Pricing interest rate derivatives in an illiquid market

Abstract

Globally, one-factor diffusion processes have been popular models for the short rate by virtue of their analytically tractable features. However, due to shortcomings of these models in certain markets a number of models, such as two-factor diffusion and jump diffusion models, have been developed over time. Interest rate models for the South African market have not been researched thoroughly. As a consequence, one-factor diffusion models remain the popular choice in South African interest rate markets. We will investigate, by empirical means, whether one-factor diffusion models are suitable for the modelling of domestic short dated low risk interest rate data. We will show evidence that the South African short rate should be modelled by a pure jump process. The evidence is found through empirically analysing and applying hypothesis tests for jumps on historical 3-month Johannesburg Interbank Agreed Rate (JIBAR) data. We fit a nonstationary compound Poisson process with stably distributed jumps and rate dependent intensities to the 3-month JIBAR. As a result we use a slightly altered model to price options on the 3-month forward JIBAR. We find potentially large changes of these option prices compared to prices derived from a nonparametric one-factor diffusion short rate model. In order to fit a distribution from the family of stable distributions we show how to estimate its parameters. We apply two methods and compare the results with each other. To calculate maximum likelihood estimators (MLEs) we develop a method to estimate stable density function values. We compare these estimators to integrated least squared estimators (ILSEs). ILSEs are asymptotically less efficient than MLEs. However, we develop an algorithm to calculate the ILSEs that is quicker to apply than the method used to find MLEs.