Rational Term-Structure Models and Geometric Levy Martingales

Abstract

One of the most important problems in modern finance is to understand how best to model the occurrence of jumps in asset pricing models. With this issue in mind, the main topic of this thesis is the development of a set of asset pricing models, driven by Levy uncertainty, applicable across a wide range of asset classes. In particular, we model the term structure of interest rates in a Levy setting, by use of the so-called positive interest models of Flesaker and Hughston. We begin with a brief review of the term-structure literature. We then introduce elements of the theory of Levy processes and develop a rather general theory of geometric Levy models (GLMs) for dynamic asset pricing, paying attention in particular to issues concerning the relation between risk and return for the models under consideration. The special case of a GLM with constant parameters can be regarded as a natural generalisation of the standard geometric Brownian motion used in the Black-Scholes theory. General conditions are established under which assets show a positive risk premium in such a setting. The Flesaker-Hughston approach has the advantage that positive nominal interest rates are built in from the beginning. The resulting models are rational in the sense that the price of a discount bond is given by a ratio of integrals of families of positive martingales. We develop a class of models of this type, where the martingale families are modelled by parametric families of geometric Levy processes. Closed-form expressions are provided for the prices of discount bonds, the short rate of interest, and the prices of options on discount bonds, for various specific cases of Levy uncertainty. In the example of the geometric Brownian motion family we include a rather detailed discussion of the behaviour of the risk premium, and establish conditions under which it is positive. We put forward a proposal for a two-stage calibration of the rational Levy models to the market prices of options. Empirical studies are carried out on the calibration performance of (a) the rational Brownian model, and (b) the rational variance gamma model. We then develop a novel hedging strategy for a portfolio of options on discount bonds. The hedging strategy takes the form of a functional derivative of the option price with respect to the price of the underlying discount bond.