So here is our sample Short Rate Lattice again.

We introduced this in the last module and I mentioned that we're going to see this

lattice throughout these examples. So, just to remind ourselves, we start off

with r 0, 0 equals 6%. And then the interest rate grows by factor

of 1.25, or it falls by a factor of 0.9 in each period.

So this is a Short Rate Lattice. Remember, the short rate is the risk free

interest rate that applies for borrowing or lending for the next period.

What we're going to do is we're going to price an option on a zero coupon bond.

The zero coupon bond we consider will be a zero coupon bond that matures at time t

equals 4. And we actually considered this bond as

well in the last module. So, in the last module, we saw how to

price this zero coupon bond. If you recall, the way we priced it was,

we said that at any time t, zt was equal to the expected value using the

risk-neutral probabilities of zt plus 1 over 1 plus rt.

And this is our risk-neutral pricing. So we started off at time t equals 4.

Where we know the value of the zero coupon bond at this period, z subscript 4

superscript 4, it's equal to the maturity of the bond.

So you got a face value of 100 dollars back at that period.

And then you just work backwards in the lattice using this expression here to

compute the value in the previous period. So we start off here with 100.

We come back here and we work backwards. We said for example, the value of 83.08 at

this node is given to us by 1 over 1 plus the short rate to the product of this node

which is 9.38%. Times the expected value of the zero

coupon bond one period ahead. And one period ahead was either 89.51 or

92.22. So we work backwards and we get a price of

77.22. And that's equal to the zero coupon bond

price at time 0 for maturity t equals 4. More generally, of course, we could have

priced this bond as follows. We could have said, we know from our

risk-neutral pricing, we could have said Z, Z0 over B0 equals the expected value

times 0 using the risk-neutral probabilities of Z4, 4 over B4.

So if we recall, this is our risk-neutral pricing expression for any security that

did not have intermediate cash flows. And certainly a zero coupon bond is such a

security. We know that B0 equals 1.

So in fact we would just get that Z04 is equal to the expected value at time 0 of Z

4, 4 over B4. So this translates to Z04 equals the

expected value of 100 divided by B4. And so we could have actually calculated

the zero coupon bond price at time 0 just using this expression and just ca, doing

one single calculation instead of working backwards period by period, we could've

done it all in one step by evaluating this and figuring out the probabilities of the

various values before and summing these quantities appropriately weighted by those

probabilities. So let's get to pricing a European Call

Option on this zero coupon bond. The maturity, the expiration of the

option, would be t equals 2. We're going to assume a strike of $84.

So therefore the option payoff would be the maximum 0 and Z24 minus 84.

This little dot here I've used just to denote the fact that actually this is a

random variable, it will depend on what state we're in, so for example the state

at time 2 will either be 0, 1, or 2. So the underlying zero coupon bond matures

at time t equals 4. So what we need to do is to figure out the

value of this at time 2. But we've already done that in the

previous slide. We know the option value at time equals 2

is given to us by these numbers here. The strike is 84, so in that case, if the

strike is 84 we would not exercise here but we would exercise here and get $3.35

and we'd exercise here and get $6.64. And that's where these value come from

here. So 0, 3.35 and 6.64 are the value of the

option at expiration. So all we're going to do now is use our

usual risk-neutral pricing. Risk-neutral pricing tells us how to

evaluate this, so we can simply work backwards in the lattice one period of

time to get the initial value of the option.

So for example the 1.56 we see here is equal to 1 over 1 plus the interest rate

that prevails at this node. That interest rate is given to us here at

7.5%. So we get 1 over 1 plus 0.75 times a half

times 0 plus a half times 3.35 and that equals 1.56.

And then after calculating that number and 4.74 down at this node, we go back to time

t equals 0, and get the initial value of the option.

We can see it's going to be 2.97. If we want to price an American option on

the same zero coupon bond, we can do the exact same thing.

The only difference being that at each node we stop to see whether or not it was

optimal to early exercise at that note or not.

So here's an example, this time the expiration is t equal to 3, it's going to

be an American put option on the same zero coupon bond and it has the strike of $88.

So if we go back to the zero coupon bond, price at t equals 3, well these are the

prices at t equals 3. Now you can see that $88 is actually less

than all of these prices. So in fact, at t equals 3 it would never

be optimal to exercise because $88 is less, as I said, than all of these prices.

So therefore, in fact, the payoff of the put option at maturity at t equals 3 is

indeed 0. And that's why we have zeros all along

here. So now we just work backwards in the

ladders using our risk-neutral pricing as usual, but also checking in each period

whether or not it is optimal to early exercise in that period.

So for example, the 4.92 here is equal to the maximum of the value of exercising at

that period 88 minus 83.04. Where does that 83.04 come from?

Well, that's the value of the zero coupon bond, at that node at time 2.

Here it is 83.08. So that's the value we get if we exercise

then and then we alter to compare it to 1, with 1 over 1 plus 9.38%, the expected

value under q, the value of the option one period ahead.

Well the value of the option one period ahead, as we said, is 0, so therefore at

this point the value of the option is 4.92.

And in fact, we just worked backwards, doing that in every note.

It turns out that in this example, it's optimal to early exercise everywhere, so

it's a very realistic example, but that's fine.

We just want to see the mechanism of how the American put option works and how we

can use risk-neutral pricing to, to price it.

So, here is the Excel spreadsheet. I hope you have this open with you when

you're going through these, these video modules.

Because you can see how to price all of the securities that we will discuss in the

spreadsheet. So up here, we have the parameters of a

binomial lattice model. And it begins at 6%, the short rate does,

it grows by a fact of u equals 1.25 or falls by factor t equals 0.9, and the

risk-neutral probabilities of 0.5 and 0.5. So what you can see here is the first

lattice we've built is the short rate. It starts off at 6%, and then it grows by

a certain amount, or falls by a certain amount.

This cell is in bold because by getting the correct formula into this cell, I can

actually copy this cell forward and across throughout the lattice to populate the

rest of the cells. So this is our short rate.

If I want to price the zero-coupon bond, I come over here.

I know the value of the bond is 100 at maturity, which is why I have 100 in all

of these cells. And then I want to apply risk-neutral

pricing backwards in the, in the lattice. So here I've highlighted in bold this

cell, because this is the cell where I entered the formula.

And then I can drag and copy this formula back through the rest of the lattice to

get the zero coupon bond prices every note.

So I get the value of 77.2, which 77.22 we have seen before.

Over here, down here I compute the price of the American zero, of the American

option on the zero coupon bond. Over here, I compute the value of the

European call option on the zero coupon bond.

So you can see again, we've highlighted and bold the cells, the important cells we

should input the Excel formula. Once you've got that in there, you can

drag and copy that formula to the other cells.

So this is the spreadsheet. We can see how to both construct the short

rate lattice, price zero coupon bonds, and compute European and American option

values on those zero coupon bonds.