So in this spreadsheet we're actually going to perform such a calibration. We have here, a 14 period model. You can think of periods corresponding to years or months, whatever you like. We have the market term structure of interest rates. So we have 7.3 percent for one year or one period, 7.62 percent for two years, 8.1 percent for three years or three periods, and so on.The important thing here is that we see the term structure of interest rates in the market place, as it's given to us here in these highlighted cells. We need to assume some starting values for the a values. So we'll just put in the value of five for all of these values, for all of these values of a. The calibration is going to actually determine what the correct values of a should be so that the model prices of zero-coupon bonds or the term structure matches the market term structure. Right now for example, we can see here is the market spot rates, 7.3 percent up to 12.32 percent. Down here, we actually correspond. Down here, we have our short rate lattice. This short rate lattice is the Black-Derman-Toy lattice. You can actually see that we're constructing this lattice using the b and a parameters from Black-Derman-Toy. Very important to note over here, we've actually fixed the volatility parameter, b. We've assumed that b is a constant for all periods I, and we've set it equal to 0.005. We will certainly be returning to this issue in a while. Now, we've got the risk neutral probabilities, which we've assumed to be q, and 1 minus q and that they're both equal to a half. We compute the elementary prices down here using the forward equations. So we actually copy in our expression here for the Excel formula in the cell. So you actually copy in our formula in this cell, and then we can drag the formula in the cell throughout the lattice to complete the elementary prices at all other nodes. So we have our short rate lattice, we have our elementary prices. Note that these elementary prices are a function of b, and these a's, which at the moment are all set equal to five. Below that, we can actually sum the elementary prices for each period, to get zero-coupon bond prices. So for example, if I sum these three guys here, I'm going to get 0.907. If I sum all of these, I'm going to get 0.709. So we're just using what we've seen in an earlier module. That is how we can use the elementary security prices to compute zero-coupon bond prices. Once I have the zero-coupon bond prices in the model, I can invert them in the usual way to get the spot rates in my model. So I'm seeing here that the spot rates of five percent, 5.01 percent, 5.06 percent, and so on. So a very flat term structure for all the interest rates are approximately five percent, and this is following because I've actually assumed that the a's are all five. That's to begin with. So what I want to do, is to do a simple calibration. What I want is I want to choose the AI's so that these spot rates down here that I've highlighted in row 48, I want to choose them so that they match the market spot rates in row four. Right now, they certainly don't. So how am I going to force them to match that? Well, I'm just going to use Solver. I'm going to compute the square differences between those cells. So this number here for example, would be the difference between 5.04, which is the model spot rate, and the market spot rate which is up in cell I4. I'm going to square it, and I get this number here. I'm going to do that for all the periods from i equals 1 up to 14, and then down here I sum them. So here is the sum of the squared differences, and I want this to be equal to zero. So this is what I'm going to do with Solver. I'm going to try and choose the AI's up there in row five. I'm going to tell Solver, to choose the A's in row five, so as to minimize this sum here. I want this as close to zero as possible. So we can do that in Solver. So let's do that. So we have Solver installed, Solver is in Excel add-in. We've set the objective to be cell D51, and I want to minimize D51 by changing the variable cells C5 to P5. These are the cells that contain the A's, A0 up to A13. So let's run it. For some reason, we're using a Mac here. Solver seems to be quite slow using a Mac. We can see the solution we're getting. I'm getting an objective function of 0.001. Now, this looks pretty small, it looks pretty good. But actually, I expect it to be even smaller. So I'm going to actually run Solver again and try to get a better solution. When I do this on Excel for Windows, I don't need to run it again. I seem to get a very good solution when I run it in Windows. So as you can see, I'm actually getting it much smaller now. I'm getting a number by 8.7 by 10 to the minus 11. So if you notice, the values of a have now changed. I no longer have five and all of the cells. By running Solver to minimize the square differences between the market spot rates and the model spot rates, I have succeeded in matching the market spot rates to the model spot rates. So 7.3 percent. Let's pick a value. So 9.64 percent is the market spot rate. My model spot rate is 9.64 percent for the same period. So now, the term structure of interest rates in my model, matches the term structure of interest rates in the market. You might also ask the question at this point, well, we've just calibrated our model to the term structure of interest rates in the marketplace, is that enough? Should we not also be calibrating to other security prices? The answer to that is absolutely yes, and we'll return to that issue in the next module