Prisfastsættelse af Obligationer i Kontinuert Tid (The Pricing of Bonds in Continuous Time)

Abstract

In this paper is discussed and analysed a wide number of the different continuous-time interest rate models that has been suggested in the litterature. The focus is mainly on the so-called state-space models - that is models where one or more factors are being explicit modeled. However, initially we discuss the difference between arbitrage-models and equilibirum-models.Next we turn our attention to the linear one-factor models. Here we for example derive and analyse the following models: Cox, Ingersoll and Ross (1985) and Vasicek (1977). In that connection we also analyse the pricing of equity-options with a speciel focus on the Black and Scholes (1973) model.Because of our results obtained from the one-factor models section we now turn our attention to 2-factor models, and here we analyse and derive the following models: Walter (1995), Longstaff and Schwartz (1990/1992) and Vasicek and Fong (1991). These models was mainly introduced due to the limited number of shapes in the yield-curve that is possible with the linear one-factor models.Due to a number of interesting articles by Jamshidian in the late 80s the so-called Gaussian modelling assumption got some new life. One of the reasons for this new interest in Gaussian Models, was the fact that Jamshidian managed to show that it was possible to get closed-form solution for a whole range of interest rate dependent assets in this framework - which is a nice property. For that reason we analyse yield-curve modelling in the Gaussion framework using the Langetieg (1980) m-factor model - where we among other things show that the double-decay model of Beaglehole and Tenney (1991) is embedded in the Langetieg (1980) model.Lastly, we consider a fairly new class of one-factor models, namely the non-linear one-factor models. Here we analyse the following models: Constantinides (1992) and Longstaff (1989). We also consider the more general approach of Beaglehole and Tenney (1991) and Jamshidian (1993). These models also support much more flexible forms of the yield-curve, due to the fact that the term-premia can become both positive and negative - in linear one-factor models the term-premia can only become positive.