In this module we're going to discuss the cash account and the pricing of zero coupon bonds in the context of the binomial model for the short rate. The cash account and zero coupon bonds are extremely important securities and derivatives pricing in general and so we're going to spend some time now figuring out how to price them and understand the mechanics of these securities. >> So let's get started. If you recall this is our binomial lattice model for the short rate. Ri is the Short- Rate that applies for lending between times i out to i plus 1. In general it's a random variable because it can take on any of these values for example at time 2. So time t equals 2 the short rate or 2 could take on this value at state 0, this value at state 1 or this value at state 2. We also have our risk-neutral probabilities q u and q d, and of course, q u plus q d must sum to 1. These are our risk-neutral probabilities, so they're strictly positive, and what we will do is we price everything with these risk-neutral probabilities. In particular, for example, if we want to price a non coupon paying security, and I use the term, non coupon here loosely. So, a coupon could refer to any cash flow. So if we want to price in non coupon paying security at time i, state j. Well, we just do our usual discounted risk-neutral pricing. So Zij, the value of the security at time i state j, is one over one plus the short-rated time i state j, times the expected value of the security one period ahead, where we take that expected value with respect to the risk-neutral probabilities qu and qd. Now as we said as well before, there can be no arbitrage when we price using three. And the reason for that is as follows; If you recall our definition of a type a arbitrage, for example. So a type a arbitrage was a security of the form v 0 being less than 0, and its value at time 1, v 1 must be greater than or equal to 0. So we said any security like this in the one period model constituted type a arbitrage. Well this is not possible over here if we price everything according to 3. And the reason is, so Z will take the place or r v here. We see that it is not possible for this to be greater than or equal to 0, and this to be greater than or equal to 0, and yet, have this being less than or equal to 0. This is not possible, because the q's and r are all strictly positive. And if the zeds are greater than or equal to 0, then this must be greater than or equal to 0 as well. So we actually cannot get a type a arbitrage. That's not possible. It's the same for a type b. If you recall, a type b arbitrage assumed a security of the following form, V0 less than or equal to 0, V1 greater than or equal to 0, but V1 not equal to 0. Which means that V1 is greater than or equal to 0 in all states and is at least one state where it's actually strictly greater than 0. Well, again, the exact same argument over here would show that, that is not possible as well. So there can be no arbitrage when we price according to 3. I now want to talk a little bit about The Cash-Account. The Cash-Account is a particular security that in each period earns interest at the short rate. We're going to use bt to denote the value of the cash account at time t and we would be assuming without loss of generality that it starts off with a value of 1, so b 0 is equal to 1. The Cash Account is not risk free. And the reason it's not risk free is because interest rates are uncertain, they're stochastic. In particular the value of the Cash Account at the time t plus s say, is not known at time t for any value that's greater than 1. However, it is locally risk-free because I do know the value of B t plus 1 at time t. In fact, B t plus 1 will always be equal to B t times 1 plus the short rate. And I'm going to know the short rate at time t and therefore I will know B t plus 1 at time t. So again think of your bank account analogy. If I deposit money today for one month, I know what rate will apply for that one month period and so I will know how much I will get at the end of the month. But I will not know what interest rate will prevail in one months' time and therefore will not know future values of The Cash-Account beyond one month. So a quick thing to notice here, so Bt therefore has this expression here, based on the argument I just gave you, I can look at Bt plus 1 Bt and divide 1 by the other and see that I get one over 1 plus rt. And the reason I want that expression is down here, I want to derive equation four here. So how do I derive equation four? Well, again, for a non-coupon paying security, zt times z at time t, state j is equal to 1 over 1 plus rtj times the expected value of the security one period from now. So this is our familiar risk neutral pricing expression from the previous slide. I can actually rewrite this expression as the expected value under q, remember q is equal to q u and q d, the risk-neutral probabilities. And I can replace my 1 over 1 plus rtj, with this expression here bt over bt plus 1. So therefore I can write the value of the non-coupon paying security at time t as being the expected value at time t under q. On Z t plus one, multiplied by B t over B t plus one. So rewriting equation four, I can just bring the B t over to the left hand side and I get this expression here. This is an important expression but I can go a little bit further. I can actually iterate it to get the following. So for example, I can write Zt over Bt is equal to Et. Under q of, well we know it's zed t plus 1 over bt plus 1, but I can actually use this equation 5 again to write zed t plus 1 over bt plus 1 as the expected value under q. Condition on time t plus 1 information of Zt plus 2 over Bt plus 2. And using the law of iterated expectations, this is equal to the expected value condition on time t information of Zt plus 2 over Bt plus 2. And I could repeat the same trick again and again. And so, it's easy to see that this condition is hold. So this is our risk neutral pricing condition, for a non-coupon or non-dividend paying security. In particular, it's the pricing equation that we use for any security that does not pay any intermediate cash flows between times t and t plus s. When we're doing risk-neutral pricing for a coupon paying security, we use the exact same idea, so Ztj equals 1 over 1 plus Rtj times the expected value under q of the value of the security plus the cash flow at time t plus 1 that just gives us this expression here. And for the same reason as before, we can see as long as we price any coupon-paying security this way that cannot be an arbitrage. There is now way that this quantity can be greater than or equal to 0 and yet to have Ztj being less than 0. So you couldn't have a type A arbitrage because if this is greater than or equal to 0 there is neutral probabilities we know are strictly greater than 0, this is greater than 0 and so all of this expectation must be greater than or equal to 0. So in particular this is not possible. So you couldn't get a type A arbitrage, and for the same reason, you couldn't get a type B arbitrage as well. Alright, so we have seven. Well, it's easy to rewrite seven using the same ideas we used in the previous slide. I can replace 1 over 1 plus RTJ with Bt over Bt plus 1, bring the Bt over to the other side, and I get Expression eight. Now I can iterate, I can substitute in for example, if I substitute in the following; I know that Zt plus Bt plus 1 is equal to the expected value under q conditional on time t plus one information. Ct plus 2 over Bt plus 2. Plus Zt plus 2 over Bt plus 2. So if I substitute that in down here, I'm going to get this expression here when s equals 2. Now it's easy to see that this expression holds more generally for general values of s, or for integer values of s greater than t. So this is an extremely important condition. We're going to use this throughout this section on term-structure models and pricing fixed-income derivatives that tells us how to price fixed-income derivatives using risk-neutral pricing and this ensures that there's no arbitrage. In other words, it ensures that we're pricing fixed-income derivatives in a manner which is consistent with no arbitrage. Note also that equation six is actually a special case of nine, because we get equation six From nine by just taking all of these cjs equal to 0. So this is an extremely important result, it guarantees that we can price everything with no arbitrage. Here is a sample short-rate lattice. It starts off with the short rate r00 being equal to 6%. And then the short rate will grow by a factor of u equals 1.25 or fall by a factor of d equals 0.9 in each period. It's not very realistic. These interest rates, as you can see, grow quite large here. And given the current economy, global economy, where interest rates are very low. This example wouldn't be very realistic. But it is more than sufficient for our purposes. In fact, it's good to have such a wide range of possible interest rates. As it makes it easier to distinguish them in the examples that we'll see in the future. At this point I should also mention. That you should look at the spreadsheet that is associated with these modules. The spreadsheet you'll see this particular example there. And we're going to be using this example throughout this section to price various types of fixed income derivatives. We're going to be looking at caps, floors, swaptions. Options on zero coupon bonds and so on. So we're going to use this particular short rate example as our model in all of these pricing examples. So the first thing we're going to do is we're going to see how to price a zero coupon bond that matures a time t equal to 4. So if we want to do that we're going to use our risk neutral pricing, our risk neutral pricing result If you recall, states that Zt over Bt is equal to the expected value conditional on time t information of Zt plus 1 over BT plus 1, and this is a risk-neutral pricing identity for securities that do not pay coupons or do not have intermediate cash flows. And certainly that is true of a zero coupon bond. If you recall, a zero coupon bond does not have any intermediate cash flows. It only pays off its face value at maturity. And this example matures at t equal to 4. This face value is 100 and this indeed is if you like using our notation for 0 zero coupon bonds Z44. So what we're going to do is we're going to bond rate the price to zero coupon bond is to use this expression here by just working backwards in the lattice 1 period at a time. We know Z44, it's 100. At maturity the, the bond is worth 100, so let's work back and compute its value at time t equals 3. Well, to do that, we can just use this expression. Another way of saying this, and we saw this as well before, this is equivalent to saying that Zt Is equal to the expected value time t of Zt plus 1 over 1 plus or t. And so in fact it's this version that we're going to use. We're going to work backwards computing the values t at every node by discounting and computing the expected value one period ahead so that's all we're doing here. So, for example, the 83.08 that we've highlighted here is equal to 1 over 1 plus the short rate value at that node, and if we go back one slide we'll see the short rate value at that node was 9.38%. So that's where the 0.0938 comes from here. And then it's the expected value under q of the value of the zero coupon bond one period ahead. There are two possible values, 89.51, 92.22 and that's what we have here. So we just work backwards in the lattice, one period at a time, until we get its value here at time 0 and this is Z04. The time 0 value of the zero coupon bond that matures the time 4. Having calculated the zero coupon bond price at time0, we can now infer from that the actual interest rate that corresponds to, to t equals 4. In particular if we assume part period compounding and we let S4 denote the, the interest rate that applies to borrowing or lending for four periods then we know that 77.22 times 1 plus S moved to the power of 4 must be equal to 100. So of course we can invert that to get that S4. Is equal to 100 over 77.22, all to the power of 1 quarter minus 1 and so that's how we get S4. So there's always a one to one correspondence between seeing the zero coupon bond prices, and seeing the corresponding interest rate. Therefore it means that we can actually compute all of the zero coupon bond prices for the four different maturities. So we can compute the zero coupon bond price for maturity t equals 1, t equals 2, 3, and 4. And from this we can actually back out, back out the actual interest rates that apply to these periods. So, for example, we will get a term structure of interest rates that looks like the following; We have t down here and we've got the spot rate St here and maybe we'll see something like the following or maybe it's an inverted curve but this point here, so for example t1, that point corresponds to there and it corresponds to some spot rate st1 there. So we can actually use this model to price all the zero coupon bonds. And from all of these zero coupon bond prices, we can invert the message in the previous slide to get the term structure. The term structure is the term structure of interest rates. We can see what interest rate applies to each time t. At time t equals 1, for example, we will then compute a new set of zero coupon bond prices, and obtain a new term structure. So for example at time t equals 0 were down here. But at time equals 1 may be I'm up in this state of the world. So if I'm up in this state of the world I could recompute the time structure of interest rates, I could do that by a pricing all of the zero coupon bonds at this point. I'm going to get a different set of prices at the set of prices ahead at time t equals 0 and I can invert this new set of prices to get the new time structure and may be that new time structure will look different, may be will look something like this. So I will get a new term structure times t equals 1, moreover the term structure I see will depend on whether I'm up here or down here. So what we've actually succeeded in doing is defining a stochastic model or a random model for the term structure of interest rates by just focusing on the short rate. So the short-rate or t is just a scalar random variable or scalar process by focusing on modeling this short rate, as we've done, we've actually succeeded in defining a stochastic or random model for the entire term structure. And that's actually a very significant point to keep in mind when working with these short-rate models.
