In the last module, we saw how to price caplets and floorlets. Well there are other liquidly traded fixed income derivatives securities in the market place and there are swaps and swaptions. So in this module how to price swaps and then how to price options and swaps which are called swaption. Again we will be focusing on the binomial lattice model. And by pricing them in this model, we will also understand and reinforce our learning behind the mechanics of these securities. So here again is our familiar short-rate lattice. We've seen it now many times beginning at r equal to 6%. Rising by a factor of u equals 1.25 or falling by a factor of d equals 0.9. What we want to do initially in this module is the price an interest rate swap with a fixed rate of 5% that expires at t equals to 6. The first payment will occur at t equal to 1, and the final payment will occur at t equal to 6. Now the payment at each node will be the following. It will be plus or minus depending on whether you're long or short the swap. Of the short rate that prevails off that node minus k. But that payment is made in a rears at time t = i + 1. So again, like the caplet that we saw in the last module, we're going to assume that the cash flows in our swap are paid in a rears which is a typical situation that occurs in practice. So just to give you an idea. So for example up at this node here, 18.31%. Well the cash flow at this node will actually be 18.31%-5%. But actually I should say that this is the cash flow that is determined out of this node. The cashflow itself will not take place until one period later time equal to six. Similarly the cash flow at this node will be 11.2%-5%. That is the cash determined at that node, it would be paid a time t equal to 4. And so all of the nodes here will have cash flows equal to these values minus 5% to be paid one period ahead. How do we price this swap? Well let's go to that. So what we're going to do is to figure out again how much the swap is worth at the final node. Now the final node again is t equal to 6. But we don't want to record the final values of t equal to 6. Because for example, if we find ourselves at this node here. Well it won't be clear how we got here. Is the payoff at this node because we were at this prior node or we were at this prior node? And depending on which node we were at in the previous period, we'll get a different cash flow up here. So this is the exact theme situation as we saw the cash in previous module. It's actually much easier to record the cash flows after nodes at which they occur. But discount them suitably to reflect the fact that they paid in the rears or in one period later. And that's what we're going to do here, so for example, this 0.723 that we have highlighted up here. Well that is equal to the cashflow, r5 5-K divided by 1 + r5 5. So r5 5-K is the payoff of the swap, but this swap takes place at time 6. So we have to discount it by the short rate that prevails at this node. And so we get this value which is 0.0723 at time t=5 and stays 4. Here's another sample calculation. 0.1686, so where do we get this from? Well we're going to get that from our standard risk neutral pricing. Maybe we should write our risk mutual pricing again here. So in this case, risk mutual pricing takes the following form. Let st be the generic value of some security that we want to price. Then we know that st is equal to the expected value, conditional on time t information under the risk mutual probabilities. The payoff or the value of the security one period ahead ST plus 1. But also this swap actually has intermedia cash flows or coupons if you like. We must also include these cash flows CT + 1, discounted by 1 + rt. So this our risk mutual pricing or familiar risk mutual pricing for security that has intermedia cash flows. So this 1.686, for example, how is that calculated? Well it is as follows. It is 1 over the short rate that prevails at that node, so that short rate is 9.38%. And we can check that by going backwards. Here it is, 9.38%. And we have the cash flow, so this is our CT + 1. This is the payment that takes place one period ahead. It is the cash flow determined at this node, but it is paid one period ahead. And these terms here, give us the possible values of ST + 1. So either 0.1793 if we go up or 0.1021 if we go down. So this is just risk neutral pricing. We work backwards in the ladders until we get the time t equals 0. That gives us the initial arbitrary value of the swap, and we find this 0.0990. And again to keep in mind here, this is the fair value of the swap for a notion of $1, right? We're multiplying each of these pay offs, these cash flows by $1. In practice maybe you'd have $100,000 or $1 million and so you should be multiplying this value then by that notion of a 1 million or $100,000. Now let's consider how to price swaptions. Swaptions like swaps are very liquid securities that are traded all the time used by many market participants to both hedge interest rate risk as well as speculate on interest rate risk. A swaption is an option on a swap. So what we're going to do is we're going to consider a swaption on the swap of the previous slide. We're going to assume that the option strike is 0%. This should not be confused with the fixed rate of the underlying swap. Remember the fixed rate of the underlying swap was, 5%. So we're going to assume that the option strike is 0% and that the swaption expiration is t = 3. In other words, the option on the swap expires at t = 3. Therefore a time t = 3, if you own the swaption, you have the right to exercise or enter into this swap for a value or a strike value of 0. So the payoff at t equals to 3 is going to be the maximum of 0 and S3. You will only exercise this swaption at t equals to 3 if S3 is positive. If S3 is negative, then you won't exercise the maximum would be 0 and you will do nothing. So how are we going to price this swaption? Well actually it's going to be very simple. We're going to use our calculations from the previous slide which we see here, to get the value of the swaption or order to get the value of the swap a t equals a 3. We're going to replace this column of prices with the maximum of 0 on this column, that will gives us a pay off of swap option at maturity, t = 3. And then we will just calculate the fair value of this pay off by working backwards in the lattice using risk neutral pricing. One thing to keep in mind though however, is that when we're working back from t equal 3 back to 0, we no longer include the underlying cash flows of the swap. And that's because the holder of the swaption does not get those underlying cash flows. The holder of the swaption will only get the cash flows of associated with the swap if they exercise the t equals 3. And then they will get this swap cash flows for all time periods greater than t equals 3. They will not get them for the periods before t=3. So here are some sample calculations again, for example, so if you look at t equals to 3, this is the pay off of, This swaption. So in fact all of these values here from t equals 4 and 5 are simply the values that we see that we calculated earlier for the underlying swap. So we see 0.0512 here, well that's the same 0.0512 here. Down here however, is the value of the swaption at maturity. And now this is equal to the maximum of 0 and S3. So if you recall, the only change is actually down here. Because in the underlying swap value, we had a negative value minus 0.0085 at this node. But now, the holder of the swaption would not choose to exercise at this node. Why would they buy something for 0 when it has a negative value? They won't do that. So they will exercise however at all of these three nodes, because these three nodes the swap has a positive value. So certainly the whole of the swaption will exercise at these three nodes and receive these three payoffs. So what we have is here are the values, the payoff values of the swaption at maturity, at t = 3. It's equal to the maximum of 0 and S3. And now we actually just compute the fair value of the swaption by computing the value of this payoff backwards in the lattice. So here what we're using in our risk-neutral pricing again, if you like I won't use s here, I'll say Z. So Zt = the expected value of Zt + 1 divided by 1 + over t. This is our risk neutral pricing for the swaption. Note, I don't have any intermediate coupon or cash flow in here. And that's because the swaption doesn't pay any intermediate coupons or cash flows between t = 0 and t = 3. So I just iterate this backwards to get the value of 0.0620 as being the value of the swaption at t = 0. So here's a sample calculation, 0.0908. Well 0.0908 is just 1 over 1 plus, or 1 over 1 is going to be 7.5% of this node times the expected value of the swaptions one period head while the risk mutual probabilities are half in a half. These are the values of the swaption one period ahead 0.1286 and 0.0665 and so that's how I get 0.0908. You can see all of this calculations performed in the spreadsheet, they're very easy to do. We see again here that we have our usual short rate lattice that we're using throughout. So these are the details here, I won't go through them again. Down here, we calculate the value of the swap, the underlying swap. So there's a fixed rate of 5%, here, we see the payoffs of the swap at time t equals the 5. In fact these are the values of the swap, because remember the payoffs occur time t equals to 6. But we discount them back one period to reflect that the swap cash flows occur in the rears. So we do that at time t equals 5, we now work backwards in the lattice using risk mutual pricing. So for example down here, we see that the value of the swap at this node is the payoff that's determined at that point. So that's the cash flow identified up here plus the expected value of the swap one period ahead. And that's given to us by these terms here. And of course, I have to discount both of these quantities. The cash flow because of in the rears and the value of the swap one period ahead by 1 plus the short rate at that node. And that's how I get this expression here. It's in bold again because you can actually enter the formula into the cell. And then drag this formula throughout the rest of the lattice to compute the prices of the swap at every node. We find an initial swap value of 0.0990. To compute the fair value of the swaption, well all we do is we just repeat really what we did for the swap. We just calculate the value of the swaption at maturity, so we see it down here. In this case, it's the maximum of the swap value which is given to us here and 0. So that's how we get these values down here. And now we just use risk mutual pricing to work backwards on the lattice. Note that any cell in this lattice, we don't have any intermediate cash flow or we don't have the cash flow associated with the swap at that time. And that's because the holder of the swaption does not get any of those cash flows or pay any of those cash flows until such time as they exercise the swaption. So we just work backwards to find the initial value of 0.0620.
