In this thesis we model the term structure of zero-coupon bonds. Firstly, in the static setting by norm optimization Hilbert space techniques and starting from a set of benchmark fixed income instruments, we obtain a closed from expression for a smooth discount curve. Moving on to the dynamic setting, we describe the stochastic modeling of the fixed income market. Finally, we introduce the Heath-Jarrow-Morton (HJM) methodology. We derive the evolution of zero-coupon bond prices implied by the HJM methodology and prove the HJM drift condition for non arbitrage pricing in the fixed income market under a dynamic setting.
Knowing the current discount curve is crucial for pricing and hedging fixed income securities as it is a basic input to the HJM valuation methodology. Starting from the non arbitrage prices of a set of benchmark fixed income instruments, we find a smooth discount curve which perfectly reproduces the current market quotes by minimizing a suitably defined norm related to the flatness of the forward curve. The regularity of the discount curve estimated makes it suitable for use as an input in the HJM methodlogy.
This thesis includes a self-contained introduction to the mathematical modeling of the most commonly traded fixed income securities. In addition, we present the mathematical background necessary for modeling the fixed income market in a dynamic setting. Some familiarity with analysis, basic probability theory and functional analysis is assumed.
