Rational Term-Structure Models and Geometric Levy Martingales
Author(s)
Mackie, Ewan Thomas Braid
Type
Thesis
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.
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.
Date Issued
2011
Date Awarded
2012-03
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Hughston, Lane
Perraudin, William
Sponsor
EPSRC
Creator
Mackie, Ewan Thomas Braid
Publisher Department
Imperial College Business School
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)