Pricing of stochastic interest bonds using affine term structure. Models: a comparative analysis

The aim of this work is to use one-factor stochastic term structure models to evaluate stochastic interest bonds, that are bonds bundled together some interest rate derivative, and to compare them with the theoretical value that the issuer indicates in the prospectus for the public offering. Stochastic interest bonds are a sub-set of the big family of structured bonds, the latter being bonds that present specific algorithms driving coupons computation and payment at maturity, mainly due to the presence of one or more derivative components embedded in their financial structure. Structured bonds are mainly issued by banks. Over the last two decades the offering of structured bonds to retail investors has consistently increased, with a contextual rise in the variety of the payoff structures. Chapter 1, after a brief exposure of the evolution of term structure models and their classification, is devoted to analyze several one-factor affine term structure models: the Vasicek model, the Ho-Lee model and the Hull-White model. Chapter 2 shows how to use the above models to price some typical interest rate derivatives (namely caps and floors) that are often embedded in the structure of stochastic interest bonds like those that will be considered in Chapter 5, which in fact, will include either a cap or a floor or both these two types of interest rate derivatives. Chapter 3 is devoted to analyze some key concepts about credit risk in order to take into account the impact of this risk factor on the bond value. To this aim, we will illustrate some key results regarding credit derivatives, and, specifically, credit default swaps whose market quotes allow to infer reliable estimates of the cumulative and intertemporal default probabilities of an issuer at various maturities by using the so-called bootstrapping technique. Once these default probabilities are estimated they can be used to derive a general pricing formula for defaultable bonds which will be used to perform the fair evaluation of the ten stochastic interest bonds analyzed in Chapter 5. Chapter 4 is devoted to study in detail the financial engineering of a specific kind of stochastic interest bonds, namely the so-called collared floaters, which are floating-rate coupon bonds whose coupons are subject to both an upper and a lower bound, hence embedding two interest rate derivatives, either a long cap and a short cap or a long floor and a short cap depending on the specific unbundling choice we make. In particular, the unbundling of a generic collared floater into its various elementary components is examined, as it will be useful to the pricing of many bonds included in the set of securities analyzed in Chapter 5. Chapter 5 is focused on the pricing of ten stochastic interest bonds recently issued by four of the major Italian banks: six of them are pure collared floaters, two of them are mixed fixed-floating coupon bonds, whose floating coupons have the typical structure of collared floaters, one bond is a floating-rate coupon bond embedding a floor, and one bond is a floating-rate coupon bond embedding a floor for the first half of its life and a cap for the second half of its life. After the illustration of their unbundling, these bonds are priced by means of two alternative pricing methodologies. The first methodology is based on the unbundling of their financial structure which reveals how these bonds can be seen as the composition of one or more pure bond components and of one or more interest rate derivatives, namely caps and/or floors, whose closed formulas - in the framework of the one-factor affine term structure models of Chapter 1 developed under the risk neutral probability measure - have been presented in Chapter 2. The second methodology relies instead on Monte Carlo simulations, performed again under the risk neutral probability measure; in this case the fair value of a bond is determined by discounting back at the evaluation date the final value of the security over each simulated trajectory and, then, by averaging these discounted values. The two pricing methodologies are implemented both in the framework of the Vasicek model and in that of the Hull and White model. Their results turn out to be consistent and, compared with the theoretical value indicated in the final terms of the prospectus published by the issuers, are a useful instrument to explore the reliability and the accuracy of the informative set included in this document that investors use to take their financial decisions.