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by Raquel Medeiros Gaspar
This thesis consists of two papers that study forward price term structure models.
Forward prices differ from futures prices in stochastic interest rate settings and become in their own right an interesting object of study.
Forward prices with different maturities are martingales under different forward measures. This
mathematical property makes the term structure of forward prices always connected with the
term structure of bond prices, and this dependence makes forward price terms structure models relatively harder to handle.
For finite dimensional factor models, the first paper (Chapter 1) studies general quadratic
term structures.
These term structures include as special cases the affine term structures and the Gaussian
quadratic term structures, previously studied in the literature. We show, however, that there
are other, non-Gaussian, quadratic term structures and derive sufficient conditions for the existence of these general quadratic term structures for bond, futures and forward prices.
We exploit the connection with the term structure of bond prices and show that even in
quadratic short rate settings we can have affine term structures for forward prices.
Finally, we show how the study of futures prices is naturally embedded in a study of forward
prices, that the difference between the two prices has to do with the correlation between bond
prices and the price process of the underlying to the forward contract and that this difference
may be deterministic in some (non-trivial) stochastic interest rate settings.
In the second paper (Chapter 2) we study a fairly general Wiener driven model for the term
structure of forward prices.
The model, under a fixed martingale measure, Q, is described by using two infinite dimensional
stochastic differential equations (SDEs). The first system is a standard HJM model for (forward)
interest rates, driven by a multidimensional Wiener process W. The second system is an infinite
SDE for the term structure of forward prices on some specified underlying asset driven by the
same W. Since the zero coupon bond volatilities will enter into the drift part of the SDE for
these forward prices, the interest rate system is needed as input into the forward price system.
Given this setup we use the Lie algebra methodology of Bj