This thesis studies a dynamic Select Sector SPDRs ETFs portfolio optimization problem. The objective of the optimization model is to maximize the risk-adjusted expected return of a portfolio similar to a logarithmic utility maximization. The conditional value-at-risk measure is chosen to be an additional risk exposure constraint. The vector auto-regression (1) regime-switching economic factor model estimated with the expectation-maximization algorithm is employed to identify different market regimes over time. The expected ETFs returns and their variance-covariance matrix used in the objective function of the optimization model are generated by a regime-switching asset pricing model. Both regime-switching models have proven to be superior to respective single-regime models due to their greater predictive ability. The optimized portfolio performance evaluated by Sharpe ratio, Treynor ratio and Jensen’s alpha are all statistically significant compared to those of the equally weighted ETFs portfolio and S&P 500 stock index. This illustrates that incorporating the regime-switching technique, the portfolio optimization model is effective and successful under both bull and bear market conditions.

One of the most famous problem of finding optimal weight to maximize an agent's expected terminal utility in finance literature is Merton's optimal portfolio problem. Classic solution to this problem is given by stochastic Hamilton-Jacobi-Bellman Equation where we briefly review it in chapter 1. Similar idea has found many applications in other finance literatures and we will focus on its application to the high-frequency trading using limit orders in this thesis. In [1], major analysis using the constant volatility arithmetic Brownian motion stock price model with exponential utility function is described. We re-analyze the solution of HJB equation in this case using different asymptotic expansion. And then, we extend the model to the regime-switching volatility model to capture the status of market more accurately.

One of the most famous problem of finding optimal weight to maximize an agent's expected terminal utility in finance literature is Merton's optimal portfolio problem. Classic solution to this problem is given by stochastic Hamilton-Jacobi-Bellman Equation where we briefly review it in chapter 1. Similar idea has found many applications in other finance literatures and we will focus on its application to the high-frequency trading using limit orders in this thesis. In [1], major analysis using the constant volatility arithmetic Brownian motion stock price model with exponential utility function is described. We re-analyze the solution of HJB equation in this case using different asymptotic expansion. And then, we extend the model to the regime-switching volatility model to capture the status of market more accurately.

This thesis explores option pricing and hedging in a discrete time regime-switching environment. If the regime risk cannot be hedged away, then we cannot ignore this risk and use the Black-Scholes pricing and hedging framework to generate a unique pricing and hedging measure. We develop a risk neutral pricing measure by applying an Esscher Transform to the real world asset price process, with the focus on the issue of incompleteness of the market. The Esscher transform turns out to be a convenient and effective tool for option pricing under the discrete time regime switching models. We apply the pricing measure to both single variate European options and multivariate options. To better understand the effect of the pricing method, we also compared the results with those generated from two other risk neutral methods: the Black-Scholes model, and the natural equivalent martingale method. We further investigate the difference in hedging associated with different pricing measures. This is of interest when the choice of pricing method is uncertain under regime switching models. We compare four hedging strategies: delta hedging for the three risk neutral pricing methods under study, and mean variance hedging. We also develop a more general tool of tail ordering for hedging analysis in a general incomplete market with the uncertainty of the risk neutral measures. As a result of the analysis, we propose that pricing and hedging using the Esscher transform may be an effective strategy for a market where the regime switching process brings uncertainty.

Although globally used in option pricing, the Black-Scholes model has not been able to reflect the evolution of stocks in the real world. A regime-switching model which allows jumps in the underlying asset prices and the parameters of the corresponding stochastic process is more accurate. We evaluate the analytical solution for pricing of European options under a two-state regime switching model. Both the convergence of the analytical solution and the feature of implied volatility are investigated through numerical examples. We develop a number of techniques for pricing American options by solving the system of partial differential equations in a general \mathcal{K}-state regime-switching model. The linear complementarity problem is replaced by either the penalty or the direct control formulations. With an implicit discretization, we compare a number of iterative procedures (full policy iteration, fixed point-policy iteration, and local American iteration) for the associated nonlinear algebraic equations. Specifically, a linear system appears in the full policy iteration which can be solved directly or iteratively. Numerical tests indicate that the fixed point-policy iteration and the full-policy iteration (using a simple iteration for the linear system), both coupled with a penalty formulation, results in an efficient method. In addition, using a direct solution method to solve the linear system appearing in the full policy iteration is usually computationally very expensive depending on the jump parameters. A Fourier transform is applied to the system of partial differential equations for pricing American options to obtain a linear system of ordinary differential equations that can be solved explicitly at each timestep. We develop the Fourier space timestepping algorithm which incorporates a timestepping scheme in the frequency domain, in which the frequency domain prices are obtained by applying the discrete Fourier transform to the spatial domain. Close to quadratic convergence in time and space is observed for all regimes when using a second order Crank-Nicolson scheme for approximation of the explicit solution of the ordinary differential equation.

In contrast to the studies of constant or time-varying correlations between stock and bond returns, in this thesis, I explore the regime-dependent correlations between stock and bond returns. Specifically, I start with a comprehensive asset pricing model, i.e., a regime-switching multifactor model, and then investigate the regime-dependent correlations between stock and bond returns. Based on the BIC, the number of regimes in the regime-switching model is optimally determined to be two. For the two regimes, the directions of the regime-dependent correlations appear to be significantly different. Also, the magnitudes of the regime-dependent correlations are substantially larger in these two regimes than the correlation in the single regime. With my findings in the regime-dependent correlations, I then examine the performance of portfolio strategies. Throughout the in-sample and out-of-sample tests, I find that the two portfolio strategies, regime inferred portfolio and probability implied portfolio, can outperform the benchmark, S&P 500.

Although globally used in option pricing, the Black-Scholes model has not been able to reflect the evolution of stocks in the real world. A regime-switching model which allows jumps in the underlying asset prices and the parameters of the corresponding stochastic process is more accurate. We evaluate the analytical solution for pricing of European options under a two-state regime switching model. Both the convergence of the analytical solution and the feature of implied volatility are investigated through numerical examples. We develop a number of techniques for pricing American options by solving the system of partial differential equations in a general \mathcal{K}-state regime-switching model. The linear complementarity problem is replaced by either the penalty or the direct control formulations. With an implicit discretization, we compare a number of iterative procedures (full policy iteration, fixed point-policy iteration, and local American iteration) for the associated nonlinear algebraic equations. Specifically, a linear system appears in the full policy iteration which can be solved directly or iteratively. Numerical tests indicate that the fixed point-policy iteration and the full-policy iteration (using a simple iteration for the linear system), both coupled with a penalty formulation, results in an efficient method. In addition, using a direct solution method to solve the linear system appearing in the full policy iteration is usually computationally very expensive depending on the jump parameters. A Fourier transform is applied to the system of partial differential equations for pricing American options to obtain a linear system of ordinary differential equations that can be solved explicitly at each timestep. We develop the Fourier space timestepping algorithm which incorporates a timestepping scheme in the frequency domain, in which the frequency domain prices are obtained by applying the discrete Fourier transform to the spatial domain. Close to quadratic convergence in time and space is observed for all regimes when using a second order Crank-Nicolson scheme for approximation of the explicit solution of the ordinary differential equation.

This thesis explores option pricing and hedging in a discrete time regime-switching environment. If the regime risk cannot be hedged away, then we cannot ignore this risk and use the Black-Scholes pricing and hedging framework to generate a unique pricing and hedging measure. We develop a risk neutral pricing measure by applying an Esscher Transform to the real world asset price process, with the focus on the issue of incompleteness of the market. The Esscher transform turns out to be a convenient and effective tool for option pricing under the discrete time regime switching models. We apply the pricing measure to both single variate European options and multivariate options. To better understand the effect of the pricing method, we also compared the results with those generated from two other risk neutral methods: the Black-Scholes model, and the natural equivalent martingale method. We further investigate the difference in hedging associated with different pricing measures. This is of interest when the choice of pricing method is uncertain under regime switching models. We compare four hedging strategies: delta hedging for the three risk neutral pricing methods under study, and mean variance hedging. We also develop a more general tool of tail ordering for hedging analysis in a general incomplete market with the uncertainty of the risk neutral measures. As a result of the analysis, we propose that pricing and hedging using the Esscher transform may be an effective strategy for a market where the regime switching process brings uncertainty.

For the empirical macroeconomist, accounting for nonlinearities in data series by using regime switching techniques has a long history. Over the past 25 years, there have been tremendous advances in both the estimation of regime switching and the incorporation of regime switching into macroeconomic models. In this dissertation, I apply techniques from this literature to study two topics that are of particular relevance to the conduct of monetary policy: asset bubbles and the Federal Reserve’s policy reaction function. My first chapter utilizes a recently developed Markov-Switching model in order to test for asset bubbles in simulated data. I find that this flexible model is able to detect asset bubbles in about 75% of simulations. In my second and third chapters, I focus on the Federal Reserve’s policy reaction function. My second chapter advances the literature in two important directions. First, it uses meeting- based timing to more properly account for the target Federal Funds rate; second, it allows for the inclusion of up to 14 economic variables. I find that the long-run inflation response coefficient is larger than had been found in previous studies, and that increasing the number of economic variables that can enter the model improves both in-sample fit and out-of-sample forecasting ability. In my third chapter, I introduce a new econometric model that allows for Markov-Switching, but can also remove variables from the model, or enforce a restriction that there is no regime switching. My findings indicate that the majority of coefficients in the Federal Reserve’s policy reaction function have not changed over time.

In this paper, focus is on the global vector autoregressive (GVAR) model. Its attractiveness stems from an ability to incorporate global interdependencies when modeling local economies. The model is based on a collection of local models, which in general are estimated as regular VAR models. This paper examines alternative specifications of the local models by estimating them as regime-switching VAR models, where transition probabilities between different states are studied using both constant and time-varying settings. The results show that regime-switching models are appealing as they yield inferences about the states of the economy, but these inferences are not guaranteed to be reasonable from an economic point of view. Furthermore, the global solution of the model is in some cases non-stationary when local models are regime-switching. The conclusion is that the regime-switching alternatives, while theoretically reasonable, are sensitive to the exact specification used. At the same time, the issue of specifying the regime-switching models in such a way that they perform adequately speaks in favor of the simpler, yet functional, basic GVAR model.