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So here is our sample Short Rate Lattice again.

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

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

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

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

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

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

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

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

Â equals 4. And we actually considered this bond as

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â security. We know that B0 equals 1.

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

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

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

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

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

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

Â various values before and summing these quantities appropriately weighted by those

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â time to get the initial value of the option.

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

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

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

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

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

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

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

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

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

Â optimal to early exercise at that note or not.

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

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

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

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

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

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

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

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

Â here. So now we just work backwards in the

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

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

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

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

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

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

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

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

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

Â this point the value of the option is 4.92.

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

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

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

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

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

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

Â you're going through these, these video modules.

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

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

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

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

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

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

Â a certain amount, or falls by a certain amount.

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

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

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

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

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

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

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

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

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

Â get the zero coupon bond prices every note.

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

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

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

Â European call option on the zero coupon bond.

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

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

Â drag and copy that formula to the other cells.

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

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

Â values on those zero coupon bonds.

Â