In this dissertation, an approximate version of the Barndorff-Nielsen and Shephard model, driven by a Brownian motion and a Lévy subordinator, is formulated. The first-exit time of the log-return process for this model is analyzed. It is shown that with a certain probability, the first-exit time process of the log-return is decomposable into the sum of the first exit time of the Brownian motion with drift, and the first exit time of a Lévy subordinator with drift. Subsequently, the probability density functions of the first exit time of some specific Lévy subordinators, connected to stationary, self-decomposable variance processes, are studied. Analytical expressions of the probability density function of the first-exit time of three such Lévy subordinators are obtained in terms of various special functions. The results are implemented to empirical S&P 500 dataset. After this exit time analysis, in this dissertation, we propose a model for the soybean export market share dynamics and analyze the empirical data using machine and deep learning algorithms. We justify the proposed general model and provide several theoretical analyses related to a special case of the general model. The empirical data set is a time series with weekly observations over the period January 6, 2012, through January 3, 2020. This is a period of growing intense competition, and during which a trade war had influenced the results. The target variable is the share of soybean exports made from the US Gulf to China. We implement machine and deep learning-based techniques to analyze the empirical data. Various numerical results are obtained. The results indicate that export market shares, which are otherwise highly volatile, can be effectively explained (predicted) using machine/deep learning methodologies and a set of logical feature variables. We conclude this dissertation with an analysis of option pricing and implied volatility in the case when the market is driven by a jump-stochastic volatility model. We find the price of the European call option in this case. In addition, we implement Malliavin calculus to analyze the implied volatility.