# Are there closed formulas for non-callable defaultable floating rates in a reduced form models?

currently, I am evaluating for my company the possibility to price defaultable bonds with stochastic default intensity. Precisely, I am considering using the G2++ model where one factor is the riskless rate and the other the default intensity. Hereby, it is an essential goal to allow correlation between the two factors. While it is rather easy to find literature on tree models for this situation (e.g. the article A TREE IMPLEMENTATION OF A CREDIT SPREAD MODEL FOR CREDIT DERIVATIVES by Schönbucher) so far I failed to find any closed formulas. For the beginning, I would already be happy to know if there is a closed formula for a standard floater (i.e. reference period is interest rate period) under this situation, although in the long term I am also interested in pricing bond options in this models.

Does anybody know any literature where this matter is discused?

• I don't know of a closed form. AFAIK, many people use some version the 2-dimensional tree as in Schönbucher. Is your goal simply to get a fair value and its sensitivity to small perturbations in interest rates, credit spreads, and correlations and LGD assumptions? That should be legit and fast enough. However I am a little skeptical that the interplay of interest rates and credit spreads under bigger bumps, i.e. market stress scenario testing, is simple enough to be expressed by one correlation coefficient. Commented Jul 10 at 13:42
• My main goal is indeed the first point. The problem is that we also have some perpetual bonds. For them I derived a geometric series which approximates the closed formulas I had in the case of a one factor model. I used this at the fictional end of my tree to price the perpetual rest payments. For a two factor tree I would like to do the same thing, that's why I am looking for closed formulas. Concerning market stress scenario testing do you have additional approaches/literature you can suggest? Commented Jul 11 at 13:38
• For perps yields and other bond maths, Bloomberg pretends that the maturity date is something like 2149, rather than closed form. Commented Jul 11 at 15:37