A volatility-varying and jump-diffusion Merton type model of interest rate risk

According to many recent studies, Lévy processes with stochastic volatility seem to be the best candidates for replacing geometric Brownian motion (GBM) as a price process model. This means that the GBM model has to be generalised by introducing the possibility of jumps and allowing the volatility to be a stochastic process. In this paper, we present a generalisation of the traditional Lévy-Merton jump-diffusion model, allowing discrete stochastic volatility. In order to estimate jump instants and jump amplitudes, we use, and improve on, a method based on quadratic variation. We apply this method to two time series provided by the “Banco de España” comprising daily observations of interest rate for operations of 1 day and 1 year (from 4 January 1988 to 31 December 1998).