# How to price a credit-risky zero-coupon bond?

I recently received the following exercise:

Construct a $$n=10$$-period binomial model for the short-rate, $$r_{i,j}$$.

The lattice parameters are:

• $$r_{0,0}=5\%$$,
• $$u=1.1$$,
• $$d=0.9$$ and;
• $$q=1−q=1/2$$.

This I did do at this point without any problems.

Assume that the 1-step hazard rate in node $$(i,j)$$ is given by $$H_{i,j}=ab^{j-i/2}$$ where $$a=0.01$$ and $$b=1.01$$. Compute the price of a zero-coupon bond with face value $$F=100$$ and recovery $$R=20%$$.

Now this is a novel part for me as I till now did not price zero-coupon bonds with a default risk.

My question from here is how do I, incorporate the values of the hazard lattice as well as the recovery rate into a zero-coupon bond pricing lattice. (Thus what formula would I use in order to do so correctly)?

• Hi Noir, to get the best answers to your problem, you should format your question a bit differently. It's a good question, but at first glance, looks just like "how do I do this exercise from start to finish". – AfterWorkGuinness Oct 29 '15 at 12:55
• Hey AfterWorkGuinness thanks for the comment, I reworded the last part of the exercise a bit, as I am indeed only interested in the very last step of the exercise. – Noir Oct 29 '15 at 13:38
• Your notation is super confusing, can you double check as I think you have errors in the problem statement for example " q=1−q=1/2" I assume you mean p = 1-q and q = 0.5 where p is probability of an up-jump and q is the probability of a down-jump ? – AfterWorkGuinness Oct 29 '15 at 18:04
• You should make the notation clearer by not using single letter variables, this will make your question easier to understand and get you an answer faster. – AfterWorkGuinness Oct 29 '15 at 18:51
• Your credit-risk text book should have a formula. It's really the normal price - (prob of default * (portfolio - recovery rate)). My notation is informal (I don't have the book with me), but it's really saying getting the probability of default and how much is lost once it defaults. – HelloWorld Oct 30 '15 at 3:53