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

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

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

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

Â and j. i went from 0 through n.

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

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

Â on. So, each of these nodes were being

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

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

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

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

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

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

Â q of d. And the probability of reaching any other

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

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

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

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

Â whether or not default has occurred before date i.

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

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

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

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

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

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

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

Â be. Again, these transition probabilities are

Â going to be risk neutral transition probabilities.

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

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

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

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

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

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

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

Â occurred. And one, the black nodes are referring to

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

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

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

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

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

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

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

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

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

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

Â What about the down probabilities? They are the same.

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

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

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

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

Â times hij. Okay?

Â So, in this table emphasizes the same thing.

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

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

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

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

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

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

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

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

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

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

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

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

Â state. So, notice that there are no blue arrows

Â or black arrows going to the state with no default.

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

Â It's the same as the short rate probabilities.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â eta. When a default-free zero-coupon.

Â just by context, this is what I mean here.

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

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

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

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

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

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

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

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

Â these states, are going to be exactly the same.

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

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

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

Â 8:58

When we do risk neutral pricing for this default-free zero-coupon bond, it's the

Â same as that was seen in the terms structure modules.

Â ZTij is simply 1 over 1 plus rij. This is the short rate in node ij.

Â quZTi plus 1j plus 1qdZTi plus 1j. So, this is the expectation of the prices

Â once time step later according to the risk-neutral probabilities, discounted

Â back to time i. And as before, as you saw in the modules

Â corresponding to term structures, we can calibrate the short-rate lattice, that

Â is, we can compute qd, qu and rij, using the prices of this default-free

Â zero-coupon bonds and other default-free instruments.

Â In the modules, we emphasize the fact that there are too many variables, and

Â often one sets qu equals to qd equal to a half, and then calibrate the rij to make

Â sure that prices of all the default-free instruments work out to be correct.

Â What about extending this idea to defaultable zero-coupon bonds?

Â So now, the bonds that are, that we are considering, have the possibility that

Â they are going to default. And when they default, we're going to

Â start with the instrument that pays no recovery.

Â So, what does it mean? It means that these bonds pay $1 in every

Â state at the expiration, provided default has not occurred on any date T less than

Â or equal to T. If a default occurs at some point, the

Â bond pays 0. There's no recovery.

Â Again, let's see what the pricing works out to be.

Â Z bar T i j eta will denote the price of a defaultable zero-coupon bond maturing

Â at, on date T in node i j eta. Since there is no recovery, we know that

Â as soon as the bond gets into a default state, its price is going to be zero.

Â The moment a default occurs, you don't get anything, and therefore, the no

Â arbitrage price must be 0. So, ZTij1 is always going to be 0 for all

Â default notes ij1. What happens to the price in the new

Â default node? Now, we can use risk-neitral pricing.

Â We know what happens to the transitions under the risk-neutral probability.

Â So, the price at iT, iJ0 is going to be 1 over 1 plus rij.

Â This is simple discounting. So, there are four possible states that

Â you can go to. You can go to state i plus 1, j plus 1

Â without default. You can go to state i plus 1 j plus 1

Â with default. You can go to state 1 plus 1j without

Â default. And you can go to state i plus 1j with

Â default. What are the corresponding probabilities?

Â It's qu1 minus hij, qd1 minus hij, quhij, and qdhij.

Â We already know that if you're going to a default state, the price of a zero-coupon

Â bond with no recovery is going to be 0, so this term drops away.

Â This other term also goes away because it's also a default state.

Â So, essentially the risk-neutral pricing for a zero-coupon bond with no recovery

Â is, take the prices one times step later and discount them using the risk-neutral

Â probability. But the risk-neutral probability now has

Â two components. One that corresponds to the interest rate

Â dynamics. And another that corresponds to the

Â default dynamics. Once you have the prices for all

Â defaultable zero-coupon bonds with no recovery, you can can calibrate hij using

Â those prices. Now, we want to interpret the prices of

Â defaultable zero-coupon bonds with no recovery, and try to understand or give

Â an interpretation for what hij is. The risk-neutral pricing, by ignoring the

Â two, default state. Essentially happens to be Z bar ij0 is 1

Â minus hij divided by 1 plus rij qu zt, i plus 1, j plus 1, no default, qdZT i plus

Â 1j, no default. This fraction here, can be approximated

Â by e to the power minus rij, hij. And the quantity in the bracket here is

Â actually an expected value with respect to a different probability match up which

Â we're going to call the default free risk neutral probability measure.

Â What do I mean by default free? What I mean is that all the default rates

Â have been pulled out. So, the price of a defaultable

Â zero-coupon bond is set by discounting the expected value by rij plus hij.

Â And that's just coming from the expected, this expectation involves both the rates.

Â So, hij has the interpretation of a one median credit spread.

Â It's state dependent, it's time dependent, and it's a credit spread

Â because if the bond was not defaultable, then I would have discounted only by rij.

Â But because the bond is def, defaultable, I'm going to discount the price by extra

Â interest rate which is hij, which is exactly the interpretation of a credit

Â spread. The conditional probability of default

Â hij is also called a hazard rate. It's the probability of default given no

Â default has occurred up to time i. Now, let's extend this to the notion of

Â recovery. I'm going to assume that the random

Â recovery, R tilde, is independent of the default and interest rate dynamics.

Â And let R denote the expected value of R tilde.

Â As before, let Z bar tij eta denote the price of a defaultable bond maturing on

Â date i, in node ij eta. But now, I'm going to emphasize the fact

Â that this is after recovery, after whatever recovery amount has been paid.

Â It's the x dividend or the x coupon, here's it's the x recovery price.

Â Again, once the recovery has been done, there are no cash flows available from

Â these bonds. We already know that Z bar ij1 is going

Â to be equal to 0 for all the default notes.

Â What about the no default nodes? The price there, Z bar tij0, is going to

Â be the discounted value of the prices in the future nodes as before.

Â But now, if a default occurs, you get an expected value R, which ends up playing a

Â role. And therefore, the new risk-neutral

Â pricing for what happens to zero-coupon bonds which are defaultable, but have a

Â recovery, is that this extra term shows up.

Â And we'll stop here. And in the next module, we're going to

Â talk about how to price general bonds and how to use these general bonds or coupon

Â paying bonds to calibrate the hazard rate.

Â