THE BOND MARKETS: INTRODUCTION


This paper examines the role of jump-enhanced diffusions (i.e., Poisson-Gaussian processes) in modelling the term structure of interest rates. Theoretical work 011 jump-difFusion term structure models exists,1 and is of recent origin, but little in the way of empirical examination of these models has been undertaken so far. This paper (i) develops jump-diffusion analytics for a wide class of models, and (ii) empirically examines the value of these models.

Motivation: Why should we expect jumps to be a satisfactory modelling device? Stylized facts from the bond markets suggest that jump behavior is ubiquitous. Exogenous interventions in the markets by the Federal Reserve causes jumps. Supply shocks are another factor, as regular debt refundings inject sufficient volume to magnify price effects. Demand shocks such as market behavior at Treasury auctions often result in jumps, as do economic news announcements. As Merton [36] emphasizes, routine trading information releases are well depicted by smooth changes in interest rates, yet bursts of information are often reflected in price behavior as jumps. Jump effects tend to be prevalent in regulated “intervention” environments such as the interest rate and foreign exchange markets.

Raw statistical evidence is strongly suggestive of jumps. Interest rate volatility is very high at the short end of the term structure, and changes in interest rates demonstrate considerable skewness and kurtosis. Poisson-Gaussian processes can flexibly accommodate a wide range of skewness and kurtosis effects. Kurtosis can substantially affect the pricing of derivative securities. Table 1 provides summary statistics for the short rate of interest. The presence of leptokurtosis in interest rate changes in undeniable and makes a strong case for jump models.

The extent of the volatility “smile”, symptomatic of excess kurtosis (fat tails) in the conditional distribution of changes in interest rates, cannot theoretically be sustained by Gaussian models. One way to model the exccss kurtosis is by means of Poisson-Gaussian processes. Other approaches which capture leptokurtosis are stochastic volatility models or simpler time-varying volatility models, such as ARCH processes. The paper examines these various alternative models. The degree of conditional leptokurtosis varies with the time interval between data observations (see Das and Sundaram [24]), and jump-diffusion models allow for parameter choices which match conditional skewness and kurtosis at varying maturities. Chan, Karolyi, Longstaff and Sanders [18] found that interest rates display level dependent volatility to an extent not accounted for by existing theoretical models.