Credit spread (or yield spread) of a given corporate bond is defined to be the difference between its yield and the yield of a government bond, or more generally (and more accurately after the recent debt crisis) a reference bond, which is assumed to be risk-free and has the same time to maturity. The raison d'etre of credit spreads is the risk of default inherent in corporate bonds, in which case the bond holders receives only partial payment or no payment at all. Therefore, in order to price corporate bonds, or more generally any defaultable bond and other credit sensitive instruments, it is necessary to consider the evolution of credit spreads and the risk-free term structure, as well as the correlation structure between these assuming that both are given stochastically.

In a structural credit risk framework, as it is documented by Longstaff and Schwartz (1995), there is an unambiguous economic relation between the credit spread and the risk-free rate, manifesting itself as the negative correlation. On the other hand, in a reduced-form setting, this kind of negative correlation is captured by imposing negative instantaneous correlation between the state variables that drives the defaultable and non-defaultable term structures. However, in a setting where the risk-free rate and credit spread are given by affine diffusions although one has the analytical tractability of the bond prices, due to the admissibility conditions, one cannot simultaneously have a positive spread (or intensity) and risk-free rate while sustaining negatively correlated increments of both. Non-negativity of the spread is a great concern due to the impossibility to construct Cox process with a negative intensity rate. Similarly, it can be shown that in the presence of negative nominal interest rates, arbitrage opportunities arise. Although, non-positivity of the interest rates and intensity process are ignored in the literature by assuming that its probability is close to zero, it might be a concern especially in term-structure modeling and complex derivative pricing. Motivated by the above discussion, the project has these main objectives:

• to come up with a tractable defaultable term-structure model in a reduced-form setting that takes care of empirical stylized facts (negative instantaneous correlation between credit spread and risk-free rate) coherent with the mathematical and economical facts (non-negative intensity and risk-free rates) with an aim towards better understanding of the credit markets.
• to understand better in a general setting the notion of instantaneous correlation in term-structure and credit risk models and its implications in credit sensitive derivative pricing.