Coherent chaos interest rate models and the wick calculus in finance
File(s)
Author(s)
Hadjipetri, Stala
Type
Thesis
Abstract
This thesis develops new tools in stochastic analysis with applications to finance. The first
part presents novel developments in the Wiener chaos approach to the modelling, calibration,
and pricing of interest rate derivatives. To price financial instruments it suffices to
specify the pricing kernel, which in Brownian models can be represented as the conditional
variance of a square-integrable random variable which serves as the “generator" of the pricing
kernel. The coefficients of the chaos expansion of the generator act as the parameters of
a generic interest-rate model. A special class of generators, arising from “coherent" chaos
expansions, is considered, and the resulting interest rate models are investigated. Coherent
representations are important since a kernel generator can be expressed as a linear superposition
of coherent generators. This property is exploited to derive general expressions
for the pricing kernel, along with the associated discount bond and short rate processes.
Pricing formulae for bond options and swaptions are obtained in closed form. The pricing
kernel of a generic incoherent model is then obtained by use of the underlying coherent elements.
Finite-dimensional representations of coherent chaos models are investigated, and
used to construct a class of tractable models having the feature that discount bond prices
are piecewise-flat processes. In the second part of the thesis, a general theory of the Wick
calculus is developed. Novel results concerning the Wick orders of random variables are
derived. In the case where the underlying process is a Brownian motion the Wick calculus
reduces to the Ito calculus, but the former is not restricted to the Gaussian class, and is
applicable to other cases, such as Lévy processes. With financial applications in mind, the
Wick calculus is extended to a wider class of stochastic processes. The thesis concludes
with a change of measure analysis for Wick exponentials of Lévy processes, indicating that
the Wick calculus can be used as a tool for modelling the dynamics of asset prices.
part presents novel developments in the Wiener chaos approach to the modelling, calibration,
and pricing of interest rate derivatives. To price financial instruments it suffices to
specify the pricing kernel, which in Brownian models can be represented as the conditional
variance of a square-integrable random variable which serves as the “generator" of the pricing
kernel. The coefficients of the chaos expansion of the generator act as the parameters of
a generic interest-rate model. A special class of generators, arising from “coherent" chaos
expansions, is considered, and the resulting interest rate models are investigated. Coherent
representations are important since a kernel generator can be expressed as a linear superposition
of coherent generators. This property is exploited to derive general expressions
for the pricing kernel, along with the associated discount bond and short rate processes.
Pricing formulae for bond options and swaptions are obtained in closed form. The pricing
kernel of a generic incoherent model is then obtained by use of the underlying coherent elements.
Finite-dimensional representations of coherent chaos models are investigated, and
used to construct a class of tractable models having the feature that discount bond prices
are piecewise-flat processes. In the second part of the thesis, a general theory of the Wick
calculus is developed. Novel results concerning the Wick orders of random variables are
derived. In the case where the underlying process is a Brownian motion the Wick calculus
reduces to the Ito calculus, but the former is not restricted to the Gaussian class, and is
applicable to other cases, such as Lévy processes. With financial applications in mind, the
Wick calculus is extended to a wider class of stochastic processes. The thesis concludes
with a change of measure analysis for Wick exponentials of Lévy processes, indicating that
the Wick calculus can be used as a tool for modelling the dynamics of asset prices.
Version
Open Access
Date Issued
2014-08
Date Awarded
2014-10
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Brody, Dorje
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)