 Abstract | This thesis examines finite dimensional representability of Forward Rate and
LIBOR models. A new approach is examined. This approach is more general,
elementary, and relevant to finance when compared with existing approaches.
This new approach is applied to the following infinite dimensional equations
used in finance:
?Gaussian Heath, Jarrow and Morton model;
?Free 1 Heath, Jarrow and Morton model;
?Brace, G?atarek and Musiela’s LIBOR model.
Stronger results have been achieved using this approach. The results are
as follows:
?The Gaussian HJM model can be represented in finite dimensions if and
only if the volatility satisfies a particular differential equation. In which
case the finite dimensional representation can be explicitly written;
?The Brace, G?atarek and Musiela’s LIBOR model with one dimensional
Wiener process cannot be represented in finite dimensions (other than
in a trivial case);
?The Brace, G?atarek and Musiela’s LIBOR model with multidimen-sional
Wiener process, and Free HJM have a finite dimensional repre-sentation
only if the initial yield curves satisfy a restrictive differential
equation.
This thesis is arranged as follows
?Chapter 1 is an introduction to this thesis and derivative pricing in
general. The reader is referred to section 1.4 titled ”This Thesis?for a
more detailed description of the approach of this thesis and its results.
?Chapter 2 contains a brief summary of results from the theory of
stochastic processes, stochastic calculus and stochastic equations in
infinite dimensions
?Chapter 3 contains an overview of spot market pricing models including
the Cox, Ross and Rubinstein and Black and Scholes models.
?Chapter 4 contains an overview of the fixed income market pricing
models including the Heath, Jarrow and Morton model; Musiela’s re-formulation
of the HJM model; the Goldys, Musiela and Sondermann
model; and the Brace, G?atarek and Musiela LIBOR model.
?Chapter 5 contains the primary results of this thesis. Finite Dimen-sional
Representability is defined formally and applied to the Musiela
reformulated Gaussian HJM model; Musiela reformulated free HJM
model; and the Brace, G?atarek and Musiela LIBOR model. This ap-proach
and results are compared with the literature.
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