Risk-Neutral Pricing Formula for Zero-coupon bonds with Default Risk

I am looking for the equations or papers showing the risk-neutral pricing for zero-coupon bonds including default risk. I already tried Googling and searching SSRN and Jstor.

A brief educational note and then where you can find the info...

As a first step, set the expected payoff equal to 0 where prob_D = probability of default, cur_Px = current price, mat_Px = maturity payment, and R = recovery.

Therefore

prob_D * (recovery - cur_Px) + (1 - prob_D) * (mat_Px - cur_Px) = 0

results in

prob_D = (cur_Px - mat_Px) / (R - mat_Px)

As an example, say a zero is trading at 75, matures at 100 and would recover 25. Then: prob_D * (25-75) + (1 - prob_D) * (100-75) = 0 solves to prob_D = 1/3

You can think of the first half of the above equation, the part prior to the plus sign, as your return (which will be negative) if there is a default prior to maturity. The second half of the equation can be thought of as your return if there is no default prior to maturity. You probability weight those returns and they sum to 0. Of course that's not a real world probability of default as there should be some risk for which the holder is being compensated, which means that calculated prob_D is an overstatement of the real world probability of default.

And yes, this is vastly simplified, but just understanding the above will help as you read more about this in the 2004 JPM credit derivatives handbook.

Directions: Google "Morgan Stanley Credit Derivatives Handbook" In the first few results you will see old (2004-2008) versions of credit handbooks from both MS and JPM. Find the JPM one. There is way more in there than you need, but they are great. Pages 15-23 and then from 45-54. Pay special attention to the examples on pgs 50 and 51. If you can understand how to calculate hazard rates from CDS and then how to calculate CDS-bond basis, you'll be able to answer your question. You can understand those things by looking at the above. This is not easy stuff to wrap your head around for the first time.

• Thanks. Do you know any models which account for the unamortized original discount not being recoverable in bankruptcy court? Apr 4 '20 at 19:28
• You can build whatever you want into your own model. For my example above if you had 25 recovery on 100 par amount and the amortized portion of par on the zero is say 88, then you would change the recovery in my equation to 25 * 88% = 22 and then solve. It's just changing how much you lose if they default because you recover less. Does that make sense? Apr 4 '20 at 19:47
• @JakeFreeman I forgot to at you in my previous comment Apr 4 '20 at 20:22