Bootstrap Results from State Space Form Representation of the Health-Jarrow-Morton Model

Ramaprasad Bhar and Carl Chiarella
School of Finance and Economics, University of Technology, Sydney
R.Bhar@uts.edu.au
Carl.Chiarella@uts.edu.au

Abstract

A major advance in the modelling of the term structure of interest rates and of interest rate derivatives more generally was achieved by Health, Jarrow and Morton (HJM, 1992). Their approach has three features which made it an advance on previous approaches of Merton (1973), Vasicek (1977) and Ball and Torous (1983) to the evaluation of derivatives security prices in a stochastic interest rate environment. Firsly they choose as the driving dynamic variable the instantaneous forward rate of interest instead of the instantaneous spot rate of interest or the bond price as in the alternative approaches refereed to earlier. Since these various rates are related it should in principle make no difference which one chooses as the driving dynamics however in practice the choice of the instantaneous forward rate proves to be far more convenient that the other two. Secondly HJM make use of the concepts of change probability measure and Girsanov's theorem in particular to express prices of interest rate derivative securities in a form which does not involve the empirically troublesome market price of interest-rate risk. Hence this approach avoids making assumptions about investor preferences. Thirdly, because of its choice of the instantaneous forward rate as the driving dynamics the term structure of interest rates that it implies is consistent with the currently observed yield curve. The key input variable to the HJM modelling framework is the function which describes the volatility of the instantaneous forward rate.

To date there has not been a great deal of work on the empirical estimation of the HJM term structure model. Perhaps one reason for the paucity of empirical studies is the difficulty in expressing the equations of the HJM model in a form to which standard statistical estimation methodology can be applied. This difficulty stems in part from the fairly general specification of the instantaneous forward rate volatility that the HJM theory allows, which means that in general the system dynamics are non-Markovian. Bhar and Chiarella (1995a) have shown that by a specific (but nevertheless fairly general) choice of functional form for the volatility of the instantaneous forward rate it is possible to express the system dynamics in Markovian form. More importantly it is then possible to recast the system dynamics in state-space form. It then becomes possible to approach the estimation problem by use of the Kalman filtering methodology outlined for example by Harvey (1989). In Bhar and Chiarella (1995b) this technique has been applied to estimate the HJM model in the Australian market using SFE data on 90 day bank bill and 3 year treasury bond data. Due to the short sample sizes involved it may not be possible to rely for statistical inference upon the standard errors of the parameter estimates obtained by maximum likelihood estimation. Whilst these are asymptotically normally distributed it may be problematic to make this assumption when the samples are of small length. Hence the Monte-Carlo bootstrat methodology was employed to place statistical confidence intervals around the estimates. This approach to bootstrapping state space models had been proposed by Stoffer and Wall (1991). These authors had also suggested comparing the bootstrap sample estimates with an approximation to the true small sample distribution obtained by a parametric Monte Carlo experiment.

In this paper we compare the bootstrap distribution of Kalman filter estimators of the HJM model with an approximation to the true small-sample distribution. We do this by simulating the model for known sets of parameters, and thereby generating time-series of bond prices of typical length dealt with in actual market studies. The Kalman filter estimates of the known parameters are obtained and a bootstrap distribution for them is generated. The approximation to the true small-sample distribution is also generated by a parametric Monte Carlo Experiment. We are thereby able to draw some conclusions about the statistical reliability of the Kalman filter and associated bootstrapping methodology as applied to the HJM model.

It is shown later that the bootstrap resampling technique is a viable option for precise estimation of the parameters of the volatility function that charaterises the HJM model. Also, not all combination of the parameters are equally likely. The approach presented here helps to gain some insight into the behaviour of the instantaneous spot rate of interest implied by the HJM model.