A stochastic volatility model is studied where the volatility process is modeled using a fractional Brownian motion. The Hurst parameter of the fractional Brownian motion is estimated from discrete observations of the price process. The method estimates the magnitude of the fractional Brownian motion path at a lesser frequency. Three Hurst parameter estimators are constructed based on the 2-variation of the absolute fractional Brownian motion process. Empirical results are provided and one of the estimators is proven to be strongly consistent.
A more general model is considered in which the volatility process is a function of a fractional Brownian motion. The magnitude estimation method is adapted to recover the volatility process for this model. Existing 2-variation estimators are adapted to estimate the Hurst parameter. Empirical results are shown for the cases where the function is an exponential and standard logistic function of a fractional Brownian motion.
The magnitude estimation method is applied to stock trading data to recover the volatility process of several stock price processes. Bear Stearns stock is considered during its collapse in 2008. The recovered volatility approximations are compared to the results of a time series filtration method.
The magnitude estimation method and Hurst parameter estimators are used to estimate the daily Hurst parameter behavior of Dow Jones Index stocks in 2008 during the collapse of Bear Stearns and on Dow stocks in a more stable economic time in early 2015. Two models are considered in which the volatility process is assumed to be an affine transformation or an exponential function of a fractional Brownian motion. The resulting observed Hurst parameter behavior is compared between the time periods and between the assumed volatility models.
|Commitee:||Chatterjee, Rupak, Lonon, Thomas, Suffel, Charles, Zabarankin, Michael|
|School:||Stevens Institute of Technology|
|School Location:||United States -- New Jersey|
|Source:||DAI-B 77/11(E), Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics, Statistics|
|Keywords:||Fractional brownian motion, Hurst parameter, Quadratic variation, Statistical estimation, Stochastic volatility model|
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