An Interest Rate Tree Driven by a Lévy Process

Abstract

The lognormal diffusion process is mathematically tractable and incorporates the kind of continuous random evolution of the price by small increments that seems to characterize most security prices. But market microstructure studies have shown that a lognormal diffusion does not describe very well price formation at the shortest intervals. This is especially true of short-term bond returns. Bond price changes are mostly small, but the tails of the distribution are fatter than the lognormal allows and occasional non-diffusive jumps do seem to occur. Also, the intervals between price changes vary considerably in length. Alternative distributions have been proposed, but they do not have the convenient mathematical properties of the lognormal, so implementation can be challenging. Hainaut and MacGilchrist propose using the normal inverse Gaussian (NIG) distribution that arises from a particular Lévy process and develop a lattice implementation for pricing. A pentanomial tree incorporates the NIG by matching its first four moments. In a simulation exercise, the NIG consistently outperforms the lognormal, largely due to its ability to capture skewness in returns.