option pricing model and regression analysis.Different functions(logarithmic,quadratic and trigonometric)are used to fit the analytical expression of the volatility and the evaluation metric is the adjusted coefficient of determination.Firstly,we build the model on the cross sectional data,which provides the analytic expression of volatility to strike prize and the volatility smile curve for a fixed expectancy.Then,we arrange volatility smile curves with different expectancies as a time series which makes it a panel dataset and the model can be visualised by a volatility surface.However,due to the complexity of panel data,the goodness of fit decreased significantly comparing to the models on cross sectional data,but after we added the interactive term of expectancy and strike prize,there is a significant increase on adjusted R-squared statistics and it increases the goodness of fit of the panel data models.