Modeling Defaultable Bonds

In this lesson, we looked at defaultable bonds and how to model them.  In a later lesson, we will look at pricing them.

A defaultable bond is characterized by a coupon rate, c, a face value, F, and a recovery value, R, which is a random fraction of the value value recovered when the bond defaults.  We will model the term-structure of default using a 1-step default probability:



and we will calibrate h(t) to market prices. h(t) is the conditional probability that the bond will default over the period t to t+1, given the information available at time t: (F(t)).


When we previously modeled the binomial lattice, we had two parameters for each node: i and j.  The i parameter indicated the date: i=0...n and the j parameter indicated the state j=0...i for each time parameter.  For defaultable bonds we will "split" each node (i,j) by introducing a variable that encodes whether or not default has occurred before time i.  For example:

• (i,j,0) <- state j at time i with default time tau > i
• (i,j,1) <- state j at time i with default time tau ≤ i

Now we need to define the new risk-neutral transition probabilities for the binomial lattice.  

Here is one example of the possible transitions for one lattice:

and here are the transitions from no-default state (i,j,0):

and here are the transitions from default state (i,j,1):

Notice that there are much fewer transitions because a bond that has already defaulted cannot be un-defaulted.


We started modeling these bonds with the most simple example:
a default-free zero-coupon bond with expiration date T.  This bond pays $1 in every state at the expiration date T, and no default is possible.
The price of such a bond is Z(i,j,eta,T) where i is the date, j is the state, eta is the default state, and T is the expiration date.  Since default events do not effect default-free bonds: Z(i,j,1,T) = Z(i,j,0,T) = Z(i,j,T) with no default state.
By risk-neutral pricing:

which is exactly what we had previously.

We can calibrate the short-rate lattice using the market prices of default-free zcbs and other default-free instruments.


We then moved on to zero coupon bonds that are defaultable, but have no recovery.  These bonds pay $1 in every state at expiration T, provided that default has not occurred at any date t ≤T.  If default occurs at t≤T, then the bond pays $0.
By risk-neutral pricing:

which means that:

and approximately:

In this case E[i,Qbar] is the expectation of risk-neutral default-free probability.
The price of a zcb is set by discounting the expected value by (r(i,j) + h(i,j)), where h(i,j) is the 1-period credit spread.  The conditional probability of default h(i,j) is also called the hazard rate.  This is the probability of default given that no default has occurred up to time i.


We then moved on to zero coupon bonds that are defaultable and have recovery.  We assumed that the recovery R is random and independent of the default and the interest dynamics.
By risk-neutral pricing:

This makes sense, because here we will recover R instead of zero (like we did one example above).

In the next lesson we will talk about general bonds and how to use these general bonds to estimate the hazard rate.