The Statistical and Economic Role of Jumps in Continuous‐Time Interest Rate Models

Abstract

This paper analyzes the role of jumps in continuous-time short rate models. I first develop a test to detect jump-induced misspecification and, using Treasury bill rates, find evidence for the presence of jumps. Second, I specify and estimate a nonparametric jump-diffusion model. Results indicate that jumps play an important statistical role. Estimates of jump times and sizes indicate that unexpected news about the macroeconomy generates the jumps. Finally, I investigate the pricing implications of jumps. Jumps generally have a minor impact on yields, but they are important for pricing interest rate options.

A new technique to estimate the risk-neutral processes in jump–diffusion commodity futures models

“…Bandi and Nguyen [1] and Johannes [14] showed how to estimate the functions of a jump-diffusion process by means of their moment equations for interest rate models. However, this approach does not allow us to estimate the market prices of risk, which are necessary to price commodity derivatives but not observable.…”

“…We also assume that the jump magnitude and the jump arrival time are uncorrelated with the diffusion parts of the processes. We suppose that the functions µ S , µ δ , σ S , σ δ , J, λ and Π satisfy suitable regularity conditions: see [5] and [14]. Under the above assumptions, a commodity futures price at time t with maturity at time T , t ≤ T , can be expressed as F (t, S, δ; T ) and at maturity it is…”

Operator Methods for Continuous-Time Markov Processes

“…The drift and diffusion will only contribute to the first two conditional moments. Johannes (2004) used this observation to infer the importance of jump components in interest rates.…”

“…First, we consider the identification scheme advocated by Johannes (2004). Consider first linear test functions parameterized as φ(x) = a · (x − x * ) for some a ∈ R m and some x * .…”

A Jump-Diffusion Nominal Short Rate Model

“…Nevertheless, since LP consider the case of pure diffusion processes, the obtained dynamics of the various short-term interest rates do not meet empirical evidence on the presence of discontinuities in the process of the interest rate. For instance, Das (2002) and Johannes (2004) provide evidence of the presence of jumps in the dynamics of interest rates. Similarly, Guan et al (2005) find evidence that a jump-diffusion setting is required to efficiently explain the Libor rates dynamics.…”