This is the first of two modules where we will introduce you to the modeling of defaultable bonds. In this first module, we will introduce a modified binomial lattice for pricing short rates with default. We will show you how to price zero coupon bonds that are default free and zero coupon bonds that are, have a default and have a recovery associated with them on this binomial lattice. A defaultable bond is characterized by a coupon payment c, the face value F and the recovery value R, where R is defined as the random fraction of the face value recovered upon default. We also have to specify the probabilities of default. When would the default events happen, what is the rate at which they're going to happen, and so on. As before, we will work directly with the risk neutral probability, Q. We will model the term structure of default that is the time dependence of default probabilities using the one step default probability h of t. Which is defined as the conditional probability that the bond will default over the interval t to t plus 1, given the information at time t. So, Ft will represent the information at time t. This is the conditional probability that the bond is going to default over the next period. 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. When we do risk neutral pricing for this default-free zero-coupon bond, it's the same as that was seen in the terms structure modules. ZTij is simply 1 over 1 plus rij. This is the short rate in node ij. quZTi plus 1j plus 1qdZTi plus 1j. So, this is the expectation of the prices once time step later according to the risk-neutral probabilities, discounted back to time i. And as before, as you saw in the modules corresponding to term structures, we can calibrate the short-rate lattice, that is, we can compute qd, qu and rij, using the prices of this default-free zero-coupon bonds and other default-free instruments. In the modules, we emphasize the fact that there are too many variables, and often one sets qu equals to qd equal to a half, and then calibrate the rij to make sure that prices of all the default-free instruments work out to be correct. What about extending this idea to defaultable zero-coupon bonds? So now, the bonds that are, that we are considering, have the possibility that they are going to default. And when they default, we're going to start with the instrument that pays no recovery. So, what does it mean? It means that these bonds pay $1 in every state at the expiration, provided default has not occurred on any date T less than or equal to T. If a default occurs at some point, the bond pays 0. There's no recovery. Again, let's see what the pricing works out to be. Z bar T i j eta will denote the price of a defaultable zero-coupon bond maturing at, on date T in node i j eta. Since there is no recovery, we know that as soon as the bond gets into a default state, its price is going to be zero. The moment a default occurs, you don't get anything, and therefore, the no arbitrage price must be 0. So, ZTij1 is always going to be 0 for all default notes ij1. What happens to the price in the new default node? Now, we can use risk-neitral pricing. We know what happens to the transitions under the risk-neutral probability. So, the price at iT, iJ0 is going to be 1 over 1 plus rij. This is simple discounting. So, there are four possible states that you can go to. You can go to state i plus 1, j plus 1 without default. You can go to state i plus 1 j plus 1 with default. You can go to state 1 plus 1j without default. And you can go to state i plus 1j with default. What are the corresponding probabilities? It's qu1 minus hij, qd1 minus hij, quhij, and qdhij. We already know that if you're going to a default state, the price of a zero-coupon bond with no recovery is going to be 0, so this term drops away. This other term also goes away because it's also a default state. So, essentially the risk-neutral pricing for a zero-coupon bond with no recovery is, take the prices one times step later and discount them using the risk-neutral probability. But the risk-neutral probability now has two components. One that corresponds to the interest rate dynamics. And another that corresponds to the default dynamics. Once you have the prices for all defaultable zero-coupon bonds with no recovery, you can can calibrate hij using those prices. Now, we want to interpret the prices of defaultable zero-coupon bonds with no recovery, and try to understand or give an interpretation for what hij is. The risk-neutral pricing, by ignoring the two, default state. Essentially happens to be Z bar ij0 is 1 minus hij divided by 1 plus rij qu zt, i plus 1, j plus 1, no default, qdZT i plus 1j, no default. This fraction here, can be approximated by e to the power minus rij, hij. And the quantity in the bracket here is actually an expected value with respect to a different probability match up which we're going to call the default free risk neutral probability measure. What do I mean by default free? What I mean is that all the default rates have been pulled out. So, the price of a defaultable zero-coupon bond is set by discounting the expected value by rij plus hij. And that's just coming from the expected, this expectation involves both the rates. So, hij has the interpretation of a one median credit spread. It's state dependent, it's time dependent, and it's a credit spread because if the bond was not defaultable, then I would have discounted only by rij. But because the bond is def, defaultable, I'm going to discount the price by extra interest rate which is hij, which is exactly the interpretation of a credit spread. The conditional probability of default hij is also called a hazard rate. It's the probability of default given no default has occurred up to time i. Now, let's extend this to the notion of recovery. I'm going to assume that the random recovery, R tilde, is independent of the default and interest rate dynamics. And let R denote the expected value of R tilde. As before, let Z bar tij eta denote the price of a defaultable bond maturing on date i, in node ij eta. But now, I'm going to emphasize the fact that this is after recovery, after whatever recovery amount has been paid. It's the x dividend or the x coupon, here's it's the x recovery price. Again, once the recovery has been done, there are no cash flows available from these bonds. We already know that Z bar ij1 is going to be equal to 0 for all the default notes. What about the no default nodes? The price there, Z bar tij0, is going to be the discounted value of the prices in the future nodes as before. But now, if a default occurs, you get an expected value R, which ends up playing a role. And therefore, the new risk-neutral pricing for what happens to zero-coupon bonds which are defaultable, but have a recovery, is that this extra term shows up. And we'll stop here. And in the next module, we're going to talk about how to price general bonds and how to use these general bonds or coupon paying bonds to calibrate the hazard rate.
