THE BOND MARKETS: INTRODUCTION 2

Recent work by Brenner, Harjes and Kroner [15] and Koedijk, Nissen, Schotman and Wolff [32] provides strong evidence that in addition to level dependence, time varying volatility (i.e. ARCH) models provide a better empirical fit. In this article, we explore simple versions of these models with jumps, and find improvements in fit. We essentially conclude that mixed models with stochastic volatility and jumps are predicated. We briefly summarize the results of the paper, theoretical and empirical.

Analytics: The following are developed. First, characteristic functions for a range of jump-diffusion stochastic processes (irrespective of jump distribution) are derived, thereby obtaining the primary tool for pricing and statistical analysis of our models. Second, probability density functions are obtained for estimation by maximum-likelihood and for derivative security pricing. Third, the first four moments for jump-diffusion stochastic processes are calculated in closed form so as to enable method of moments estimation methods. Fourth, analytical expressions for bond prices are derived. Hence, the paper offers a comprehensive set of tools for the application of jump-diffusion processes to term structure models. These methods do not rely on specific choices of the jump distribution, but apply to any jump distribution with finite moments.

Empirical Work: The content of the paper is as follows. First, exact estimation of specific jump-diffusion models is possible in continuous time, and is undertaken, evidencing a good fit for this class of models. Second, a more easily implementable and analogous discrete time method is used to integrate ARCH type models with jump-diffusions, and estimation results show that the best models we consider are those that contain features of jumps as well as time varying volatility.

Third, the flexibility of the estimation approach is exploited by enhancing the model for day of the week effects, wherein we find that jumps are most likely on Wednesdays and Thursdays, probably on account of option expiry effects. Fourth, the model is enhanced to make jumps dependent on Federal Reserve activity, and we find that the two-day meetings of the Federal Open Market Committee appear to have an information effect. Fifth, estimation is also undertaken using the analytically derived moments, and the presence of jumps is confirmed in a general model, where no jump distribution is imposed a priori. Sixth, an examination of the structure of empirical moments confirms that they would not be generated by diffusion processes alone, indicating strongly that jumps be added to diffusion-based term structure models.