THE BOND MARKETS: CONCLUDING COMMENTS

The objective of this paper is twofold. One, we develop technical methods for jump-diffusion term structure models. Two, we examine whether it. is worthwhile to enhance existing diffusion models with jump processes.

The methodological innovations of the paper are as follows, (i) We derived the characteristic function for a general jump-diffusion stochastic process for the short rate. The process admits any jump distribution, and hence allows a wide range of possible specifications. This provides the raw material for further analysis, (ii) From the characteristic function we obtained the conditional moments for the short rate. These are useful in examining properties of the higher-order moments and distinguishing jump processes from diffusion models, (iii) We also derived the transition probability function, which is useful in carrying out maximum-likelihood estimation of the model, (iv) Finally, we derived an analytical expression for bond prices in the model.

The methodology is useful in examining the efficacy of jump processes for interest rate models. The evidence appears overwhelming. First, enhancement of the diffusion model with jumps resulted in a significant improvement in fit. Second, the jump model lends itself easily to extended analysis of the impact of information variables, such as the meetings of the Fed Open Market Committee. We found mild evidence that the two-day meetings of the Fed were in fact information revealing to the market. Third, we were able to use the jump model to examine day-of-week effects, and found these to be quite significant.

Wednesdays and Thursdays evidence a much higher likelihood of jumps than other days of the week. This is likely to be the case since option expiry effects may result in sharp market movements. Fourth, recent research has found that the drift term in the stochastic process for interest rates appears to be non-linear. We found that this may be because of an incomplete specification of the random variation in the stochastic process. The addition of a jump process substantially diminishes the extent of non-linearity. In addition, when an ARCH model is superimposed, this provides an even greater reduction.

Finally, it is pertinent to ask whether jump processes do better than diffusion processes in modelling interest rates. We have certainly made a case for the enhancement of diffusion models with jump processes. To say that jump models do better than the best diffusion models would be going too far. For one, the literature is unclear as to what the ‘best’ diffusion model is. And two, the empirical work here clearly suggests an amalgamation of stochastic volatility cum jump-diffusion models. This paper provides in modest fashion, a comprehensive set of methods for jump processes in interest rate modelling, as well as a detailed empirical examination of the term structure using these techniques. quick payday loans