Modeling of credit risk, particularly credit risk associated with the term structure of defaultable interest rates, is one of the fundamental research topics in financial mathematics. One general approach, known as intensity-based setting, for modeling the default probability of a specific company is to model the default time as a totally inaccessible stopping time, in the terminology of the general theory of stochastic process. The main tool in this approach is an exogenous specification of the conditional probability of default, given that default has not yet occurred. In most of the cases, this is done by means of the hazard process of the default time, which is postulated to have absolutely continuous sample paths with respect to Lebesgue measure.
In this talk, after an elementary introduction to the valuation of defaultable bonds in the intensity-based framework by well known no arbitrage arguments, I will discuss a theoretical problem on the joint characterization of non-defaultable and defaultable term structures, namely characterizing the negative instantaneous correlation between stochastic processes representing short-rate and intensity of the hazard process. In the second part of my talk, I will discuss possible solutions to this problem and propose an alternative way to characterize intensity of the hazard process by using Jacobi processes. I will also present results, obtained by using spectral methods, pertaining to zero-coupon bond prices of both defaultable and non-defaultable bonds in the proposed model. Also certain properties of Jacobi processes such as their relations to Jacobi polynomials and squared Bessel processes will be discussed.