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In the last module, we saw how to calibrate the Black-Derman-Toy model to

Â the term structure of interest rates. We're now going to actually use a

Â calibrated Black-Derman-Toy model to price a swaption.

Â We've seen how to price swaptions before and we're going to do so again.

Â The difference now is that we're actually going to use a model, the Black-Derman-Toy

Â model, that has been calibrated to the term structure of interest rates that we

Â might see in the marketplace. We're also going to see what happens as we

Â change certain parameters and then reprice the swaption.

Â So, what we're going to do here is we're going to consider what's called a 2-8

Â payer swaption. So, this is some new terminology here,

Â the, the word payer. So, payer refers to the fact that the

Â owner of the option will pay fixed and receive floating.

Â A receiver swaption, the owner of such a swaption would receive fixed and pay

Â floating. So, what we have is a 2-8 pair swaption

Â with a fixed rate of 11.65%. It's an option to enter an 8-year swap in

Â 2 years time. The underlying swap payments are settled

Â in arrears so the payments would take place in years 3 through 10.

Â Each payment would be based on the prevailing short-rate of the previous

Â year. So, we therefore need a 10-period lattice

Â with one period corresponding to one year. Of course, in practice, you would have a

Â much larger lattice with many periods corresponding to one year.

Â But for our purposes, we will keep it simple with just a 10-period lattice and

Â one-period corresponding to one year. We're going to assume initially that bi is

Â a constant for all i and it's equal to b which is equal to 0.005.

Â If you recall in the Black-Derman-Toy model, in the BDT model, we assume that

Â rij the short rate of time i [unknown] j is equal to ai times e to the bi times j.

Â So, we're going to assume here that the bi's are constant for all i, and that it's

Â equal to 0.005. Later on, actually we'll change bi and see

Â what that does to the swaption price. So, our calibration parameters are

Â actually going to be the ai's. We're going to choose the ai's so that the

Â term structure in our Black-Derman-Toy model, matches the term structure in the

Â marketplace. And we actually saw on the last module how

Â to do this. We did it using the solver in Excel, that

Â was our lazy, easy way to do it. We could also have done it by using the

Â forward equations, as I also described in the last module.

Â So, returning to the swaption, we're going to assume a notional principle of $1 or

Â maybe one million dollars. Let S2 denote the value of the swap at

Â time t equals 2. We know how to compute S2 in any binomial

Â lattice, we just discount the cash flows back from t equals 10 to t equals 2 using

Â risk-neutral pricing. We also recall that it is easier to record

Â the time key cash flows after their predecessor nodes and then discount them

Â appropriately. So, this is why there would be no payments

Â recorded at t equal to 10 in this swaption lattice that we'll see in the Excel

Â spreadsheet in a few minute's time. Once we get back to time t equals 2, we

Â have the value of the underlying swap. So therefore, the value of the swaption

Â will be the maximum of 0 and S2. And then, once we have this at time t

Â equals 2, we just work backwards in the lattice using risk-neutral pricing in the

Â usual way to get the value of the swaption at time t equal to 0.

Â When we calibrate to the 0 coupon bonds in the marketplace, we find the swaption

Â price of $13,339 when b equals 0.005. So, in fact, in this case, I'm actually

Â assuming a notion of one million dollars. So, I should have had 1 million here.

Â When we change b, so here, b was 0.005. Now if we double the value of b to 0.01,

Â we actually find a different swaption price, we obtain a swaption price of

Â $19,497, which is actually approximately 50% higher than our original value of

Â 13,339. Now, when we actually change b, by the

Â way, to 0.1, we must remember to recalibrate our model.

Â We have to recaliberate recompute the a values so that the term structure of

Â interest rates in the BDT model matches the time structure of interest rates in

Â the market. So, we see significant difference in

Â swaption prices even though both models are either model with b equals 0.005 and

Â the model with b equals 0.01 were calibrated at the same zero coupon bond

Â prices at the same term structure of interest rates, we see very different

Â swaption prices, $13,000 versus $19,000. So, this is not surprising.

Â Swaption prices clearly depend on market volatility.

Â The more volatile your market is or your model is, the more valuable swaption

Â prices will be. And if you want to get some intuition for

Â this, you can think of the following. The payoff of the swaption at maturity, as

Â we said, is the maximum of 0 and S2. So, if you imagine increasing volatility,

Â well then, what you are doing is you're increasing your upside.

Â The more volatile the market is, the more upside you have in S2, the more spread out

Â it is. So, you're getting more of the upside as

Â you increase volatility but you're not getting more downside, a more negative

Â outcomes. And that's because the negative outcomes

Â are constrained by this value 0 here. You can never get something that's worth

Â less than 0. So, you actually like volatility because

Â you're getting better outcomes or better range of good outcomes with higher

Â volatility, but you are not getting a larger range of poor outcomes, because you

Â are constrained by the, this zero factor here.

Â So, very loosely, that's why when you own an option you like volatility and we'll

Â return to this again later in the course when we go back to the Black-Scholes model

Â and discuss options on stocks. But back to here, so what we're saying is

Â we shouldn't be surprised that the swaption price is increased as we've

Â increased b from 0.005 to 0.01. If you recall, we said before that in the

Â BDT model, log of rij is equal to log of ai plus bi times j.

Â J is our random variable because j is the state at time i, and bi is multiplying j.

Â So therefore, our b parameter is, if you like, a volatility parameter.

Â And the higher the value of b is, the more volatile the short rate is, and therefore,

Â we expect the swaption value to be higher as well.

Â And indeed, that is what we have seen here.

Â Now this, these observations have very important implications for the

Â calibration, calibration process in general.

Â We want our calibration securities to be close to the securities we want to price

Â with the calibrated model. So, what do I mean by close?

Â Well, I mean the following. Because a swaption, the, the value of a

Â swaption depends on volatility. I would like the securities that I'm using

Â to calibrate my mode, I would like their prices to also depend on volatility.

Â In this example, we have not done that. We've only calibrated the model to zero

Â coupon bond prices and they do not depend on volatility.

Â The value of a zero coupon bond price is, can be determined from the term structure

Â of interest rates we see in the marketplace today.

Â And actually the volatility of that term structure does not enter into the pricing

Â of these zero coupon bond prices. And so, I would say that zero coupon bonds

Â are not really close to swaptions. I would think securities like caplets and

Â flowlets are actually much closer to swaptions.

Â And that if I was to calibrate a model to caplets and flowlets, then maybe I would

Â actually get a more accurate price for my swaptions.

Â We will discuss this further in the next module.

Â So, here is our Black-Derman-Toy model again.

Â We saw in the last module how to calibrate the Black-Derman-Toy model using Excel

Â Solver. We've already gone ahead and done that

Â here, so these are our market spot rates. We can imagine having observed these in

Â the marketplace today, of course, these rates are much, much higher than you would

Â see in the financial markets today, but that's fine.

Â We can just assume that these are the actual rates that do exist today.

Â We assumed and fixed a value of B equal to 0.005.

Â So, I made the comment here that fixing the volatility parameter b, this is not a

Â good idea if we wish to use the model to price fixed income derivatives that are

Â sensitive to volatility. So, you can see that our objective

Â function is very small. We've already done the calibration at this

Â point, we see that the model spot rates actually match perfectly the market spot

Â rates that we see up here. And we actually get a pair of swaption

Â price of 0.00134. That's for a notional of $1, so if we

Â multiply it by a million dollars, we get about, a note, we get a price of a

Â swaption of about $13,000. So, we begin here at t equals 9 with the

Â value of the swap. Actually, the last payment for the swap

Â takes place at t equals the 10, but we record this t equals 9 so to be

Â discounted. We work backwards until t equals 2 and we

Â compute the value of the swaption. So, this value is equal to the maximum of

Â zero and the underlying swap. Once we obtain that maximum, we work

Â backwards in the usual way to get the price of the swaption.

Â So, what's interesting here is I've highlighted the value of the swaption is

Â $13,400 for a notion of 1 million and that's the value when we fix b to be

Â 0.005. Over here, on this worksheet, I fixed b to

Â be 0.01 so I've doubled it. Again, I've already performed the

Â calibration so that you will see that the term structure of interest rates or if you

Â like, zero coupon bond prices and the model, match those in the marketplace.

Â And yet, now, I see a swaption price of $19,600.

Â So, what I'm seeing is I, I can say that I've calibrated my model, my

Â Black-Derman-Toy model to the marketplace but I've only calibrated it to the zero

Â coupon bond prices in the marketplace. If I want to use my model to price

Â swaptions, then I need to do a better calibration.

Â I need to include more instruments in my calibration.

Â I need to include instruments whose price depends on volatility in my calibration.

Â So, the upshot of all of these is the following.

Â If I use my calibrated model to price exotic, fixed income derivative

Â securities, for example, a swaption, now in the real world, a swaption is not

Â exotic, it's actually a very liquidly traded security.

Â But for the purpose of this example, I can pretend it's an exotic security.

Â So, if I use my model to price an exotic security like a swaption, I need to be

Â aware of the fact that the price I get, very much depends on what securities I use

Â to calibrate the model in the first place. I use zero coupon bond prices and I fixed

Â certain parameters, i.e., b. If I actually use a different set of

Â securities, maybe caplet prices o flowlet prices to calibrate the model, then

Â presumably, I would get a different set of calibrated parameters and therefore, get a

Â different swaption price. Users of these models, therefore need to

Â be very careful about how they were built and how they were calibrated.

Â