Functional Data Analysis On Implied Volatility

In financial market, the option investors usually quote according to implied volatilities. Some of them even call "volatility trading" instead of "option trading". So we can see that implied volatility plays an important role in the option market. Empirical studies have shown that the volatilities implied by observed market prices are not constant as the assumption of Black-Scholes option pricing formula did, but with functional characteristics. More precisely, for one day' quote, the implied volatility is a function of two parameters: the strike price and the time to maturity, exhibiting a surfaced shape. However, in practice, to analyze such a high-dimensional surface is very complex, it is desirable to reduce the dimension of this object and characterize the surface through a small number of factors.Based on above-mentioned thinking, we introduce a new method from the perspective of function, that is, Functional Data Analysis, short for FDA. And the analysis of implied volatility is applied as an example. The output tells us that compared with traditional Multi-Statistic methods, this method is more flexible and more effective. So far, functional data analysis is just a new rising topic in foreign countries, and still in its infancy, there are many issues for further study. And the domestic research is almost blank. This paper aims at the thorough research on the theoretical system of functional data analysis, and some innovations in the theory. What's more, based on the theory, use Matlab program to provide new ideas and new methods for the analysis of implied volatility.The content of this article is divided into three parts: Chapter one and two makes the first part. Chapter 1 introduces the background and significance of this paper, the present research overview in both domestic and international, and at last put forwards the ideas of this paper. Chapter 2 focuses on the BS option pricing model and the concept of implied volatility, which is the basic financial background of the following analysis.The second part concludes Chapter three and four. The two sections are our core and innovation. The research is intended to do like this: firstly, to change the discrete data of implied volatility into function. This may involve some nonparametric methods, because it's a typical approach for functional data; Secondly, to carry out functional data analysis. In Chapter 3, two types of commonly used nonparametric methods-- local polynomial estimate and functional basis expansion—are summarized. Besides, the parameters relating to the methods are fully discussed. After that, the local polynomial method is applied to estimate the implied volatility function of CBOE S&P500 index option data. Through this process, an implied volatility surface is obtained, which is rather good, and lays the foundation for the further analysis. In Chapter 4, firstly transit from Multi-Statistic analysis to functional data analysis, and get a comparison of the two methods. And then, the setup and basic theory of functional data analysis is analyzed, especially functional principal components analysis and smoothed principal components. Finally still based on CBOE S&P500 index options, use Matlab programming to achieve the functional principal components analysis of implied volatility.Chapter 5 is our third part. It is a summary of the paper, and points out the direction for further research.