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

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

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

futures and bonds, caplets, swaps, and swaptions.

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

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

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

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

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

swaptions are all very liquidly traded.

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

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

in such a way that the model prices of the securities

match the market prices of these securities.

This process is called calibration.

What we want to do is calibrate our parameters so

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

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

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

parameters in our binomial model.

We have rij's and qij's for all ij.

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

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

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

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

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

So we actually assumed our risk neutral parameters or

risk neutral probabilities were half at all notes.

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

were also given to us up until this point.

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

or fall by a factor of 0.9.

Well, we can no longer do that.

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

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

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

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

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

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

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

then there is a volatility component bi.

Why do I call this a volatility component?

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

j is a random variable.

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

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

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

bi for i equals 0 up to n minus 1.

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

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

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

model at all.

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

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

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

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

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

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

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

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

the log of rij is equal to the log of

ai plus bi times j.

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

the log of the short rate.

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

the log of the short rate.

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

the stage j.

And that's why I can interpret bi which multiplies

the random variable as been in volatility parameter.