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In this module we're going to discuss the cash account and the pricing of zero

Â coupon bonds in the context of the binomial model for the short rate.

Â The cash account and zero coupon bonds are extremely important securities and

Â derivatives pricing in general and so we're going to spend some time now

Â figuring out how to price them and understand the mechanics of these

Â securities. >> So let's get started.

Â If you recall this is our binomial lattice model for the short rate.

Â Ri is the Short- Rate that applies for lending between times i out to i plus 1.

Â In general it's a random variable because it can take on any of these values for

Â example at time 2. So time t equals 2 the short rate or 2

Â could take on this value at state 0, this value at state 1 or this value at state 2.

Â We also have our risk-neutral probabilities q u and q d, and of course,

Â q u plus q d must sum to 1. These are our risk-neutral probabilities,

Â so they're strictly positive, and what we will do is we price everything with these

Â risk-neutral probabilities. In particular, for example, if we want to

Â price a non coupon paying security, and I use the term, non coupon here loosely.

Â So, a coupon could refer to any cash flow. So if we want to price in non coupon

Â paying security at time i, state j. Well, we just do our usual discounted

Â risk-neutral pricing. So Zij, the value of the security at time

Â i state j, is one over one plus the short-rated time i state j, times the

Â expected value of the security one period ahead, where we take that expected value

Â with respect to the risk-neutral probabilities qu and qd.

Â Now as we said as well before, there can be no arbitrage when we price using three.

Â And the reason for that is as follows; If you recall our definition of a type a

Â arbitrage, for example. So a type a arbitrage was a security of

Â the form v 0 being less than 0, and its value at time 1, v 1 must be greater than

Â or equal to 0. So we said any security like this in the

Â one period model constituted type a arbitrage.

Â Well this is not possible over here if we price everything according to 3.

Â And the reason is, so Z will take the place or r v here.

Â We see that it is not possible for this to be greater than or equal to 0, and this to

Â be greater than or equal to 0, and yet, have this being less than or equal to 0.

Â This is not possible, because the q's and r are all strictly positive.

Â And if the zeds are greater than or equal to 0, then this must be greater than or

Â equal to 0 as well. So we actually cannot get a type a

Â arbitrage. That's not possible.

Â It's the same for a type b. If you recall, a type b arbitrage assumed

Â a security of the following form, V0 less than or equal to 0, V1 greater than or

Â equal to 0, but V1 not equal to 0. Which means that V1 is greater than or

Â equal to 0 in all states and is at least one state where it's actually strictly

Â greater than 0. Well, again, the exact same argument over

Â here would show that, that is not possible as well.

Â So there can be no arbitrage when we price according to 3.

Â I now want to talk a little bit about The Cash-Account.

Â The Cash-Account is a particular security that in each period earns interest at the

Â short rate. We're going to use bt to denote the value

Â of the cash account at time t and we would be assuming without loss of generality

Â that it starts off with a value of 1, so b 0 is equal to 1.

Â The Cash Account is not risk free. And the reason it's not risk free is

Â because interest rates are uncertain, they're stochastic.

Â In particular the value of the Cash Account at the time t plus s say, is not

Â known at time t for any value that's greater than 1.

Â However, it is locally risk-free because I do know the value of B t plus 1 at time t.

Â In fact, B t plus 1 will always be equal to B t times 1 plus the short rate.

Â And I'm going to know the short rate at time t and therefore I will know B t plus

Â 1 at time t. So again think of your bank account

Â analogy. If I deposit money today for one month, I

Â know what rate will apply for that one month period and so I will know how much I

Â will get at the end of the month. But I will not know what interest rate

Â will prevail in one months' time and therefore will not know future values of

Â The Cash-Account beyond one month. So a quick thing to notice here, so Bt

Â therefore has this expression here, based on the argument I just gave you, I can

Â look at Bt plus 1 Bt and divide 1 by the other and see that I get one over 1 plus

Â rt. And the reason I want that expression is

Â down here, I want to derive equation four here.

Â So how do I derive equation four? Well, again, for a non-coupon paying

Â security, zt times z at time t, state j is equal to 1 over 1 plus rtj times the

Â expected value of the security one period from now.

Â So this is our familiar risk neutral pricing expression from the previous

Â slide. I can actually rewrite this expression as

Â the expected value under q, remember q is equal to q u and q d, the risk-neutral

Â probabilities. And I can replace my 1 over 1 plus rtj,

Â with this expression here bt over bt plus 1.

Â So therefore I can write the value of the non-coupon paying security at time t as

Â being the expected value at time t under q.

Â On Z t plus one, multiplied by B t over B t plus one.

Â So rewriting equation four, I can just bring the B t over to the left hand side

Â and I get this expression here. This is an important expression but I can

Â go a little bit further. I can actually iterate it to get the

Â following. So for example, I can write Zt over Bt is

Â equal to Et. Under q of, well we know it's zed t plus 1

Â over bt plus 1, but I can actually use this equation 5 again to write zed t plus

Â 1 over bt plus 1 as the expected value under q.

Â Condition on time t plus 1 information of Zt plus 2 over Bt plus 2.

Â And using the law of iterated expectations, this is equal to the

Â expected value condition on time t information of Zt plus 2 over Bt plus 2.

Â And I could repeat the same trick again and again.

Â And so, it's easy to see that this condition is hold.

Â So this is our risk neutral pricing condition, for a non-coupon or

Â non-dividend paying security. In particular, it's the pricing equation

Â that we use for any security that does not pay any intermediate cash flows between

Â times t and t plus s. When we're doing risk-neutral pricing for

Â a coupon paying security, we use the exact same idea, so Ztj equals 1 over 1 plus Rtj

Â times the expected value under q of the value of the security plus the cash flow

Â at time t plus 1 that just gives us this expression here.

Â And for the same reason as before, we can see as long as we price any coupon-paying

Â security this way that cannot be an arbitrage.

Â There is now way that this quantity can be greater than or equal to 0 and yet to have

Â Ztj being less than 0. So you couldn't have a type A arbitrage

Â because if this is greater than or equal to 0 there is neutral probabilities we

Â know are strictly greater than 0, this is greater than 0 and so all of this

Â expectation must be greater than or equal to 0.

Â So in particular this is not possible. So you couldn't get a type A arbitrage,

Â and for the same reason, you couldn't get a type B arbitrage as well.

Â Alright, so we have seven. Well, it's easy to rewrite seven using the

Â same ideas we used in the previous slide. I can replace 1 over 1 plus RTJ with Bt

Â over Bt plus 1, bring the Bt over to the other side, and I get Expression eight.

Â Now I can iterate, I can substitute in for example, if I substitute in the following;

Â I know that Zt plus Bt plus 1 is equal to the expected value under q conditional on

Â time t plus one information. Ct plus 2 over Bt plus 2.

Â Plus Zt plus 2 over Bt plus 2. So if I substitute that in down here, I'm

Â going to get this expression here when s equals 2.

Â Now it's easy to see that this expression holds more generally for general values of

Â s, or for integer values of s greater than t.

Â So this is an extremely important condition.

Â We're going to use this throughout this section on term-structure models and

Â pricing fixed-income derivatives that tells us how to price fixed-income

Â derivatives using risk-neutral pricing and this ensures that there's no arbitrage.

Â In other words, it ensures that we're pricing fixed-income derivatives in a

Â manner which is consistent with no arbitrage.

Â Note also that equation six is actually a special case of nine, because we get

Â equation six From nine by just taking all of these cjs equal to 0.

Â So this is an extremely important result, it guarantees that we can price everything

Â with no arbitrage. Here is a sample short-rate lattice.

Â It starts off with the short rate r00 being equal to 6%.

Â And then the short rate will grow by a factor of u equals 1.25 or fall by a

Â factor of d equals 0.9 in each period. It's not very realistic.

Â These interest rates, as you can see, grow quite large here.

Â And given the current economy, global economy, where interest rates are very

Â low. This example wouldn't be very realistic.

Â But it is more than sufficient for our purposes.

Â In fact, it's good to have such a wide range of possible interest rates.

Â As it makes it easier to distinguish them in the examples that we'll see in the

Â future. At this point I should also mention.

Â That you should look at the spreadsheet that is associated with these modules.

Â The spreadsheet you'll see this particular example there.

Â And we're going to be using this example throughout this section to price various

Â types of fixed income derivatives. We're going to be looking at caps, floors,

Â swaptions. Options on zero coupon bonds and so on.

Â So we're going to use this particular short rate example as our model in all of

Â these pricing examples. So the first thing we're going to do is

Â we're going to see how to price a zero coupon bond that matures a time t equal to

Â 4. So if we want to do that we're going to

Â use our risk neutral pricing, our risk neutral pricing result If you recall,

Â states that Zt over Bt is equal to the expected value conditional on time t

Â information of Zt plus 1 over BT plus 1, and this is a risk-neutral pricing

Â identity for securities that do not pay coupons or do not have intermediate cash

Â flows. And certainly that is true of a zero

Â coupon bond. If you recall, a zero coupon bond does not

Â have any intermediate cash flows. It only pays off its face value at

Â maturity. And this example matures at t equal to 4.

Â This face value is 100 and this indeed is if you like using our notation for 0 zero

Â coupon bonds Z44. So what we're going to do is we're going

Â to bond rate the price to zero coupon bond is to use this expression here by just

Â working backwards in the lattice 1 period at a time.

Â We know Z44, it's 100. At maturity the, the bond is worth 100, so

Â let's work back and compute its value at time t equals 3.

Â Well, to do that, we can just use this expression.

Â Another way of saying this, and we saw this as well before, this is equivalent to

Â saying that Zt Is equal to the expected value time t of Zt plus 1 over 1 plus or

Â t. And so in fact it's this version that

Â we're going to use. We're going to work backwards computing

Â the values t at every node by discounting and computing the expected value one

Â period ahead so that's all we're doing here.

Â So, for example, the 83.08 that we've highlighted here is equal to 1 over 1 plus

Â the short rate value at that node, and if we go back one slide we'll see the short

Â rate value at that node was 9.38%. So that's where the 0.0938 comes from

Â here. And then it's the expected value under q

Â of the value of the zero coupon bond one period ahead.

Â There are two possible values, 89.51, 92.22 and that's what we have here.

Â So we just work backwards in the lattice, one period at a time, until we get its

Â value here at time 0 and this is Z04. The time 0 value of the zero coupon bond

Â that matures the time 4. Having calculated the zero coupon bond

Â price at time0, we can now infer from that the actual interest rate that corresponds

Â to, to t equals 4. In particular if we assume part period

Â compounding and we let S4 denote the, the interest rate that applies to borrowing or

Â lending for four periods then we know that 77.22 times 1 plus S moved to the power of

Â 4 must be equal to 100. So of course we can invert that to get

Â that S4. Is equal to 100 over 77.22, all to the

Â power of 1 quarter minus 1 and so that's how we get S4.

Â So there's always a one to one correspondence between seeing the zero

Â coupon bond prices, and seeing the corresponding interest rate.

Â Therefore it means that we can actually compute all of the zero coupon bond prices

Â for the four different maturities. So we can compute the zero coupon bond

Â price for maturity t equals 1, t equals 2, 3, and 4.

Â And from this we can actually back out, back out the actual interest rates that

Â apply to these periods. So, for example, we will get a term

Â structure of interest rates that looks like the following; We have t down here

Â and we've got the spot rate St here and maybe we'll see something like the

Â following or maybe it's an inverted curve but this point here, so for example t1,

Â that point corresponds to there and it corresponds to some spot rate st1 there.

Â So we can actually use this model to price all the zero coupon bonds.

Â And from all of these zero coupon bond prices, we can invert the message in the

Â previous slide to get the term structure. The term structure is the term structure

Â of interest rates. We can see what interest rate applies to

Â each time t. At time t equals 1, for example, we will

Â then compute a new set of zero coupon bond prices, and obtain a new term structure.

Â So for example at time t equals 0 were down here.

Â But at time equals 1 may be I'm up in this state of the world.

Â So if I'm up in this state of the world I could recompute the time structure of

Â interest rates, I could do that by a pricing all of the zero coupon bonds at

Â this point. I'm going to get a different set of prices

Â at the set of prices ahead at time t equals 0 and I can invert this new set of

Â prices to get the new time structure and may be that new time structure will look

Â different, may be will look something like this.

Â So I will get a new term structure times t equals 1, moreover the term structure I

Â see will depend on whether I'm up here or down here.

Â So what we've actually succeeded in doing is defining a stochastic model or a random

Â model for the term structure of interest rates by just focusing on the short rate.

Â So the short-rate or t is just a scalar random variable or scalar process by

Â focusing on modeling this short rate, as we've done, we've actually succeeded in

Â defining a stochastic or random model for the entire term structure.

Â And that's actually a very significant point to keep in mind when working with

Â these short-rate models.

Â