THE BOND MARKETS: ANALYTICAL METHODOLOGY

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The discussion so far begs the question: should we eschew diffusion processes in favor of jump models of the term structure? Our empirical results in the paper show that jumps are a necessary addition to existing diffusion models. We show that the jump process accounts for a large part of the total variation in interest rates, and that the patterns of higher-order moments cannot be generated by diffusion models alone, even if they are multidimensional diffusions. Therefore, rather than view jump models as competing with the best diffusion models, we demonstrate the strong complimentarity of these two stochastic process choices in modelling the term structure.

Thus, this article provides a comprehensive toolkit for the application of jump-diffusion methods to term structure models. The detailed content of the paper comprises two sections. Section 2 provides analytics, and Section 3 contains the empirical implementation. We conclude in Section 4.

ANALYTICAL METHODOLOGY

This section deals with the derivation of the analytics required for maximum-likelihood and method of moments estimation, as well as the analytics of the term structure.

Estimating mean-reverting interest rate processes with jumps entails complications beyond those encountered for processes without mean reversion. There are two main reasons for this:

• For interest rates, no common modelling approach seems to be adopted, and a wide variety of stochastic processes are used, where often, transition density functions are unavailable in closed form.

• Mean reversion substantially complicates the derivation of the conditional transition density function, used for maximum-likelihood estimation. With the lognormal form used for stocks, the exact time at which a jump occurs in any time interval does not matter in the determination of the transition den* sity function, while with interest rate processes, the presence of mean reversion is important, as it affects the drift of the process differentially, depending on where in time the jump occurs.

In this paper, a generalized derivation of the probability function and the moments of the jump-diffusion process surmounts these issues in a framework where estimation is undertaken with the exact densities from the continuous time stochastic process.

The plan for this section is as follows. The stochastic process for the jump-diffusion model is presented, followed by a derivation of the characteristic function. This provides two important by-products: (i) the conditional moments of the process, in particular the kurtosis, and (ii) the transition density functions.