Pricing Defaultable Bonds

In this lesson, we extended the last lesson on pricing defaultable bonds to pricing coupon-bearing defaultable bonds as well as calibrating them.

We are assuming that the hazard rates, h(i,j) are state independent. This ensures that the default event is independent of the interest rate dynamics. We also let q(t) be the risk-neutral probability that the bond will survive until date t.  We then combine q(t) and h(i,j) into;

so as we can define the probability for survival in the next period based on survival up to that period.

Let I(t) denote the indicator variable that the bond survives up to time t;

The indicator variable that the default will occur at time t is I(t-1) - I(t).  It therefore follows that;
E[I(t)] = q(t).

Again, we assume that the random recovery rate, R, is independent of the interest rate dynamics as we did previously.  R denotes the fraction of the face value, F, paid on default.

We are going to assume the following information for the pricing process;

1. The current date is 0: t=0.
2. {t(1)...t(n)} are the futures dates at which the coupons are paid out
3. The coupon is paid on date t(k) only if I(t(k)) = 1.  Therefore the random cash flows associated with the coupon payment on dates t(k) is c*I(t(k))
4. The face value F is paid on date t(n) only if I(t(n)) = 1. Therefore the random cash flow associated with the face value payment on date t(n) is F*I(t(n)). 
5. The recovery R(t(k))*F is paid on date t(k) if the bond defaults on date t(k).  Therefore the random cash flow associated with the recovery on date t(k) is R(t(k))*F*[I(t(k-1)) - I(t(k))], where R is the recovery fraction, F is the face value, and the quantity [I(t(k-1)) - I(t(k))] denotes that the bond has defaulted.

Now that we have all these cash flows, we can price the bond by simply discounting all of the cash flows with respect to the correct risk-neutral probabilities. 

The price of the defaultable fixed coupon bond at time t=0 is given by:

The first term in the expectation, 

denotes the cash flow paid at time t(k) so they must be discounted at time t(k), which is what the B(t(k)) in the denominator represents (remember that B(t) is simply the value of the cash account at time t.

The second term in the expectation,

denotes the cash flow from the face value payment of the bond at time t(n), which is the B(t(n)) in the denominator.  B(t) is simply 1 since at time t=0, the value of the cash account is 1.

The third term in the expectation, 

is the random cash flow associated with the recovery if the bond defaults, discounted at time B(t(k)).  This quantity will equal 0 unless the bond defaults at time t(k).  

Since we assumed that the default is independent of interest rate dynamics, we can split the expectation up according to whether it is the expectation of default or the expectation of the interest rate dynamic.  In the following equation, each term in the equation above has been split into two expectations multiplied by each other.  The first expectation is the expectation of default and the second is the expectation of interest rate dynamics;

We can then substitute q for the expectation of default and we can substitute Z(0,t(k)) (zcb price) for the expectation of the short rate:

We can further simplify this equation by simply substituting out the zcb prices (Z(0,t(k)) for the discount rate up to time t(k):

Next we are going to calibrate the hazard rates.  First, we assume that the interest rate is deterministic and known (and even if it was not, we are able to calibrate it, as we did in previous lessons).  

We then denote the model price of the defaultable bonds as P(h) as a function of h = {h(0)...h(n-1)}.

We then denote the market price for the defaultable bonds as P(market).

For model calibration, take the model prices and the market prices, get the pricing error between them:

and minimize it:

We then did this numerically in an excel spreadsheet.