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FIGURE 4.12 Use of spot and options to create a callable bond.


1. It may simply decide to keep them as outright investments.
2. It may decide to sell them. That is, it may decide to take a pool of home
mortgages and sell them into the open market as tradable fixed income
securities. When this is done, the organization that purchased the mort-
gages is transferring the embedded short options to other investors who
purchase the home mortgages.
3. It may decide to keep the mortgages, but on a hedged basis. One way
they could hedge the mortgages would be to issue callable bonds (a bond
with a short call option).

How would issuing callable bonds help serve as a hedge against home
mortgages? Recall that the creditor of a home mortgage (a bank, a mort-
gage company, or whatever) is holding a product that has a short call option
embedded in it. It is a short call option because it is the homebuyer who has
the right to make the choice of whether or not to refinance the mortgage
when interest rates decline; the homebuyer is long the call option. A callable
debenture (bond) consists of a bond with an embedded short call option.
Anyone who purchases a callable bond subjects him- or herself to someone
else deciding when and if the embedded option will be exercised. That
“someone else” is the issuer of the callable bond, or in our story, Fannie Mae
or Freddie Mac. Fannie Mae and Freddie Mac can attempt to hedge some
of the short call risk embedded in their holdings of mortgage product by issu-
ing some callable bonds against it.
Figure 4.13 borrows from the pictorial descriptions in Chapter 2 to pre-
sent a callable bond.

If discrete, bond is callable
only at payment of 18-month
Callability period

If continuous, bond is callable
p4 p5
Cash Flow anytime after 12-month
+ Lockout period
The p™s represent probability
values that are assigned to
p1 p2 p3 p5 each cash flow after purchase.


FIGURE 4.13 Conceptual presentation of a callable bond.

Financial Engineering

The callable shown in our diagram has a final maturity date two years
from now and is callable one year from now. To say that it is callable one
year from now is to say that for its first year it may not be called at all; it
is protected from being called, and as such investors may be reasonably
assured that they will receive promised cash flows on a full and timely basis.
But once we cross into year 2 and the debenture is subject to being called
by the issuer who is long the call option, there is uncertainty as to whether
all the promised cash flows will be paid. This uncertainty stems not from
any credit risk (particularly since mortgage securities tend to be collateral-
ized), but rather from market risk; namely, will interest rates decline such
that the call in the callable is exercised? If the call is exercised, the investors
will receive par plus any accrued interest that is owed, and no other cash
flows will be paid. Note that terms and conditions for how a call decision
is made can vary from security to security. Some callables are discrete, mean-
ing that the issue could be called only (if at all) at coupon payment dates;
for continuous callables, the issue could be called (if at all) at any time once
it has lost its callability protection.
Parenthetically, a two-year final maturity callable eligible to be called
after one year is called a two-noncall-one. A 10-year final maturity callable
that is eligible to be called after three years is called a 10-noncall-three, and
so forth. Further, the period of time when a callable may not be called is
referred to as the lockout period.
Figure 4.13 distinguishes between the cash flows during and after the period
of call protection with solid and dashed lines, respectively. At the time a callable
comes to market, there is truly a 50/50 chance of its being called. That is because
it will come to market at today™s prevailing yield level for a bond with an embed-
ded call, and from a purely theoretical view, there is an equal likelihood for
future yield levels to go higher or lower. Investors may believe that probabili-
ties are, say, 80/20 or 30/70 for higher or lower rates, but options pricing the-
ory is going to set the odds objectively at precisely 50/50.
Accordingly, to calculate a price for our callable at the time of issuance
(where we know its price will be par), if we probability weight each cash
flow that we are confident of receiving (due to call protection over the lock-
out period) at 100 percent, and probability weight the remaining uncertain
(unprotected) cash flows at 50 percent, we would arrive at a price of par.
This means p1 p2 100% and p3 p4 p5 50%. In doing this calculation
we assume we have a discrete-call security, and since both principal and
coupon are paid if the security is called, we adjust both of these cash flows
at 50 percent at both the 18- and 24-month nodes. If the discrete callable is
not called at the 18-month node, then the probability becomes 100 percent
that it will trade to its final maturity date at the 24-month node, but at the
start of the game (when the callable first comes to market), we can say only
that there is a 50/50 chance of its surviving to 24 months.


Incremental yield is added when an investor purchases a callable,
because she is forfeiting the choice of exercise to the issuer of the callable.
If choice has value (and it does), then relinquishing choice ought to be rec-
ompensed (and it is). We denote the incremental yield from optionality as
Is, the incremental yield from credit risk as Ic, and the overall yield of a
callable bond with credit risk as

Y Yield of a comparable-maturity Treasury Ic Is.

Next we present the same bond price formula from Chapter 2 but with
one slight change. Namely, we have added a small p next to every cash flow,
actual and potential. As stated, the p represents probability.

C p1 C p2
11 11
Price 1
Y>22 2
1C & F2 p5
C p3&F p4
11 11
Y>22 3 Y>22 4

p1 probability of receiving first coupon
p2 probability of receiving second coupon
p3 probability of receiving third coupon
p4 probability of receiving principal at 18 months
p5 probability of receiving fourth coupon and principal at 24 months

Let™s now price the callable under three assumed scenarios:

1. The callable is not called and survives to its maturity date:

p1 p2 p3 p4 p5 100%.

2. The callable is discrete and is called at 18 months:

p1 p2 p3 p4 100%.

3. The callable is discrete and may or may not be called at 18 months:

p1 p2 100%, and p3 p4 p5 50%.

Assuming Y C 6%, what is the price under each of these three sce-
narios? “Par” is correct. At the start of a callable bond™s life, Y C (as with
a noncallable bond), and it is a 50/50 proposition as to whether the callable

Financial Engineering

will in fact be called. Accordingly, any way we might choose to assign rele-
vant probability weightings, price will come back as par, at least until time
passes and Y is no longer equal to C.
Another way to express the price of a callable is as follows:

Pc Pb Oc,

Pc price of the callable
Pb price of a noncallable bond (bullet bond)
Oc call option

By expressing the price of a callable bond this way, two things become
clear. First, we know from Chapter 2 that if price goes down then yield goes
up, and the Oc means that the yield of a callable must be higher than a
noncallable (Pb). Accordingly, Y and C for a callable are greater than for a
noncallable. Second, it is clear that a callable comprises both a spot via Pb
and an option (and, therefore, a forward) via Oc.
As demonstrated in Chapter 2, when calculating a bond™s present value,
the same single present yield is used to discount every one of its cash flows.
Again, this allows for a quick and reasonably accurate way to calculate a
bond™s spot price. When calculating a bond™s forward value in yield terms (as
opposed to price terms), a separate and unique yield typically is required for
every one of the cash flows. Each successive forward yield incorporates a chain
of previous yields within its calculation. When these forward yields are plot-
ted against time, they collectively comprise a forward yield curve, and this
curve can be used to price both the bond and option components of a bond
with embedded options. By bringing the spot component of the bond into the
context of forwards and options, a new perspective of value can be provided.
In particular, with the use of forward yields, we can calculate an option-
adjusted spread (or OAS). Figure 4.14 uses the familiar triangle to highlight
differences and similarities among three different measures of yield spread:
nominal spreads, forward spreads, and option-adjusted spreads.
In our story we said that a second possibility was available to Fannie
Mae and Freddie Mac regarding what they might do with the mortgages they
purchased: Sell them to someone else. They might sell them in whole loan
(an original mortgage loan as opposed to a participation with one or more
lenders) form, or they could choose to repackage them in some way. One
simple way they can be repackaged is by pooling together some of the mort-
gages into a single “portfolio” of mortgages that could be traded in the mar-
ketplace as a bundle of product packaged into a single security. This bundle
would share some pricing features of a callable security. Callable bonds, like
mortgages, embody a call option that is a short call option to the investor
in these securities. Again, it is the homeowner who is long the call option.


• Spread between a benchmark bond™s
spot yield and a (non)benchmark
• The difference in yield between a
bond™s forward yield.
benchmark bond and a
• Spread is expressed in basis points.
nonbenchmark bond.
• When the spot curve is flat, the
• Spread is expressed in basis points.
forward curve and spot curve are
• The two bonds have comparable
equal to one another, and a
maturity dates.
nominal spread is equal to a
forward spread.
Nominal Forward

Option adjusted

• Spread between a benchmark bond™s forward yield (typically without
optionality) and a (non)benchmark bond™s forward yield (typically
with optionality).
• Spread is expressed in basis points.
• When an OAS is calculated for a bond without optionality, and when
the forward curve is of the same credit quality as the bond, the
bond™s OAS is equal to its forward spread.
When an OAS is calculated for a bond with optionality, the
bond™s OAS is equal to its forward spread if volatility is zero.
This particular type of OAS is also called a ZV spread (for zero
• When an OAS is calculated for a bond with optionality, if the
spot curve is flat, then the bond™s OAS is equal to its forward
spread as well as its nominal spread if volatility is zero.


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