As in the case with other fixed income securities, we are going to calibrate age

of t to market prices. And we will modify the binomial lattice

to include defaults. Just as a review, let's look at the

binomial lattice for short rates. The nodes in this lattice were labeled i

and j. i went from 0 through n.

G went from 0 through i. So, at time zero, you have one node.

At time one, you have two nodes. At time two, you have three nodes and so

on. So, each of these nodes were being

labeled i, j. There was a short rated, all of these

nodes r, i, j. And the transition probability was given

by the binomial lattice probability. So, the, at time i plus 1, you could only

reach states j and j plus 1. The probability that you went from node

i, j to node i plus 1, j plus 1 was qu. The upper probability still is qu.

The probability that you went from node i, j to node i plus 1j, it's going to be

q of d. And the probability of reaching any other

state is going to be 0. This was the short rate lattice that we

worked with when we were constructing, the short rate lattice for default free

bonds. Now, in order to model default, we are

going to split the node i, j by introducing a new variable that encodes

whether or not default has occurred before date i.

i, j, 0 will denote the fact that the state is j on date i, and the default

time tau is greater than i. Which means that that time i or date i,

the default has not happened. i, j, 1 will encode the fact that the

system is in state j on date i and the default has already occurred at some

point before i. And when I say before i, I mean it

includes i as well. Now, for the split lattice, for these

split nodes, I'm going to tell you what the transition probabilities are going to

be. Again, these transition probabilities are

going to be risk neutral transition probabilities.

So, lets work through this figure slowly. Here's my state at date i.

And here, the possible states at date i plus 1.

So, here is date i plus 1. When we looked at the short rate lattice,

we had nodes i, j. Now, we've split each of those nodes into

two nodes, i, j, 0, means default has not occurred, i, j, 1, which means default

has occurred. So, the red nodes here, indicate the fact

that these are the nodes associates with states where the default has not

occurred. And one, the black nodes are referring to

the fact that these are the states at which the default has occurred.

So, what are the transitions out of the state i, j, 0?

This is the state where the interest rates are in state j, the date is i, and

the bond has not yet defaulted. Then, the probability qu, you can go to

stage j plus 1. At that point, there are two

possibilities, either you can default or you do not default.

So, up here, it's qu times 1 minus hij, is the probability that you go to state j

plus 1 without default. qu times hij is going to the probability

that you go to state j plus 1. And now, you have defaulted.

So, instead of going, going from a red node, now you're going to a black node.

What about the down probabilities? They are the same.

It's going to be qd, times 1 minus hij. This is the probability that you start

from state i, j, 0. And go to state i plus 1 j 0, meaning no

default and no default at time i plus 1 as well.

What about the probability that you default at time i, it's going to be qd,

times hij. Okay?

So, in this table emphasizes the same thing.

Qi plus 1 s eta, eta is the label for whether there is a default or not.

s is the label for the state. You can only go, if you go to stage j

plus 1 and eta equals 1, it means default the probability is quhij.

If you go to stage j plus 1 eta equals 0, which mean no default, the probability is

qu 1 minus hij. This is exactly the same probability that

I just wrote on the figure. What happens with the transition out of

the default state? So, the transition from the default state

are actually very simple. So, once the bond has defaulted, we are

going to assume that there is exactly one default event between dates 0 through n.

So, once a bond has defaulted, it's always in the default state.

So, from this particular default state, you can go to state j plus 1 and default

and you can go to state j and default. You can never go to the non-default

state. So, notice that there are no blue arrows

or black arrows going to the state with no default.

What is the probability of going to the state j plus 1?

It's the same as the short rate probabilities.

So, this is going to be qu and qd. The hazard rates or hij's, which are the

conditional probabilities of default, are not going to play a role in this

transition, because once the bond has defaulted, it never reappears and always

stays in the default state. The conditional probability of default,

hij, is state dependent. It's labelled by both i and j.

And therefore, this is date i and this is the state on that particular date.

Okay, once we have specified the transition probabilities, we can now

start thinking about how we are going to price simple securities using this

binomial lattice. And also, how we can use these simple

securities to calibrate both the intrastate lattice, as well as this

conditional probability hij. So, the first thing we're going to start

with our default-free zero-coupon bonds. The default-free zero-coupon bonds with

expiration capital T, pays $1 in every state on the expiration date capital T.

These are default-free, so no default is possible.

Let Z, super tij eta denote the price of the bond, maturing on date i in node ij

eta. When a default-free zero-coupon.

just by context, this is what I mean here.

So, it's the same story. Earlier, we had one node.

Now, we have two nodes. This is i, j, 1 and this is i, j, 0.

I need to specify what is going to be the price off of a zero-coupon bond in both

of these states. The first thing we recognize is the fact

that default events do not affect the default-free bonds.

So, whether the, the default event has happened or weather the default event has

not happened, the price of a default-free zero-coupon bond is going to remain the

same. So, the price of that bond, in both of

these states, are going to be exactly the same.

So, ZTij1 is going to be the same as ZTij0.

And, we're going to drop the state corresponding to default, since it does

not matter to a zero-coupon bond, and just call it ZTij.