Longitudinal Data Analysis On The Model-Free Implied Volatility

One of the most important study of the Black—Scholes option pricing model is about the only unobserved variable, volatility. There are a lot of methods to model the volatility, including the ARCH,SV and implied volatility model.There are different advantages and disadvantages of those models. The ARCH model has the ability of capturing the change of volatility and estimating the parameters in a simple way.However,its prediction about one month shows unreliable and unstable.What is more, the ARCH model can not display the randomness and jumping of volatility,which will not be the problem of SV model.But the computation of SV model is complicate,so it is difficult to apply SV model to the real market.We can calculate the BS implied volatility easily,however,due to the incorrect assumptions of BS option pricing model,the BS implied volatility can not subsume all the information contained in options.Compared with the BS implied volatility,the model-free volatility is independent of any option pricing model and it extracts information from options across all strike prices,thus it is more efficient in forecasting volatility.Nevertheless, the model-free volatility is not a perfect model. The strike prices we observed are discrete,so the major problem is how to deal with the integral of formula using the discrete data. Jiang and Tian (2005) suggested cubic spline to solve this problem, however, its accuracy will be doubted when the data are sparse and irregularly spaced. Luckily, we know this sparse data as longitudinal data in statistics, so we introduce longitudinal data analysis in this case. We hope that the new method will bring us a more reasonable result.The remainder of the article is organized as follow.The section2is literature review. In section3we discuss the derivation process of Black—Scholes option pricing model and the definitions of volatility, BS implied volatility and model-free volatility. We recommend the functional data analysis and local polynomial method in the section4.The next two sections are the core of this article. We introduce the details of the method we used,and it is concluded that our method will be better than cubic spline in sparse situation from the simulation result.We will apply the method we mentioned to the study on the Hang-Seng Index options in section7.The last part is a summary.