Affine futures and forward prices

This thesis is a contribution to the theory of mathematical finance, and is concerned with pricing three contingent claims: zero coupon bonds, futures contracts on a risky asset, and forward contracts on a risky asset. Adapting techniques from the term structure of interest rates we consider factors process models for the asset price and dividend yield as well as the interest rate.; The factors processes we consider are either Gaussian stochastic differential equations or square-root affine stochastic differential equations. Following the approach of Elliott and van der Hoek (2001), based on stochastic flows and the forward measure, we characterize the term structure of bond prices as exponential affine when the short interest rate is an affine function of the factors process. A new aspect of the flows based approach, in the context of bond prices, is the introduction of forward-backward stochastic differential equations. By linking the existence and uniqueness of certain forward-backward stochastic differential equations with the solvability of Riccati ordinary differential equations we are able to unify the results of Elliott and van der Hoek (2001) with the results of Duffle and Kan (1996). Namely, we prove that provided certain Riccati equations are solvable the term structure of bond prices is exponential affine.; We generalize the stochastic flows approach to consider the forward price of a risky asset and the futures price of a risky asset. By the introduction of appropriate measure changes we are able to provide results for forward and futures prices that are analogous to the results for bond prices. For forward prices we employ the risk-neutral measure for the forward price reinvested in the zero-coupon bond as numeraire. For futures prices we employ the risk-neutral measure for the futures price reinvested in the bank account as numeraire. Provided the risk-free interest rate and dividend yield are affine functions of the factors process and the asset price is an exponential affine function of the factors process we are able to characterize the forward and futures prices are exponential affine functions of the factors process.; As in the case of bond prices we introduce forward-backward stochastic differential equations to the analysis of forward and futures prices in order to provide a link with existing techniques which require the solvability of Riccati equations. We prove that provided certain Riccati equations are solvable the forward and futures prices are exponential affine functions of the factors. Our results include as special cases many recent models from the literature and provide more complete results from both a technical standpoint and explanatory standpoint.