In this section and the next sequence of modules we are going to discuss fix income derivative pricing in the context of term structure lattice models. We are going to see how the philosophy behind fix income derivative pricing is different to the philosophy behind the binomial model and pricing equity derivative. Fix income markets are enormous and in fact they are bigger than equity markets. According to SIFMA, in Q3 2012, the total outstanding amount of US bonds was $35.3 trillion. In comparison, the size of the US equity markets was only approximately $26 trillion in comparison. Fixed income derivatives markets are also enormous, they include interest-rate and bond derivatives. Credit derivatives, mortgage backed securities. And asset back securities, more generally in this section, we're going to be focusing mainly on interest rate and bond derivatives. And we're going to use binomial lattice models to understand these securities. The mechanics of these securities, and also how to price them using risk neutral pricing. The slides and excel spreadsheets should be sufficient. But chapter 14 of the Lunberger text is also an excellent reference for the material in this section. We're going to use binomial lattice models as a vehicle for introducing both the mechanics of fixed income derivative securities as well as the philosophy behind fixed income derivatives pricing. We'll talk more about the philosophy soon. But let's look at some of the most important derivative fixed income securities. We're going to talk about bond futures and also forwards. We'll also talk about caplets and cap, floorlets and floors, and swaps and swaptions. Now I should mention at this point that Lipor rates are the interest rates, that underlie these securities here. Whereas, these securities typically have government rates underlying them. And so these are different interest rates, but we're not going to make that distinction here in these modules. We're going to assume that it's the same underlying rates. And the reasons for doing this are two-fold. We want to focus mainly on the mechanics of these securities, how they work and how they're priced. We're going to price them using risk neutral pricing. And it would only distract us if we had to focus on different interest rates when we were pricing different securities. Now fixed-income models are inherently more complex than security models. The problem with fixed-income models is that we need to model the evolution of the entire term structure of interest rates. So, for example, let's come down here and let's see a little plot of the term structure of interest rates. So we have time, t, here and we have st here. St is going to stand for the spot rate, spot interest rate at time T. And maybe you've got some function like this. So this tells us what the time structure of interest rates look like. So for example, if this is t1, then we have, over there, and that's St1. And St1 is then the spot interest rate that applies to borrowing or lending at the time t one. So when we want to build a fixed income model or term, term structure model. We need a model which will, which will model how this entire curves moves through time, note the distinction. When we have a model for stock prices, we just need to model the evolution of a single stock. A scale of random variable, here we need to model the entire evolution of this term structure. And so term structure models are inherently more complicated. But actually we're going to see there's some easy ways to get around this problem. One of the classical ways to get around this problem is to focus on what's called the short-rate. The short-rate, r little t, is the variable of interests and many fixed income models, including binomial lattice models. It is the risk-free rate that applies between periods t, and t plus 1. So r of t is going to be the risk-free rate that applies from period t out to period t plus 1. It's a random process, R is random. Remember, interest rates in the real world are random, and so this short rate will also be random. However, rt is known to us by time t. So at time t, we know. What we're going to get at time t plus 1 if we deposit $1 in the bank account at time t. What about the philosophy of fixed income derivatives pricing? Well, what we're going to do here is as follows. We will simply specify risk mutual probabilities for the short rate r of t, and we will do this without any reference whatsoever to the true probabilities of the short rate. This is in contrast to the binomial model for stocks where we specified p and 1 minus p and then used replication arguments to get q and 1 minus q, the risk neutral probabilities. What we're going to do is we're going to price securities, fixed income derivatives, in such a way that guarantees no arbitrage. We're going to match the market prices of liquid securities via a calibration procedure. This is often the most challenging part of the entire exercise. And we will see that derivatives pricing in practice is really about extrapolating from liquid security prices to illiquid security prices. So just summarizing here the philosophy of fixed income derivatives pricing is not to focus on the true probabilities but to go directly to risk neutral probabilities. Do your pricing in such a way that guarantees no arbitrage. Once you do that we'll then pick the unknown parameters in the model, so that the model prices will match market prices. That's the calibration exercise, and that is how we would price fixed income derivatives in practice. So here's an example of a binomial model for the short rate. We're actually going to be using binomial models throughout these modules on term structure modeling, and fixed income derivatives pricing. We have time as usual, down here t equals 0, t equals 1, and so on. We're going to use the notation rij to specify the short rate at node Nij. So, for example, this point here is node N22. So rij, i refers to time, and j refers to the state. So we're going to number the states from 0, 1, 2 and so on. So what we have here is the binomial model for the short rate. We're going to take 0 coupon bond prizes and we're going to use zcb as a shorthand for 0 coupon bond, throughout. We'll take 0 coupon bond prices to be our basic securities. We will use the notation Z subscript i,j superscript k to denote the time i state j price of a zero coupon bonds that matures at time k. Now its important to get comfortable and familiar with this notation. So again Zijk, its the time i state j price of a zero coupon bond that matures at time k. So for example, if met this point here. Then this z213 is the price, at time 2 state 1 up here of a zero coupon bond that matures at time 3 which is here. What we would like to do is we would like to specify binomial model by specifying all the Zijks at all nodes. Now this is possible, but it's actually very awkward if you want to insure no arbitrage. Moving on more generally, we're going to let Zij be the date i, state, j, price of some non-coupon paying security. And what we're going to do, is we're going to use risk neutral pricing to price every security, every sub security. So for example, let qu and qd be the probability of an up move and a down move. Of course here a down move more, looks more like an across move, but we'll still stick with qu and qd. So we're going to assume that qu and qd are given to us, and we know them at every node. So at every node, qu is the same, qd is the same at every node and of course qu plus qd equals 1 and they're both strictly greater than 0. So what we'll do is we will use risk mutual processing to price every security. So for example, Zij this is the price of the security at time i state j, is going to be 1 over the discount factor times the probability, risk [inaudible] probability of an up move, in which case we have the price at times i plus 1 and state j plus 1 plus the risk neutral probability with down move times the price of the security its in time i plus 1 state j. So if we price every non-coupon paying security like this there can be arbitrage when we priced using 1 and very loosely when we're trying to this in the next slide but very loosely the reason is as follows. If you recall it's not possible for example for this to be greater than or equal to 0, and this to be greater than or equal to 0 and yet have this be less than 0, why? Well qu and qd are both strictly positive. So if qu and qd are strictly positive and the interest rate is strictly positive, the short rate which would always be the case, then you cannot have this being the case. And this being less than zero, so it's not possible to construct arbitrages when we price like this. More generally, if the security pays the coupon, c i plus 1 comma j, at date i plus 1 and state j, then we have the following from risk neutral pricing again: zed i j is equal to this quantity on the right hand side, where Zi plus 1 dot. So this dot here means it can take on the value j plus 1, or j as we see here. So, if we price our coupon paying security according to this, then again we'll see that there can not be any arbitrage. And the reason is as follows. If you recall the definition of our type a arbitrage. So, a type A arbitrage goes as follows. So, we have some security or portfolio whose initial value was less than 0 and whose final value v1 is greater than or equal to 0. Well, we see that this is not possible here. So, for the same reason i gave in the previous slide, this would be our v1, so these are the two possible values of v1. So v1 in this state j plus 1 is greater than or equal to 0. And v1 in this state of j is greater than or equal to 0. Then Zi plus Zij must also be greater than or equal to 0. So this is not possible. Similarly, what we have for type b arbitrage, if you recall. So a type b arbitrage was one where v0 was less that or equal to zero. And v1 was greater than or equal to zero, But v1 was not equal to zero. So in this case what would we have? We would have the following. Let's just erase some of this here. So for type B arbitrage we would have say that v1 is greater than or equal to zero. And say v1 over in this state j is strictly greater than zero. And that this must actually be less than or equal to zero. Well again this isn't possible, qu and qd are both strictly positive. So if all of this is greater than or equal to zero, then qu times this is greater than or equal to zero, and qd times this is strictly greater than zero. So Zij must be strictly greater than zero. This situation's not possible. So again, we see that a type B arbitrage is not possible when we price like this. Now, before continuing I just want to mention one other item. That is I'm losing, I'm using the word coupon very loosely. Obviously it, it denotes the idea of a bond, and the coupon one gets more bond. But it could be the cash flow from a swap. The fixed rate minus the floating rate cash flow for example. So I'm going to use the word coupon to denote any intermediate cash flow throughout these modules.
