The Asymptotic Behavior of the Term Structure of Interest Rates (English shop)

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Long-term interest rates are essential for the valuation and hedging of various fixed income products and derivatives as well as for the pricing of payments in a distant future, such as long-term infrastructure projects or compensatory adjustments in the course of an accident or a divorce. In the aftermath of the 2008 financial crisis the modeling of interest rate curves with a long time horizon became more and more important due to increased investments in long-term products. Therefore, the study of the asymptotic behavior of the term structure of interest rates has recently achieved new relevance.
In this dissertation we investigate long-term interest rates, i.e. interest rates with maturity going to infinity, in the post-crisis interest rate market. Three different concepts of long-term interest rates are considered for this purpose: the long-term yield, the long-term simple rate, and the long-term swap rate. We analyze the properties as well as the interrelations of these long-term interest rates. In particular, we study the asymptotic behavior of the term structure of interest rates in some specific models. First, we compute the three long-term interest rates in the HJM framework with different stochastic drivers, namely Brownian motions, Lévy processes, and affine processes on the state space of positive semidefinite symmetric matrices. The HJM setting presents the advantage that the entire yield curve can be modeled directly. Furthermore, by considering increasingly more general classes of drivers, we were able to take into account the impact of different risk factors and their dependence structure on the long end of the yield curve. Finally, we study the long-term interest rates and especially the long-term swap rate in the Flesaker-Hughston model and the linear-rational methodology.

Long-term interest rates are required for the valuation and hedging of fixed income financial products and derivatives with long maturities, as well as for the pricing of payments that lie far in the future. Such payments may arise, for example, in long-term infrastructure projects or in compensation arrangements in the event of an accident or a divorce. Particularly in the wake of the global financial crisis of 2008, investors’ interest in investments with a long time horizon grew, and with it the necessity of modeling interest rate curves further into the future and of determining the behavior at the long end of the curves as precisely as possible. The present work is devoted to the investigation of the asymptotic behavior of interest rate curves.
To this end, three different long-term interest rates are analyzed: the long-term yield, the long-term simple rate, and the long-term swap rate. These long-term rates are defined as interest rates whose maturity tends to infinity, within the framework of an interest rate market based on the insights gained from the financial crisis. All relevant model-independent properties of these rates are explained, and the connections between them are examined in detail with respect to their interrelations. Moreover, an important part of this dissertation is devoted to the description of the asymptotic behavior of interest rate curves in specific interest rate models. These models comprise the term structure model of Heath, Jarrow and Morton, known as the HJM framework, the Flesaker-Hughston model, and the linear-rational model. The HJM framework is used for the analysis because of the possibility of modeling the entire term structure curve and all associated forward rates directly. The stochastic component is first described by means of Brownian motion, then by a Lévy process, and finally with the help of an affine process on the state space of positive semidefinite symmetric matrices. The use of these stochastic processes can be understood as a step-by-step further development of the HJM framework, since in each case more factors influencing the term structure can be incorporated into the modeling. The other two models presented, the Flesaker-Hughston model and the linear-rational model, are applied in the analysis of the asymptotic behavior of interest rate curves because of a number of attractive properties, such as simple formulas for all interest rates, which cannot take negative values.