In the last module we saw how to price zero coupon bonds. In this module we're now going to go a step further and see how to price options on zero coupon bonds and options on bonds in general. We're going to discuss European and American column put options on zero coupon bonds and see that you can price these securities by just using risk- neutral pricing and working backwards in a binomial lattice model. 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.
