In this paper we consider the possible dependence of the market price of risk on time and interest rates. This fact gives as a result that the risk-neutral drift, which is one of the coefficients of the pricing equation, also depends on time and interest rates. Then, we estimate the risk-neutral drift directly from the slope of the yield curve. This approach is very accurate as we show with a numerical experiment. In order to obtain the term structure we also propose a suitable finite difference method, which converges to the true solution. Finally, we obtain and compare the yield curves with data from the US Treasury Bill market.
The term structure of interest rates has fascinated generations of researchers, [1], [2], [3] and [4]. This fact should not come as a surprise. An understanding of the stochastic behavior of interest rates is important for the conduct of the monetary policy, the public debt management, the expectations of the real economy activity and inflation, the risk management of a portfolio of securities, and the valuation of interest rate derivatives, [5].
This paper focuses attention on one-factor short rate models. Although they have several shortcomings, they are still very attractive for academics and practitioners. They promise to offer stable and consistent models, with parsimonious structure for the fundamental behavior of interest rates and term structure.
In the empirical implementation of the one-factor models there is only one state variable, the instantaneous interest rate. However when we use the Theory of Arbitrage we also need to use the risk-neutral probability or equivalently, the market price of risk which is unobservable. Traditionally, this function has been considered arbitrary and even constant to find a closed-form solution. However, this fact can lead to misspecification.
The aim of this paper is to analyze the effect on the market price of risk dependence on time and interest rate on the yield curves. The cost of considering more realistic functions in the model is that a closed-form solution is not known. However this is not a problem because we propose an efficient numerical method to provide an accurate approximated solution for the term structure problems. Moreover, we propose to estimate the short rate risk-neutral drift directly from the slope of the yield curve. Therefore, we have to estimate neither the interest rate drift nor the market price of risk. This fact reduces the misspecification of these functions, the computational cost of the model and finally, the term structure errors.
The rest of the paper is organized as follows. Section 2 briefly describes a term structure model with one stochastic variable, the instantaneous interest rate. Then, we show some of the most well-known models in the term structure literature, [2] and [6]. Moreover we propose some generalizations for the market price of risk of these models. As a consequence, we obtain some new models that we call MODCIR and MODCKLS, respectively. In Section 3 we show an efficient approach to estimate the risk-neutral drift directly from data in the markets. Furthermore, we make a numerical experiment to show the properties of this approach. In Section 4 we propose a finite difference method to solve the pricing equation when a closed-form solution is not known and we show some properties as its convergence. In Section 5, we estimate the parameters of traditional [2] and [6], and new MODCIR and MODCKLS models from recent US Treasury Bill data and we obtain the yield curves. We conclude in Section 6.
Traditionally in the literature of the term structure, the interest rate process and the market price of risk have been chosen in terms of simplicity and tractability. For example, the CIR model is affine. In the literature, there is a high array of papers trying to explain which process explains the behavior of the interest rates more precisely, for example [6]. In contrast, there is scarcely evidence of papers trying to explain which process explains more accurately the behavior of the market price of risk or equivalently, the risk-neutral drift. However, the risk-neutral drift is one of the two coefficients of the pricing equation and a key element in the Theory of Arbitrage [8].
The main conclusions that we can draw from this paper are the following. First it is not necessary to estimate either the drift of the interest rates or the market price of risk; it is enough to estimate the risk-neutral drift. This approach allows us to reduce the errors and the computational cost as we showed in Section 3 by means of a numerical experiment.
Secondly, it is not necessary to choose the functions of the model in order to know a closed-form solution for the pricing equation. We propose an efficient numerical method which provides an approximated solution that converges to the exact one.
Finally, from the empirical analysis of recent US Treasury data, we show that considering the market price of risk dependence on time as well as interest rates provides more accurate yield curves when interest rates are not too low. Furthermore, we also show that the risk-neutral drift of the short rates has a higher influence than the volatility in the pricing equation.
