In the last module, we saw how to calibrate the Black-Derman-Toy model to
the term structure of interest rates. We're now going to actually use a
calibrated Black-Derman-Toy model to price a swaption.
We've seen how to price swaptions before and we're going to do so again.
The difference now is that we're actually going to use a model, the Black-Derman-Toy
model, that has been calibrated to the term structure of interest rates that we
might see in the marketplace. We're also going to see what happens as we
change certain parameters and then reprice the swaption.
So, what we're going to do here is we're going to consider what's called a 2-8
payer swaption. So, this is some new terminology here,
the, the word payer. So, payer refers to the fact that the
owner of the option will pay fixed and receive floating.
A receiver swaption, the owner of such a swaption would receive fixed and pay
floating. So, what we have is a 2-8 pair swaption
with a fixed rate of 11.65%. It's an option to enter an 8-year swap in
2 years time. The underlying swap payments are settled
in arrears so the payments would take place in years 3 through 10.
Each payment would be based on the prevailing short-rate of the previous
year. So, we therefore need a 10-period lattice
with one period corresponding to one year. Of course, in practice, you would have a
much larger lattice with many periods corresponding to one year.
But for our purposes, we will keep it simple with just a 10-period lattice and
one-period corresponding to one year. We're going to assume initially that bi is
a constant for all i and it's equal to b which is equal to 0.005.
If you recall in the Black-Derman-Toy model, in the BDT model, we assume that
rij the short rate of time i [unknown] j is equal to ai times e to the bi times j.
So, we're going to assume here that the bi's are constant for all i, and that it's
equal to 0.005. Later on, actually we'll change bi and see
what that does to the swaption price. So, our calibration parameters are
actually going to be the ai's. We're going to choose the ai's so that the
term structure in our Black-Derman-Toy model, matches the term structure in the
marketplace. And we actually saw on the last module how
to do this. We did it using the solver in Excel, that
was our lazy, easy way to do it. We could also have done it by using the
forward equations, as I also described in the last module.
So, returning to the swaption, we're going to assume a notional principle of $1 or
maybe one million dollars. Let S2 denote the value of the swap at
time t equals 2. We know how to compute S2 in any binomial
lattice, we just discount the cash flows back from t equals 10 to t equals 2 using
risk-neutral pricing. We also recall that it is easier to record
the time key cash flows after their predecessor nodes and then discount them
appropriately. So, this is why there would be no payments
recorded at t equal to 10 in this swaption lattice that we'll see in the Excel
spreadsheet in a few minute's time. Once we get back to time t equals 2, we
have the value of the underlying swap. So therefore, the value of the swaption
will be the maximum of 0 and S2. And then, once we have this at time t
equals 2, we just work backwards in the lattice using risk-neutral pricing in the
usual way to get the value of the swaption at time t equal to 0.
When we calibrate to the 0 coupon bonds in the marketplace, we find the swaption
price of $13,339 when b equals 0.005. So, in fact, in this case, I'm actually
assuming a notion of one million dollars. So, I should have had 1 million here.
When we change b, so here, b was 0.005. Now if we double the value of b to 0.01,
we actually find a different swaption price, we obtain a swaption price of
$19,497, which is actually approximately 50% higher than our original value of
13,339. Now, when we actually change b, by the
way, to 0.1, we must remember to recalibrate our model.
We have to recaliberate recompute the a values so that the term structure of
interest rates in the BDT model matches the time structure of interest rates in
the market. So, we see significant difference in
swaption prices even though both models are either model with b equals 0.005 and
the model with b equals 0.01 were calibrated at the same zero coupon bond
prices at the same term structure of interest rates, we see very different
swaption prices, $13,000 versus $19,000. So, this is not surprising.
Swaption prices clearly depend on market volatility.
The more volatile your market is or your model is, the more valuable swaption
prices will be. And if you want to get some intuition for
this, you can think of the following. The payoff of the swaption at maturity, as
we said, is the maximum of 0 and S2. So, if you imagine increasing volatility,
well then, what you are doing is you're increasing your upside.
The more volatile the market is, the more upside you have in S2, the more spread out
it is. So, you're getting more of the upside as
you increase volatility but you're not getting more downside, a more negative
outcomes. And that's because the negative outcomes
are constrained by this value 0 here. You can never get something that's worth
less than 0. So, you actually like volatility because
you're getting better outcomes or better range of good outcomes with higher
volatility, but you are not getting a larger range of poor outcomes, because you
are constrained by the, this zero factor here.
So, very loosely, that's why when you own an option you like volatility and we'll
return to this again later in the course when we go back to the Black-Scholes model
and discuss options on stocks. But back to here, so what we're saying is
we shouldn't be surprised that the swaption price is increased as we've
increased b from 0.005 to 0.01. If you recall, we said before that in the
BDT model, log of rij is equal to log of ai plus bi times j.
J is our random variable because j is the state at time i, and bi is multiplying j.
So therefore, our b parameter is, if you like, a volatility parameter.
And the higher the value of b is, the more volatile the short rate is, and therefore,
we expect the swaption value to be higher as well.
And indeed, that is what we have seen here.
Now this, these observations have very important implications for the
calibration, calibration process in general.
We want our calibration securities to be close to the securities we want to price
with the calibrated model. So, what do I mean by close?
Well, I mean the following. Because a swaption, the, the value of a
swaption depends on volatility. I would like the securities that I'm using
to calibrate my mode, I would like their prices to also depend on volatility.
In this example, we have not done that. We've only calibrated the model to zero
coupon bond prices and they do not depend on volatility.
The value of a zero coupon bond price is, can be determined from the term structure
of interest rates we see in the marketplace today.
And actually the volatility of that term structure does not enter into the pricing
of these zero coupon bond prices. And so, I would say that zero coupon bonds
are not really close to swaptions. I would think securities like caplets and
flowlets are actually much closer to swaptions.
And that if I was to calibrate a model to caplets and flowlets, then maybe I would
actually get a more accurate price for my swaptions.
We will discuss this further in the next module.
So, here is our Black-Derman-Toy model again.
We saw in the last module how to calibrate the Black-Derman-Toy model using Excel
Solver. We've already gone ahead and done that
here, so these are our market spot rates. We can imagine having observed these in
the marketplace today, of course, these rates are much, much higher than you would
see in the financial markets today, but that's fine.
We can just assume that these are the actual rates that do exist today.
We assumed and fixed a value of B equal to 0.005.
So, I made the comment here that fixing the volatility parameter b, this is not a
good idea if we wish to use the model to price fixed income derivatives that are
sensitive to volatility. So, you can see that our objective
function is very small. We've already done the calibration at this
point, we see that the model spot rates actually match perfectly the market spot
rates that we see up here. And we actually get a pair of swaption
price of 0.00134. That's for a notional of $1, so if we
multiply it by a million dollars, we get about, a note, we get a price of a
swaption of about $13,000. So, we begin here at t equals 9 with the
value of the swap. Actually, the last payment for the swap
takes place at t equals the 10, but we record this t equals 9 so to be
discounted. We work backwards until t equals 2 and we
compute the value of the swaption. So, this value is equal to the maximum of
zero and the underlying swap. Once we obtain that maximum, we work
backwards in the usual way to get the price of the swaption.
So, what's interesting here is I've highlighted the value of the swaption is
$13,400 for a notion of 1 million and that's the value when we fix b to be
0.005. Over here, on this worksheet, I fixed b to
be 0.01 so I've doubled it. Again, I've already performed the
calibration so that you will see that the term structure of interest rates or if you
like, zero coupon bond prices and the model, match those in the marketplace.
And yet, now, I see a swaption price of $19,600.
So, what I'm seeing is I, I can say that I've calibrated my model, my
Black-Derman-Toy model to the marketplace but I've only calibrated it to the zero
coupon bond prices in the marketplace. If I want to use my model to price
swaptions, then I need to do a better calibration.
I need to include more instruments in my calibration.
I need to include instruments whose price depends on volatility in my calibration.
So, the upshot of all of these is the following.
If I use my calibrated model to price exotic, fixed income derivative
securities, for example, a swaption, now in the real world, a swaption is not
exotic, it's actually a very liquidly traded security.
But for the purpose of this example, I can pretend it's an exotic security.
So, if I use my model to price an exotic security like a swaption, I need to be
aware of the fact that the price I get, very much depends on what securities I use
to calibrate the model in the first place. I use zero coupon bond prices and I fixed
certain parameters, i.e., b. If I actually use a different set of
securities, maybe caplet prices o flowlet prices to calibrate the model, then
presumably, I would get a different set of calibrated parameters and therefore, get a
different swaption price. Users of these models, therefore need to
be very careful about how they were built and how they were calibrated.
