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In this section and the next sequence of modules we are going to discuss fix income

Â derivative pricing in the context of term structure lattice models.

Â We are going to see how the philosophy behind fix income derivative pricing is

Â different to the philosophy behind the binomial model and pricing equity

Â derivative. Fix income markets are enormous and in

Â fact they are bigger than equity markets. According to SIFMA, in Q3 2012, the total

Â outstanding amount of US bonds was $35.3 trillion.

Â In comparison, the size of the US equity markets was only approximately $26

Â trillion in comparison. Fixed income derivatives markets are also

Â enormous, they include interest-rate and bond derivatives.

Â Credit derivatives, mortgage backed securities.

Â And asset back securities, more generally in this section, we're going to be

Â focusing mainly on interest rate and bond derivatives.

Â And we're going to use binomial lattice models to understand these securities.

Â The mechanics of these securities, and also how to price them using risk neutral

Â pricing. The slides and excel spreadsheets should

Â be sufficient. But chapter 14 of the Lunberger text is

Â also an excellent reference for the material in this section.

Â We're going to use binomial lattice models as a vehicle for introducing both the

Â mechanics of fixed income derivative securities as well as the philosophy

Â behind fixed income derivatives pricing. We'll talk more about the philosophy soon.

Â But let's look at some of the most important derivative fixed income

Â securities. We're going to talk about bond futures and

Â also forwards. We'll also talk about caplets and cap,

Â floorlets and floors, and swaps and swaptions.

Â Now I should mention at this point that Lipor rates are the interest rates, that

Â underlie these securities here. Whereas, these securities typically have

Â government rates underlying them. And so these are different interest rates,

Â but we're not going to make that distinction here in these modules.

Â We're going to assume that it's the same underlying rates.

Â And the reasons for doing this are two-fold.

Â We want to focus mainly on the mechanics of these securities, how they work and how

Â they're priced. We're going to price them using risk

Â neutral pricing. And it would only distract us if we had to

Â focus on different interest rates when we were pricing different securities.

Â Now fixed-income models are inherently more complex than security models.

Â The problem with fixed-income models is that we need to model the evolution of the

Â entire term structure of interest rates. So, for example, let's come down here and

Â let's see a little plot of the term structure of interest rates.

Â So we have time, t, here and we have st here.

Â St is going to stand for the spot rate, spot interest rate at time T.

Â And maybe you've got some function like this.

Â So this tells us what the time structure of interest rates look like.

Â So for example, if this is t1, then we have, over there, and that's St1.

Â And St1 is then the spot interest rate that applies to borrowing or lending at

Â the time t one. So when we want to build a fixed income

Â model or term, term structure model. We need a model which will, which will

Â model how this entire curves moves through time, note the distinction.

Â When we have a model for stock prices, we just need to model the evolution of a

Â single stock. A scale of random variable, here we need

Â to model the entire evolution of this term structure.

Â And so term structure models are inherently more complicated.

Â But actually we're going to see there's some easy ways to get around this problem.

Â One of the classical ways to get around this problem is to focus on what's called

Â the short-rate. The short-rate, r little t, is the

Â variable of interests and many fixed income models, including binomial lattice

Â models. It is the risk-free rate that applies

Â between periods t, and t plus 1. So r of t is going to be the risk-free

Â rate that applies from period t out to period t plus 1.

Â It's a random process, R is random. Remember, interest rates in the real world

Â are random, and so this short rate will also be random.

Â However, rt is known to us by time t. So at time t, we know.

Â What we're going to get at time t plus 1 if we deposit $1 in the bank account at

Â time t. What about the philosophy of fixed income

Â derivatives pricing? Well, what we're going to do here is as

Â follows. We will simply specify risk mutual

Â probabilities for the short rate r of t, and we will do this without any reference

Â whatsoever to the true probabilities of the short rate.

Â This is in contrast to the binomial model for stocks where we specified p and 1

Â minus p and then used replication arguments to get q and 1 minus q, the risk

Â neutral probabilities. What we're going to do is we're going to

Â price securities, fixed income derivatives, in such a way that guarantees

Â no arbitrage. We're going to match the market prices of

Â liquid securities via a calibration procedure.

Â This is often the most challenging part of the entire exercise.

Â And we will see that derivatives pricing in practice is really about extrapolating

Â from liquid security prices to illiquid security prices.

Â So just summarizing here the philosophy of fixed income derivatives pricing is not to

Â focus on the true probabilities but to go directly to risk neutral probabilities.

Â Do your pricing in such a way that guarantees no arbitrage.

Â Once you do that we'll then pick the unknown parameters in the model, so that

Â the model prices will match market prices. That's the calibration exercise, and that

Â is how we would price fixed income derivatives in practice.

Â So here's an example of a binomial model for the short rate.

Â We're actually going to be using binomial models throughout these modules on term

Â structure modeling, and fixed income derivatives pricing.

Â We have time as usual, down here t equals 0, t equals 1, and so on.

Â We're going to use the notation rij to specify the short rate at node Nij.

Â So, for example, this point here is node N22.

Â So rij, i refers to time, and j refers to the state.

Â So we're going to number the states from 0, 1, 2 and so on.

Â So what we have here is the binomial model for the short rate.

Â We're going to take 0 coupon bond prizes and we're going to use zcb as a shorthand

Â for 0 coupon bond, throughout. We'll take 0 coupon bond prices to be our

Â basic securities. We will use the notation Z subscript i,j

Â superscript k to denote the time i state j price of a zero coupon bonds that matures

Â at time k. Now its important to get comfortable and

Â familiar with this notation. So again Zijk, its the time i state j

Â price of a zero coupon bond that matures at time k.

Â So for example, if met this point here. Then this z213 is the price, at time 2

Â state 1 up here of a zero coupon bond that matures at time 3 which is here.

Â What we would like to do is we would like to specify binomial model by specifying

Â all the Zijks at all nodes. Now this is possible, but it's actually

Â very awkward if you want to insure no arbitrage.

Â Moving on more generally, we're going to let Zij be the date i, state, j, price of

Â some non-coupon paying security. And what we're going to do, is we're going

Â to use risk neutral pricing to price every security, every sub security.

Â So for example, let qu and qd be the probability of an up move and a down move.

Â Of course here a down move more, looks more like an across move, but we'll still

Â stick with qu and qd. So we're going to assume that qu and qd

Â are given to us, and we know them at every node.

Â So at every node, qu is the same, qd is the same at every node and of course qu

Â plus qd equals 1 and they're both strictly greater than 0.

Â So what we'll do is we will use risk mutual processing to price every security.

Â So for example, Zij this is the price of the security at time i state j, is going

Â to be 1 over the discount factor times the probability, risk [inaudible] probability

Â of an up move, in which case we have the price at times i plus 1 and state j plus 1

Â plus the risk neutral probability with down move times the price of the security

Â its in time i plus 1 state j. So if we price every non-coupon paying

Â security like this there can be arbitrage when we priced using 1 and very loosely

Â when we're trying to this in the next slide but very loosely the reason is as

Â follows. If you recall it's not possible for

Â example for this to be greater than or equal to 0, and this to be greater than or

Â equal to 0 and yet have this be less than 0, why?

Â Well qu and qd are both strictly positive. So if qu and qd are strictly positive and

Â the interest rate is strictly positive, the short rate which would always be the

Â case, then you cannot have this being the case.

Â And this being less than zero, so it's not possible to construct arbitrages when we

Â price like this. More generally, if the security pays the

Â coupon, c i plus 1 comma j, at date i plus 1 and state j, then we have the following

Â from risk neutral pricing again: zed i j is equal to this quantity on the right

Â hand side, where Zi plus 1 dot. So this dot here means it can take on the

Â value j plus 1, or j as we see here. So, if we price our coupon paying security

Â according to this, then again we'll see that there can not be any arbitrage.

Â And the reason is as follows. If you recall the definition of our type a

Â arbitrage. So, a type A arbitrage goes as follows.

Â So, we have some security or portfolio whose initial value was less than 0 and

Â whose final value v1 is greater than or equal to 0.

Â Well, we see that this is not possible here.

Â So, for the same reason i gave in the previous slide, this would be our v1, so

Â these are the two possible values of v1. So v1 in this state j plus 1 is greater

Â than or equal to 0. And v1 in this state of j is greater than

Â or equal to 0. Then Zi plus Zij must also be greater than

Â or equal to 0. So this is not possible.

Â Similarly, what we have for type b arbitrage, if you recall.

Â So a type b arbitrage was one where v0 was less that or equal to zero.

Â And v1 was greater than or equal to zero, But v1 was not equal to zero.

Â So in this case what would we have? We would have the following.

Â Let's just erase some of this here. So for type B arbitrage we would have say

Â that v1 is greater than or equal to zero. And say v1 over in this state j is

Â strictly greater than zero. And that this must actually be less than

Â or equal to zero. Well again this isn't possible, qu and qd

Â are both strictly positive. So if all of this is greater than or equal

Â to zero, then qu times this is greater than or equal to zero, and qd times this

Â is strictly greater than zero. So Zij must be strictly greater than zero.

Â This situation's not possible. So again, we see that a type B arbitrage

Â is not possible when we price like this. Now, before continuing I just want to

Â mention one other item. That is I'm losing, I'm using the word

Â coupon very loosely. Obviously it, it denotes the idea of a

Â bond, and the coupon one gets more bond. But it could be the cash flow from a swap.

Â The fixed rate minus the floating rate cash flow for example.

Â So I'm going to use the word coupon to denote any intermediate cash flow

Â throughout these modules.

Â