Hi guys,I'm a PhD-student in financ. maths currently working on (callable) convertible bonds in a reduced form model and I'm looking for a sensible but (in my model and way of (numerically) solving) tractable way to incorporate some default risk. Since it's reduced form, I basically only model the stock price S of the firm issuing the conv. bond by a jump diffusion whose parameters are all constants, also the interest rate is constant. Now I would like to model default of the issuing company by an (independent) Cox process (maybe in an easier form) whose intensity somehow depends on the current stock price (so this intensity would be the only link to the other process S in the model).What I'm looking for is hints or tips towards existing models that incorporate default risk in such a way (so with intensity somehow dependent on the stock price), so far I haven't been able to find them by myself and on advice of Philipp Schoenbucher I'm looking around on this very nice forum .Thanks in advance.

Hi,Why don't you use a first passage model (where company defaults as soon as its asset value falls below a certain level)...I know asset vaue and stock price are not the same thing but it might be worth looking into...(Do you want the deault time or the probability of deafult?)

Look for:- Ayache, Forsyth and Vetzal- Grau, Forsyth and Vetzal- Andersen and BuffumThere is a PhD thesis by Yigitbasioglu on pricing CB with volatility risk at ISMA centre, University of Reading. He uses also reduced form model for credit risk. He also has some papers. But the thesis is better, more complete.

All the ones I cited use reduced form models for credit risk.They all use finite difference schemes.There is also a paper using Monte Carlo by Lvov Yigitbasioglu and El Bachir testing the Longstaff and Schwartz method.

QuoteOriginally posted by: nyamazaniHi,Why don't you use a first passage model (where company defaults as soon as its asset value falls below a certain level)...I know asset vaue and stock price are not the same thing but it might be worth looking into...(Do you want the deault time or the probability of deafult?)Well, that's a possibility I also thought of but my supervisor would like to see something else

QuoteOriginally posted by: madmaxAll the ones I cited use reduced form models for credit risk.They all use finite difference schemes.There is also a paper using Monte Carlo by Lvov Yigitbasioglu and El Bachir testing the Longstaff and Schwartz method.Thanks so far for the refs

It still seems to be difficult to find explicit expressions that are being used. One I came across for modelling the hazard rate p as a fct. of the stock price S is p(S) = p_0 * (S/S_0)^a for some p_0 and a < -1. Any comments to that one?