Some Aspects of Statistical Volatility Analysis MIN XU A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS

Volatility is the key of the option price in the stock market. Changes in volatility will dramatically lead to changes of the option price. One of the most important volatilities is historical volatility (HV ). The HV is essentially the annualized standard deviation of the first order difference of logarithm of the asset price. Therefore, changes in HV in finance may be detected by the variance change detection methods in statistics. We propose a weighted sum of powers of variances method to detect single change in HV . It is noted that this method only examines if there is one single change-point in the data sequence. In the second part of the dissertation, we propose the empirical Bayesian information criterion (emBIC) method to detect multiple change-points simultaneously. The empirical BIC method can not only detect change-points in HV , but also in mean, and meanand-variance. Simulation study shows that both of the above methods perform very well. We also apply these methods to detect changes in HV by using real stock data. Another important volatility is the implied volatility (IV ). IV is the volatility of asset implied by the market option price based on Black-Sholes model [Black and Scholes, 1973]. The long term IV and HV have totally different behaviours. We find the optimal time range by using the emBIC method aforementioned above. We explain the long term IV behaviour by interest rate risk and capital charge in the last part of the dissertation.