Stochastic Control in the Dynamic Nelson-Siegel Framework

The dynamic Nelson-Siegel (DNS) model is one of the most popular term structure models discussed by academicians and practitioners alike. Despite its immense empirical success in the fields of finance, insurance, and economics, it often suffers from theoretical criticisms. Moreover, in the current dynamic world where minute decisions impact global markets, it is possibly restrained by the deterministic nature of its underlying factors. Hence, the primary aim of this thesis is to mathematically dissect the underlying factors of the model, and thereby propose and justify the utility of a stochastic DNS framework. One of the major contributions of this work is the detailed and elementary proof of existence of all of the possible yield curve shapes attainable by the DNS model, along with the necessary conditions required for the occurrence of each shape. All of the other main contributions of this work are made by considering a stochastic framework of the DNS model. We, thus, consider that the three β(t) factors of the model are driven by stochastic Ornstein-Uhlenbeck processes. This stochastic DNS framework allows for analytically tractable yield curve dynamics, with which we are able to efficiently demonstrate that our framework can reproduce economic stylized facts of the yield curve well-established in both literature and practice. Moreover, we show how the resultant bond price processes possess analytical representations that satisfy financial market interpretations. Furthermore, we demonstrate a real-world application of our stochastic framework: the continuous-time portfolio optimization problem using the stochastic control methodology of {korn2002} under the stochastic DNS framework. We establish sufficient conditions for assets defined in our framework in order to obtain optimal solutions. Moreover, with an appropriate verification theorem that proves the existence and uniqueness of our optimal results, we validate the significance of employing the stochastic DNS framework to solve such portfolio optimization problems. As an excursion, we discuss the long-standing criticism of the lack of consistent bond prices in the Nelson-Siegel family and its subsequent impact on the portfolio problem. We demonstrate that despite this absence of consistency, an investor operating in our stochastic DNS framework does not suffer from any unfair unbounded opportunities.