THE BOND MARKETS: Estimation 6

We now use the moments to understand the differences between the diffusion-based class of models and the jump-diffusion class. We first note that for any n-factor pure diffusion model, as the time interval for sampling the process goes to zero, i.e. T I 0, the conditional skewness and kurtosis also goes to zero. For example, in the case of a stochastic volatility diffusion model, when the time interval is very small, the volatility of volatility has little time to achieve any play, and so the higher moments are negligible.

As T increases, these moments kick in, and skewness and kurtosis increase in magnitude. As T becomes very large, the multivariate diffusion model starts returning to being asymptotically Gaussian, with the result that the skewness and kurtosis revert to normal values. More info Thus, the graph for skewness and kurtosis tends to be hurnp-shaped, beginning at normal values for small T, then increasing with T, and finally declining back to normal.

On the other hand, in a jump-diffusion model, since at small T, there is still a chance that a large jump w’ill take place, the possibility of a large outlier in comparison to normal variance is very high. This makes for substantial conditional skewness and kurtosis at short maturities. As T increases, the magnitude of the jump in relation to the diffusion shock decreases, and so skewness and kurtosis revert to normal values. Thus, the graph of skewness and kurtosis decline monotonically with T, as can be seen from Figures 3 and 4.

Therefore, an examination of the behavior of the kurtosis of the time series of changes in interest rates (dr) offers a simple way to check if the raw data itself suggests a jump model. If kurtosis declines monotonically wuth T, then it suggests that a jump process is required, since that feature would not be possible with a diffusion model, no matter how many factors it contained. The plot in Figure 5 depicts the kurtosis for interest rate changes where the time interval between observations varies from 1 day to 260 days. The plot has been generated by interfiling the data for n days, where n — 1,2, …260. When n > 1, the data set yields more than one intervalled time series; for example, when n = 2, we have two series, each 2 days apart. The reported kurtosis is the average of the kurtosis of each series. This eliminates to some extent any day-of-week effects that might affect the graph. These day-of-week effects still exists as may be seen from the jaggedness of the plot. However, the monotonic decline in kurtosis is unmistakeable.

Since the empirical kurtosis declines monotonically, as predicted by the theoretical moments from the jump-diffusion model, it confirms two aspects of term-structure models already identified previously in the empirical section: (i) that jumps exist, since the declining kurtosis plot would not arise from a pure-diffusion model alone, unless it were mixed with a jump process, and (ii) in the case of a mixed jump-diffusion model, a declining plot would only arise if jumps constituted a substantial component of the variation in the interest rate sample path. This, as we have seen from the results in Table 3, is certainly the case.

Therefore, the analysis of empirical moments adds conclusive evidence to the maximum-likelihood estimation results in confirming the presence of jumps in the data.