Term structure modelling using the Nelson-Siegel framework

Abstract

Nelson and Siegel [1987] introduce a smooth exponential function to analyze the term structure of interest rates. This greatly facilitates the literature in the Nelson-Siegel model family. In this thesis, the Nelson- Siegel framework is studied extensively in terms of arbitrage-free restrictions, pricing of interest rate derivatives and empirical performance of both in-sample fitting and out-of-sample forecasting. Chapter 1 introduces the motivations and contributions of this thesis. In the following chapter, I review the literature of interest rate models that are used in this thesis. In chapter 3, I study the performance of the dynamic Nelson-Siegel model in fitting and forecasting the government bond yields in Australia. The model is improved by utilizing a more powerful and robust state-space framework estimated with a Kalman filter. I show that this approach outperforms a random walk and a two-step estimation dynamic Nelson-Siegel model in forecasting the Australian government term structure across various forecasting horizons. Using the results from chapter 3 and Diebold and Li [2006], the dynamic Nelson-Siegel model provides exceptional in-sample fitting and out-of-sample forecasting of interest rates. However, the lack of theoretical background is criticized by academics and practitioners, such as the absence of arbitrage-free pricing. In this chapter, I develop a general arbitrage-free Nelson-Siegel model under the HJM framework, see Heath, Jarrow, and Morton [1992]. It maintains a Nelson-Siegel factor loading structure and features unspanned stochastic volatility factors. The corresponding market price of risk is derived based on Duffie, Pan, and Singleton [2000], Trolle and Schwartz [2009a], and Christensen, Diebold, and Rudebusch [2011]. The price of interest rate contingent claims, such as caps, swaptions and bond options, are derived from the generalized arbitrage-free Nelson-Siegel model based on Schrager and Pelser [2006] approximation procedure. To overcome a computationally inefficient pricing scheme in chapter 4, chapter 5 derives a consistent pricing framework of interest rate caps and swaptions based on Carr and Madan [1999] and Duffie, Pan, and Singleton [2000]. This exploits the potential to jointly model the interest rates and their derivatives. I calibrate the model on an extensive panel data including Libor/Swap rate, At-The-Money (ATM) caps and swaptions. By casting the entire model into a state-space form, the extended Kalman filter is employed to calibrate the model. The results show that the model prices both interest rates and their derivatives accurately. Finally, Chapter 6 concludes the thesis and discusses potential extensions.