Reduced form approach_Chapter 17

[asbes]

Can someone also help me with Reduced form approach.

Take the default based approach for example:

B(t,T) = P(t,T) - P(t,T)*[(1-rr)*q(t,T)]

How is it used and how does it help to price credit derivatives?
What market data is used as inputs?
What is the aim / purpose of the calculation - the output we are after?

 

[Colin McKee]

default based

This has always felt like there should be a lot more explanation behind the formulae that are given, and that these formulae are just the end product of a lot of analysis and assumptions.
However, on a high level, pricing a credit derivative is all about calculating the correct price of the bond. If you can do that, then you can calculate the "cost of default" that is embedded in the price and hence the price of a credit derivative.
the default approach assumes that the bond will either not default (with probability (1-q) ) and be worth the same as a government bond. Or it will default (with probability q and with a likely recovery rate of rr) in which case it will be worth rr. Therefore the value of the bond must be:

P(t,T) * (1 - q(t,T)) + P(t,T) * q(t,T) * rr

Many assumptions then have to be made about q and rr, and how q varies over time.
Hope this helps

 

[asbes]

So the aim is to calculate the price of the risky bond?

But the other items are not observable (and the price of a risky bond may be observable).

The notes say q(t;T) can then be modelled using a square root diffusion process. How can we find the parameter values for this process?

 

[Daleth]

I was also having issues with the reduced form approaches. I assume q(t,T) is the probability at time t that the firm defaults before time T (i.e. analogous to _(T-t)q_(t) from contingencies).

When this is extended to the rating-transition approach the notes give qi(t,T) as "the probability that default occurs given that the debt has a rating i at time t."

"B(t,T) = P(t,T)-P(t,T)[(1-rr)qi(t,T)]"​

is given as the formula for the price of the debt. To me this implies that qi(t,T) incorporates the rating transition matrix and risk-premium adjustment vector. So, qi(t,T) includes the probability that the bond changes credit rating during the period from t to T.

This does not correspond with the solution to Q&A Bank 5, question 2. That suggests that qi(t,T) "is the probability that the bond has credit rating i at time T," and that we must sum over all i to get the value of the bond.

Can anyone shed some light here? I'm inclined to go with the former approach. Also, the final answer does not help with evaluating credit derivatives whose payoff depends on changes in credit rating (although the rating transition matrix could be used to value that).