0:34

Up until now we focused on the pricing of derivative securities.

Â We've taken our model and the parameters of the model as given.

Â What we've done is we've priced everything our bonds, our options on bonds, forwards,

Â futures and bonds, caplets, swaps, and swaptions.

Â We've priced all of these securities using risk mutual pricing.

Â Using risk mutual pricing, ensured that our models were overcharge free.

Â However, a model is no good, unless at the very least, we can get the model prices

Â to agree with the market prices of liquid securities that we see in the marketplace.

Â Security such as caps and floors, caplets, floorlets, swaps and

Â swaptions are all very liquidly traded.

Â And we can see the prices of these securities in the marketplace.

Â So what we want to do with our model is to pick the parameters of our model

Â in such a way that the model prices of the securities

Â match the market prices of these securities.

Â This process is called calibration.

Â What we want to do is calibrate our parameters so

Â that the model prices agree with the market prices of liquid securities.

Â So we're going to discuss that further in this module.

Â The first thing to keep in mind is that they're actually very money-free

Â parameters in our binomial model.

Â We have rij's and qij's for all ij.

Â So in fact, there's going to be some sort of approximation to cn squared parameters

Â in our model where n is the number of periods in our binomial model.

Â So what we typically do is we fix some of these parameters, and so

Â that's in fact what we've been doing until now.

Â We just fixed our qij to be equal to q, which is equal to half for all ij.

Â So we actually assumed our risk neutral parameters or

Â risk neutral probabilities were half at all notes.

Â We've also assumed that the short rates, rij,

Â were also given to us up until this point.

Â We started off with r being 6%, we let it rise by a factor of 1.25,

Â or fall by a factor of 0.9.

Â Well, we can no longer do that.

Â If we want our model prices to match market prices, we're going to actually

Â have to have some free parameters that we can use to calibrate the market prices.

Â And so that's what we're going to do here.

Â One possible way to do this is to use a parametric form for the rij's.

Â So for example, the Ho-Lee model assumes that rij is equal to ai plus bi times j.

Â So if you recall i is time and j is state.

Â So it assumes that there is a drift component parameter ai and

Â then there is a volatility component bi.

Â Why do I call this a volatility component?

Â We remember j is the state that we're going to see a time i.

Â j is a random variable.

Â We don't know what statement will be in a time j, so we can interpret bi,

Â the multiplier of the state j as being a volatility parameter.

Â So in this case, we actually only get 2n parameters, ai and

Â bi for i equals 0 up to n minus 1.

Â You can actually check that the standard deviation of the one period rate is

Â then bi over 2 if you're conditioning on where you are at time i minus 1.

Â Now the Ho-Lee model that should be set is actually not a very realistic

Â model at all.

Â There is some real problems with the dynamics induced by this assumption here.

Â However, it has been a very influential model in the term structure literature and

Â some very interesting term structure models have grown out of some of

Â the ideas that were present in the original paper by Ho-Lee.

Â We're going to focus here instead on the Black-Derman-Toy model.

Â The Black-Derman-Toy model assumes that the interest rate

Â at node N(i,j) is given by rij equals ai times e to the b i j.

Â So actually if I take logs across here I'll see that what we're assuming is that

Â the log of rij is equal to the log of

Â ai plus bi times j.

Â So we can interpret log ai is being a drift parameter for

Â the log of the short rate.

Â I make it an interpret bi is being in volatility parameter for

Â the log of the short rate.

Â Remember, as I said it on the previous slide that the random variable here is

Â the stage j.

Â And that's why I can interpret bi which multiplies

Â the random variable as been in volatility parameter.

Â 4:55

What we want to do is we need to calibrate the model to the observed term-structure

Â in the marketplace and ideally other security prices.

Â So certainly at the very least,

Â any term structure model that you want to use should ensure that the zero-coupon

Â bond prices are equivalently the term structure of interest rates

Â matches the time structure of interest rates we see in the marketplace.

Â So we're certainly going to want to calibrate our model to zero-coupon

Â bond prices.

Â Ideally, we would also want to calibrate to other liquid security prices like

Â caplets, and swaps, and floors, swaptions, and so on.

Â So this is done in general by choosing the ai's and the bi's to match market prices.

Â 7:15

So here's how we're going to do this or

Â at least this is how we could do it if we wanted to using the forward equations.

Â We're actually going to be a little bit lazy and let Excel do all the work for

Â us or let Solver in Excel do all the work for us.

Â But this is one way we could go about doing this.

Â We know the following.

Â Well, we know that 1 over 1 plus si to the i is equal to this sum

Â here of elementary security prices.

Â What is this sum on the right inside?

Â Well if you think about it, i is fixed, so this is the time i,

Â state j elementary security price summing over all states j at time i.

Â Well we know that this is just a zero-coupon bond price at time zero,

Â which matures at time i.

Â 10:33

So we have r00, we can then use the forward

Â equations to get p10 and p01.

Â And now we can move on to i = 2.

Â We can actually use this equation we can get 1 over 1 plus s2 to

Â the power of 2 is equal to and we can sub in these terms.

Â These terms will just have the p1s,

Â p10 and p11 which we will know already.

Â They will have an a1, which we don't know, but that's what we will solve for.

Â And they will have s2 which we see in the marketplace.

Â So we'll get some quantity here which will depend on a1,

Â we will know everything else and we will there for be able to determine, a1.

Â And we can continue on, on this iterative manner working forwards to determine

Â all the ai's by comparing the si's, which we see in the marketplace,

Â using this equation to compare them with what the model says si must be and

Â figure out what the ai's are.

Â 12:15

We need to assume some starting values for the a values.

Â So we'll just put in the value of 5 for all of these values.

Â The calibration's going to actually determine what the correct values of

Â a should be so that the model prices of zero-coupon bonds or

Â the term structure matches the market term structure.

Â Right now for example, we can see here is the market spot rates, 7.3% up to 12.32%.

Â Down here, we have our short rate lattice,

Â this short rate lattice is the Blackâ€“Derman-Toy lattice.

Â You can actually see that we're constructing this lattice using the b and

Â a parameters from Black-Derman-Toy.

Â Very important to note over here, we've actually fixed the volatility parameter b,

Â we've assumed that b is a constant for all periods i,

Â and we've set it equal to 0.005.

Â We will certainly be returning to this issue in awhile.

Â 13:06

Now we've got the risk neutral probabilities which we've assumed to

Â be q and 1 minus q and that they're both equal to a half.

Â We compute the elementary prices down here using the forward equations, so

Â we actually copy in our formula in this cell.

Â And then we can drag the formula in this cell throughout the lattice to

Â complete the elementary prices at all other nodes.

Â So we have our short rate lattice, we have our elementary prices.

Â Note that these elementary prices are a function of b and

Â these a's, which at the moment are all set equal to 5.

Â Below that, we can actually sum the elementary prices for

Â each period, to get zero-coupon bond prices.

Â So for example, if I sum these three guys here, I'm going to get 0.907.

Â If I sum all of these, I'm going to get 0.709.

Â So we're just using what we've seen in an earlier module.

Â That is how we can use the elementary security prices

Â to compute the zero-coupon bond prices.

Â Once I have the zero-coupon bond prices in the model

Â I can invert them in the usual way to get the spot rates in my models.

Â So I'm seeing here that the spot rates are 5%, 5.01%, 5.06% and so on.

Â So a very flat term structure where all the interest rates are approximately 5%,

Â and this is following because I've actually assumed that the a's are all 5,

Â and that's to begin with.

Â So what I want to do is, is to do a simple calibration.

Â What I want is, I want to choose the ai's, so that these spot rates down here,

Â 14:35

that I've highlighted in row 48, I want to choose them so

Â that they match the market spot rates in row 4.

Â Right now they certainly don't.

Â So how am I going to force them to match that?

Â Well I'm just going to use Solver.

Â I'm going to compute the square differences between those cells, so

Â this number here for example will be the difference between 5.04 which is

Â the model spot rate and the market spot rate, which is up in cell I4.

Â I'm going to square it and I get this number here.

Â I'm going to do that for

Â all the periods from i = 1 out to 14, and then down here I summed them.

Â So here is sum of the squared differences, and I want this to be equal to 0.

Â So this is what I'm going to do with Solver.

Â I'm going to try and choose the ai's up there in row 5.

Â I'm going to tell Solver to choose the a's in row 5 so as to minimize this sum here.

Â I want this to as close to zero as possible.

Â So we can do that in Solver.

Â 16:28

When I do this on Excel for Windows I don't need to run it again.

Â I seem to get a very good solution when I run it in Windows.

Â So as you can see I'm actually getting it much smaller now.

Â I'm getting a number by 8.7 by 10 to the minus 11.

Â And so if you noticed, the values of A have now changed.

Â I no longer have 5 in all these cells, by running Solver to minimize the square

Â differences between the market spot rates and the model spot rates,

Â I have succeeded in matching the market spot rates to the model spot rates.

Â So 7.3% let's pick a value.

Â So 9.64% is the market spot rate.

Â My model spot rate is 9.64% for the same period.

Â So now the term structure of interest rates in my model matches the term

Â structure of interest rates in the market.

Â You might also ask the question at this point well we've just calibrated our model

Â to the term structure of interest rates in the marketplace.

Â Is that enough?

Â Should we not also be calibrating to other security prices?

Â And the answer to that is absolutely, yes, and

Â we return to that issue in the next module.

Â