0:02

In the last module, we saw how to price caplets and floorlets.

Â Well there are other liquidly traded fixed income derivatives

Â securities in the market place and there are swaps and swaptions.

Â So in this module how to price swaps and then how to price options and

Â swaps which are called swaption.

Â Again we will be focusing on the binomial lattice model.

Â And by pricing them in this model, we will also understand and

Â reinforce our learning behind the mechanics of these securities.

Â 0:26

So here again is our familiar short-rate lattice.

Â We've seen it now many times beginning at r equal to 6%.

Â Rising by a factor of u equals 1.25 or falling by a factor of d equals 0.9.

Â What we want to do initially in this module is the price an interest rate

Â swap with a fixed rate of 5% that expires at t equals to 6.

Â The first payment will occur at t equal to 1,

Â and the final payment will occur at t equal to 6.

Â Now the payment at each node will be the following.

Â It will be plus or minus depending on whether you're long or short the swap.

Â Of the short rate that prevails off that node minus k.

Â But that payment is made in a rears at time t = i + 1.

Â So again, like the caplet that we saw in the last module,

Â we're going to assume that the cash flows in our swap are paid in

Â a rears which is a typical situation that occurs in practice.

Â So just to give you an idea.

Â So for example up at this node here, 18.31%.

Â Well the cash flow at this node will actually be 18.31%-5%.

Â But actually I should say that this is the cash flow that is determined out of

Â this node.

Â The cashflow itself will not take place until one period later time equal to six.

Â Similarly the cash flow at this node will be 11.2%-5%.

Â That is the cash determined at that node, it would be paid a time t equal to 4.

Â And so all of the nodes here will have cash flows equal to these values minus

Â 5% to be paid one period ahead.

Â 2:24

Well it won't be clear how we got here.

Â Is the payoff at this node because we were at this prior node or

Â we were at this prior node?

Â And depending on which node we were at in the previous period,

Â we'll get a different cash flow up here.

Â So this is the exact theme situation as we saw the cash in previous module.

Â It's actually much easier to record the cash flows after nodes at which

Â they occur.

Â But discount them suitably to reflect the fact that they paid in the rears or

Â in one period later.

Â And that's what we're going to do here, so for example,

Â this 0.723 that we have highlighted up here.

Â Well that is equal to the cashflow, r5 5-K divided by 1 + r5 5.

Â So r5 5-K is the payoff of the swap, but this swap takes place at time 6.

Â So we have to discount it by the short rate that prevails at this node.

Â And so we get this value which is 0.0723 at time t=5 and stays 4.

Â 3:28

0.1686, so where do we get this from?

Â Well we're going to get that from our standard risk neutral pricing.

Â Maybe we should write our risk mutual pricing again here.

Â So in this case, risk mutual pricing takes the following form.

Â Let st be the generic value of some security that we want to price.

Â Then we know that st is equal to the expected value,

Â 3:56

The payoff or the value of the security one period ahead ST plus 1.

Â But also this swap actually has intermedia cash flows or coupons if you like.

Â We must also include these cash flows CT + 1, discounted by 1 + rt.

Â So this our risk mutual pricing or familiar risk mutual pricing for

Â security that has intermedia cash flows.

Â So this 1.686, for example, how is that calculated?

Â Well it is as follows.

Â It is 1 over the short rate that prevails at that node, so that short rate is 9.38%.

Â And we can check that by going backwards.

Â 4:51

And these terms here, give us the possible values of ST + 1.

Â So either 0.1793 if we go up or 0.1021 if we go down.

Â So this is just risk neutral pricing.

Â We work backwards in the ladders until we get the time t equals 0.

Â That gives us the initial arbitrary value of the swap, and we find this 0.0990.

Â And again to keep in mind here, this is the fair value of the swap for

Â a notion of $1, right?

Â We're multiplying each of these pay offs, these cash flows by $1.

Â In practice maybe you'd have $100,000 or $1 million and so

Â you should be multiplying this value then by that notion of a 1 million or $100,000.

Â Now let's consider how to price swaptions.

Â Swaptions like swaps are very liquid securities that are traded all

Â the time used by many market participants

Â to both hedge interest rate risk as well as speculate on interest rate risk.

Â A swaption is an option on a swap.

Â So what we're going to do is we're going to consider a swaption

Â on the swap of the previous slide.

Â We're going to assume that the option strike is 0%.

Â This should not be confused with the fixed rate of the underlying swap.

Â Remember the fixed rate of the underlying swap was, 5%.

Â So we're going to assume that the option strike is 0% and

Â that the swaption expiration is t = 3.

Â In other words, the option on the swap expires at t = 3.

Â Therefore a time t = 3, if you own the swaption, you have the right

Â to exercise or enter into this swap for a value or a strike value of 0.

Â So the payoff at t equals to 3 is going to be the maximum of 0 and S3.

Â You will only exercise this swaption at t equals to 3 if S3 is positive.

Â If S3 is negative, then you won't exercise the maximum would be 0 and

Â you will do nothing.

Â 6:49

So how are we going to price this swaption?

Â Well actually it's going to be very simple.

Â We're going to use our calculations from the previous slide which we see here, to

Â get the value of the swaption or order to get the value of the swap a t equals a 3.

Â We're going to replace this column of prices with the maximum of 0 on this

Â column, that will gives us a pay off of swap option at maturity, t = 3.

Â And then we will just calculate the fair value of this pay off by working backwards

Â in the lattice using risk neutral pricing.

Â 7:22

One thing to keep in mind though however, is that when we're working back from t

Â equal 3 back to 0, we no longer include the underlying cash flows of the swap.

Â And that's because the holder of the swaption

Â does not get those underlying cash flows.

Â The holder of the swaption will only get the cash flows of associated with the swap

Â if they exercise the t equals 3.

Â And then they will get this swap cash flows for

Â all time periods greater than t equals 3.

Â They will not get them for the periods before t=3.

Â 7:54

So here are some sample calculations again, for example, so

Â if you look at t equals to 3, this is the pay off of, This swaption.

Â So in fact all of these values here from t equals 4 and 5 are simply

Â the values that we see that we calculated earlier for the underlying swap.

Â So we see 0.0512 here, well that's the same 0.0512 here.

Â Down here however, is the value of the swaption at maturity.

Â And now this is equal to the maximum of 0 and S3.

Â So if you recall, the only change is actually down here.

Â Because in the underlying swap value,

Â we had a negative value minus 0.0085 at this node.

Â But now, the holder of the swaption would not choose to exercise at this node.

Â Why would they buy something for 0 when it has a negative value?

Â They won't do that.

Â So they will exercise however at all of these three nodes,

Â because these three nodes the swap has a positive value.

Â So certainly the whole of the swaption will exercise at these three nodes and

Â receive these three payoffs.

Â So what we have is here are the values,

Â the payoff values of the swaption at maturity, at t = 3.

Â It's equal to the maximum of 0 and S3.

Â And now we actually just compute the fair value of the swaption

Â by computing the value of this payoff backwards in the lattice.

Â So here what we're using in our risk-neutral pricing again,

Â if you like I won't use s here, I'll say Z.

Â So Zt = the expected value of Zt

Â + 1 divided by 1 + over t.

Â This is our risk neutral pricing for the swaption.

Â Note, I don't have any intermediate coupon or cash flow in here.

Â And that's because the swaption doesn't pay any intermediate coupons or

Â cash flows between t = 0 and t = 3.

Â So I just iterate this backwards to get the value of 0.0620 as

Â being the value of the swaption at t = 0.

Â So here's a sample calculation, 0.0908.

Â Well 0.0908 is just 1 over 1 plus, or

Â 1 over 1 is going to be 7.5% of this node times the expected value of the swaptions

Â one period head while the risk mutual probabilities are half in a half.

Â These are the values of the swaption one period ahead 0.1286 and

Â 0.0665 and so that's how I get 0.0908.

Â 10:30

You can see all of this calculations performed in the spreadsheet,

Â they're very easy to do.

Â We see again here that we have our usual short rate lattice that we're using

Â throughout.

Â So these are the details here, I won't go through them again.

Â Down here, we calculate the value of the swap, the underlying swap.

Â So there's a fixed rate of 5%, here,

Â we see the payoffs of the swap at time t equals the 5.

Â In fact these are the values of the swap,

Â because remember the payoffs occur time t equals to 6.

Â But we discount them back one period to reflect that the swap cash flows

Â occur in the rears.

Â So we do that at time t equals 5,

Â we now work backwards in the lattice using risk mutual pricing.

Â So for example down here, we see that the value

Â of the swap at this node is the payoff that's determined at that point.

Â So that's the cash flow identified up here

Â plus the expected value of the swap one period ahead.

Â And that's given to us by these terms here.

Â And of course, I have to discount both of these quantities.

Â The cash flow because of in the rears and

Â the value of the swap one period ahead by 1 plus the short rate at that node.

Â And that's how I get this expression here.

Â It's in bold again because you can actually enter the formula into the cell.

Â And then drag this formula throughout the rest of the lattice to compute the prices

Â of the swap at every node.

Â We find an initial swap value of 0.0990.

Â 11:55

To compute the fair value of the swaption,

Â well all we do is we just repeat really what we did for the swap.

Â We just calculate the value of the swaption at maturity, so

Â we see it down here.

Â In this case, it's the maximum of the swap value which is given to us here and 0.

Â So that's how we get these values down here.

Â And now we just use risk mutual pricing to work backwards on the lattice.

Â Note that any cell in this lattice, we don't have any intermediate cash flow or

Â we don't have the cash flow associated with the swap at that time.

Â And that's because the holder of the swaption does not get any of those cash

Â flows or pay any of those cash flows until such time as they exercise the swaption.

Â So we just work backwards to find the initial value of 0.0620.

Â