| This thesis applies the tools of infinite dimensional stochastic analysis to answer several questions arising in models of the fixed income markets. Because these models assume that there is a continuum of zero-coupon bonds indexed by their date of maturity, they suffer from some difficulties that are not found in the Black-Scholes theory of a stock market with a finite number of tradable assets. For instance, the classical Heath-Jarrow-Morton model [15] assumes the dynamics of the forward rate curve are driven by a finite number of factors, represented mathematically as a d-dimensional Wiener process. However, a result of this assumption is that every interest rate contingent claim can be perfectly hedged by a portfolio of any d bonds chosen a priori. This result is counter-intuitive and contrary to actual market practice. Therefore to account for this observed maturity-specific risk, Musiela [24] and other authors have proposed models of the dynamics of the forward rates driven by a Wiener process in an infinite dimensional space. We consider the problem of hedging a European interest rate contingent claim with a portfolio of zero-coupon bonds in this framework. We apply the Clark-Ocone formula to explicitly compute the martingale representation of the claim in terms of the Malliavin derivative of the payout at maturity. Under natural conditions on the model volatility and the form of the payout function, we find that there exists a hedging strategy which has the desirable property that at all times it consists of bonds with maturities that are less than or equal to the longest maturity of the bonds underlying the claim. Furthermore, under a non-degeneracy condition on the model volatility, all strategies that hedge this type of claim have this property. |
