Fiducial inference was introduced by R.A. Fisher Fisher (1930) as a response to the Bayesian approach to inference. The Bayesian paradigm begins by assuming a prior distribution on the parameter space and inference is conducted via the posterior distribution. Fisher, however, was concerned about the choice of the prior distribution, especially when there is insufficient information about the parameters of interest. To overcome this weakness, Fisher introduced the fiducial argument which is based on the following idea: randomness is transferred from the model space to the parameter space and a distribution on the parameter space can is defined that captures all of the information the data contains about these parameters. Fisher’s idea, however, soon fell into disfavor since some of the properties Fisher claimed did not hold. Recently, Fisher’s inferential framework was revived through its connection to generalized inference. Hannig (2009) generalized Fisher’s idea and introduced a framework where fiducial distributions can be defined properly. The main topic of this dissertation is to apply generalized fiducial inference methods to study intraday volatility using high frequency stock market data. In particular, we apply a generalized fiducial framework that is designed for interval data to study high frequency volatility, Hannig (2013). Our approach allows us to view the bid-ask spread as a natural interval around the latent price and use high frequency quotes for estimation. Modeling the spread in this manner allows us to take advantage of the features of the observed prices inherent to the trading process, such as rounding, and reduce the impact microstructure frictions cause to estimation. We demonstrate that our approach is very effective in estimating volatility and outperforms all alternative estimators. In chapter 2, we apply this idea, assuming that rounding errors are the only source of microstructure frictions. In chapter 3, we extend our framework to allow for additive components. In the final chapter, we perform an empirical study to compare alternative realized volatility estimators through option pricing formulas. We find that the choice of volatility estimators does matter.