The absence of diffusion in the South African short rate

Abstract

In the field of Financial Mathematics, stochastic differential equations are used to describe the dynamics of interest rates. An example is a model for the short rate, which is a mathematically defined rate not directly observable in any market. However, observable rates such as short dated Treasury rates or the Johannesburg Interbank Agreement Rate (JIBAR) can be used as proxies for the short rate. The short rate dynamics are traditionally modelled by one-factor diffusion processes. These type of models remain popular due to the analytical tractability of the pricing formulae of interest rate derivatives under these models. To capture the leptokurtic nature of interest rate returns in the South African market, two types of models can be used: a pure jump model or a jump diffusion model. In this paper we investigate whether jumps are present and whether a diffusion component is evident. Our initial investigation showed that jumps were present in the South African market, and that no diffusion component was evident at low interest rate levels. This result was found using a Monte Carlo method to test for jumps. We therefore conclude that a pure jump process is an appropriate model for the South African short rate