基于时变波动率与混合对数正态分布的50ETF期权定价

Option Pricing of Mixture of Lognormal Distributions with Time-varying Volatility in 50ETF Option

经典B-S期权定价模型经历了从常数波动率、正态分布到时变波动率、非正态分布的发展历程。对已有针对时变波动率期权定价模型效果的研究进行扩展,以时变波动率模型SSP对经典B-S期权定价公式的常数波动率进行修正,该随机条件波动率的构建充分反映了未来标的资产收益对其波动率的影响;运用广义学生t分布构建时变波动率调整后的B-S期权定价公式,并研究其风险中性概率分布形状,引入混合对数正态模型捕捉实际收益率分布相对于正态收益率分布的偏离;采用2015年2月9日、2月16日和2月25日的50ETF期权高频数据,应用严谨的参数显著性检验、样本内定价偏差和样本外预测偏差的模型选择比较标准,对提出的具有时变波动率的混合对数正态期权定价模型的定价精度进行分析。研究结果表明,中国50ETF期权的标的资产高频收益率呈现出较为明显的有偏和尖峰厚尾分布,收益波动具有明显的聚集特征和长记忆性;采用时变波动率修正后的B-S模型能够显著提高对中国50ETF期权的定价精度;在综合考虑模型对标的资产价格变化动力学的刻画效果以及对期权的定价精确性后,具有时变波动性特征的混合对数正态模型是一个相对更为合理的期权定价模型。研究结果不仅为投资者和监管机构提供了更为准确的期权定价方法,同时也丰富了有关中国50ETF期权典型统计特征的研究。

The classic Black-Scholes model has experienced the development process from the constant volatility to time-varying volatility and from the normal distribution to non-normal distribution. This paper extends prior studies on option pricing models with time-varying volatility and mixture of lognormal distributions.Two frameworks have been proposed to correct misspecification of Black-Scholes model.The first category involves relaxing the con-stant volatility assumption with sigma shape polynomial （ SSP） , in which the specification shows that conditional volatility is sto-chastic, as it is a function of future return over the life of the option.The second category involves relaxing the normality assump-tion using mixture of lognormal distributions, which can capture departures from normal returns with two subordinate lognormal distributions.Furthermore, we rebuild the Black-Scholes model based on the generalized Student t-distribution and study the risk neutral probability distribution, which is changed along with the alternative volatility parameterizations, assuming normality in stock returns.The empirical application is based on 50ETF call options′contracts traded on the selected days in the month of February 2015, a total sample of over 10000 observations.Each record in the data set comprises bid-ask quote, the synchronous-ly recorded spot price of 50ETF, the time at which the quote was recorded, and the strike price.In addition, a range of perform-ance criteria are used to evaluate the model.The first consists of conducting standard tests of significance on the parameter esti-mates.The second concentrates on comparing the relative size of mispricing errors of each model.The third focuses on comparing the relative size of forecasting errors of each model. Finally, the empirical results show that there are some significant characteristics of leverage effect, clustering, and long memory as well as conditional skewness and fat-tail in the high frequency yield of underlying assets of the 50ETF option contracts.Mean-while, the modified classic Black-Scholes model with time-varying volatility can significantly improve the pricing accuracy of 50ETF option contracts in China.Furthermore, in considering the model of the underlying asset price changes in the dynamics of depict and the pricing accuracy on the option contract, the option pricing model with the characteristics of time-varying volatility and mixture of lognormal distributions is a relatively more reasonable option pricing model selection, compared with that of cor-recting the volatility skew associated with the Black-Scholes model. This paper not only provides a more accurate option pricing method for investors and regulators, but also enriches the empirical research conclusions about the typical statistical characteristics of 50ETF option contracts in China.