This study investigates a number of issues concerning one factor interest rate models. We empirically estimate a range of models, and we compare and contrast them. We also show how to implement these interest rate models and identify which are best for forecasting and which are best for valuing fixed-income securities and derivatives. Our motivation is the importance of interest rates in the economy and the immense increasing uses of interest rate derivatives in the financial markets worldwide, as well as the mixed evidence of interest rate process documented in recent studies. Specifically, we examine two classes of one-factor interest rate models that allow the conditional mean (drift) and conditional variance (diffusion) to be functions of the current short rate. These two classes are short-term interest rate models and no-arbitrage term structure of interest rates models.
Short-term interest rate models are designed to develop a short rate process that characterises the empirical data as closely as possible. Given the knowledge of the current short rate and model parameters, we can determine the term structure of interest rates and value fixed-income securities and their derivatives. However, there is no guarantee that the theoretical term structure from the short rate model is the same as the actual observed term structure, which is a potential weakness.
No arbitrage term structure models, on the other hand, are designed to develop a short rate process that is consistent with the current observed term structure. This framework excludes arbitrage opportunities. Hence, the relative strength of this class of models is the capability to price securities and derivatives to be consistent with observed market prices. However, fitting the initial term structure reduces the capability to accommodate more complex underlying short rate dynamics.
We begin our study by examining the robustness of a range of popular short-term interest rate models over different data sets, time periods, sampling frequencies, and estimation techniques. We find that parameter estimates are highly sensitive to all of these factors in the eight countries that we examine. Since parameter estimates are not robust, these models should be used with caution in practice. In addition, we also examine the importance of non-linearities in the mean reversion and volatility of short-term interest rates. We find that different countries call for different models. In particular, we find evidence of non-linear mean reversion in some of the countries that we examine, linear mean reversion in others, and no mean reversion in some countries. For all countries we examine there is strong evidence that the volatility of interest rate changes is highly sensitive to the level of the short term interest rate. Finally, we find that more complex models have higher forecasting performance but with the tradeoffs that there is no closed-form formula for the term structure and hence require numerical approach to value fixed-income securities and their derivatives.
For the no-arbitrage term structure model, we review the arbitrage-free trinomial tree approach of Hull and White (1994), and then apply this approach to value a range of interest rate derivatives such as caps, floors, collars, and swaps. We also provide detailed numerical examples for expositional purposes. We do this so that we could confidently use the Hull and White technique in later section as the benchmark to measure the performance of the simulation technique employed in the last chapter.
Finally, we conclude our study by focusing on practical issues underlying both short-term interest rates and no-arbitrage term structure models. We demonstrate how to use a Monte-Carlo simulation to infer the entire term structure of interest rates from the short rate process and then value interest rate derivatives when a closed-form solution for the term structure is not provided. We also verify that the simulation technique is robust in pricing bond and interest rate derivatives by comparing interest rate derivative values from the simulation approach to those from the Hull and White, and the closed form formula approaches. When the closed-form formula is nor available, we value interest rate derivatives such as caps using the simulation and the Hull and White trinomial tree approaches, then compare and contrast them. By construct, we know that the Hull and White gives prices that are consistent with the market prices so we use the Hull and White price as our benchmark. Among all models we examined, we find that a more complex model such as the Aït-Sahalia (AS) model happens to provide us price that is a closer match to the arbitrage-free cap price from Hull and White approach.