Here is a sample Short-Rate lattice. It starts off with the short rate or 0,0 being equal to six percent and then the short rate will grow by a factor of U equals 1.25 or fall by 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 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 prize various types of fixed income derivatives. We can be looking at caps floors, swaptions, options and 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 zero-coupon bond that matures at 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. This is our 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. In this example, maturity is at t equal to 4, this face value is 100, and this indeed is if you like using our notation for zero-coupon bonds, Z44. So what we're going to do is, we're going to moderate the price zero coupon bond, is to use this expression here, by just working backwards in the lattice one period at a time. We know Z44 it's 100. At maturity 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. We know that Z3 over B3 will be equal to the expected value of Z4 over B4. So we can actually work backwards to do this, or another way of saying this we saw this where before this is equivalent to saying that Zt is equal to the expected value at time t of Zt plus 1 over 1 plus rt. In fact it's this version that we're going to use, we're going to work backwards computing the values at 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 percent. So that's where the 0.0938 comes from here, and then it's the expected value under queue 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 zero, and this is Z zero four the time zero value of the zero coupon bond that matures at time four. Having calculated the zero-coupon bond price at time zero, we can now infer from that, the actual interest rate that corresponds to t equals 4. In particular, if we assume per period compounding, and we let S4 denote the interest rate that applies to borrowing or lending for four periods, then we know that 77.22 times 1 plus S4 to the power 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 one-quarter minus 1, and 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. From this we can actually 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 a 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 of the zero-coupon bonds, and from all of the zero-coupon bond prices we can invert them as I did 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 one, 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 zero we're down here but at time T plus 1, maybe I'm up in this state of the world. So if I'm up in this state of the world I could recompute the term structure of interest rates. I could do that by pricing all the zero coupon bonds at this point, I'm going to get a different set of prices. So the set of prices or had at time t equals 0, and I can invert this new set of prices to get a new term structure, and maybe that new term structure will look different, maybe it will look something like this. So, I will get a new term structure at time 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 random model for the term structure of interest rates by just focusing on the short rate. So the short rate rt is just a scalar random variable, or scalar process, by focusing on modeling the 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.
