Term structure modelling : pricing and risk management
This thesis is about interest rate modelling with applications in pricing and risk management of interest rate derivatives and portfolios. The first part of the thesis is developed within the random field framework suggested by Kennedy (1994). The framework is rich enough to be used for both pricing and risk management, but we believe its real value lies in the latter. Our main objective is to construct infinite-factor Gaussian field models that can fit the sample covariance matrices observed in the market. This task has not previously been addressed by the work on field methodology. We develop three methodologies for constructing strictly positive definite covariance functions, characterising infinite-factor Gaussian fields. We test all three constructions on the sample covariance and correlation matrices obtained from US and Japanese bond market data. The empirical and numerical tests suggest that these classes of field models present very satisfactory solutions to the posed problem. The models we develop make the random field methodology a much more practical tool. They allow calibration of field models to key market information, namely the covariation of the yields. The second part of the thesis deals with pricing kernel (potential) models ofthe term structure. These were first introduced by Constantinides (1992), but were subsequently overshadowed by the market models, which were developed by Miltersen et al. (1997), and Brace et al. (1997), and are very popular among the practitioners. Our objective is to construct a class of arbitrage-free term structure models that enjoy the same ease of calibration as the market models, but do not suffer from non-Markov evolution as is the case with the market models. We develop a class of models the within pricing kernel framework. I.e., we model the pricing kernel directly, and not a particular interest rate or a set of rates. The construction of the kernel is explicitly linked to the calibrating set of instruments. Thus, once the kernel is constructed it will price correctly the chosen set of instruments, and have a low-dimensional Markov structure. We test our model on yield, at-the-money cap, caplet implied volatility surface, and swaption data. We achieve a very good quality of fit.