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zero? In this instance, the last equation shrinks to

Call value = S + SRT K.

And since we know that F S SRT, we can rewrite that equation
into an even simpler form as:

Call value = F K.

But since K is a fixed value that does not change from the time the option
is first purchased, what the above expression really boils down into is a value
for F. We are now back to the second leg of the triangle. To put this another
way, a key difference between a forward and an option is that prior to expi-
ration, the option requires a price value for V.
For our second what-if? scenario, let us assume that in addition to
volatility being zero, for whatever reason there is also zero cost to borrow
or lend (financing rates are zero). In this instance, call value S SRT
K V now shrinks to

Call value = S K.

This is because with T and R equal to zero, the entire SRT term drops
out, and of course V drops out because it is zero as well. With the recogni-
tion, once again, that K is a fixed value and does not do very much except
provide us with a reference point relative to S, we now find ourselves back
to the first leg of the triangle. Figure 2.18 presents these interrelationships
graphically.
As another way to evaluate the progressive differences among spot, for-
wards, and options, consider the layering approach shown in Figure 2.19.
The first or bottom layer is spot. If we then add a second layer called cost
of carry, the combination of the first and second layers is a forward. And if
we add a third layer called volatility (with strike price included, though “on
the side,” since it is a constant), the combination of the first, second, and
third layers is an option.




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56 PRODUCTS, CASH FLOWS, AND CREDIT



When either R or T is zero (as
with a zero cost to financing,
or when there is immediate
Forwards
Spot settlement), F = S.
Therefore, F is differentiated
from S by cost of carry (SRT)
F = S(1 + RT)
S




Call value = F “ K + V
= S(1 + RT) “ K + V

Options
When T is zero (as at the expiration of an
option), the call option value becomes
S “ K. This happens because F becomes S Special Note
(see formula for F) and V drops away; Some market participants state
volatility has no value for a security that has that the value of an option is
ceased to trade (as at expiration). In sum, really composed of two parts:
since K is a constant, S is the last remaining an intrinsic value and a time
variable. If just V is zero, then the call option value. Intrinsic value is defined
value prior to expiration is F “ K. as F K prior to expiration (for
a call option) and as S K at
Therefore, F is differentiated from an option expiration; all else is time value,
by K and V, and S is differentiated from an which, by definition, is zero
option by K, V, and RT. when T = 0 (as at expiration).


FIGURE 2.18 Key interrelationships among spot, forwards, and options.



V Volatility

Options
SRT Cost of carry
Forwards

S Spot


FIGURE 2.19 Layers of distinguishing characteristics among spot, forwards, and options.



As part and parcel of the building-block approach to spot, forwards, and
options, unless there is some unique consideration to be made, the pre-
sumption is that with an efficient marketplace, investors presumably would
be indifferent across these three structures relative to a particular underly-
ing security. In the context of spot versus forwards and futures, the decision
to invest in forwards and futures rather than cash would perhaps be influ-
enced by four things:



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57
Cash Flows



1. The notion that the forward or future is undervalued or overvalued rela-
tive to cash; that in the eyes of a particular investor, there is a material dif-
ference between the market value of the forward and its actual worth
2. Some kind of investor-specific cash flow or asset consideration where
immediate funds are not desired to be committed; that the deferred exchange
of cash for product provided by the forward or future is desirable
3. The view that something related to SRT is not being priced by the mar-
ket in a way that is consistent with the investor™s view of worth; again,
a material difference between market value and actual worth
4. Some kind of institutional, regulatory, tax, or other extra-market incen-
tive to trade in futures or forwards instead of cash

In the case of investing in an option rather than forwards and futures
or cash, this decision would perhaps be influenced by four things:

1. The notion that the option is undervalued or overvalued relative to for-
wards or futures or cash; that in the eyes of a particular investor, there
is a material difference between the market value of the forward and its
actual worth
2. Some kind of investor-specific cash flow or asset consideration where
the cash outlay of a strategy is desirable; note the difference between
paying S versus S K
3. The view that something related to V is not being priced by the market
in a way that is consistent with the investor™s view of worth; again, a
material difference between market value and actual worth
4. Some kind of institutional, regulatory, tax, or other extra-market incen-
tive to trade in options instead of futures or forwards or cash

It is hoped that these illustrations have helped to reinforce the idea of inter-
locking relationships around the cash flow triangle. Often people believe that
these different cash flow types somehow trade within their own unique orbits
and have lives unto themselves. This does not have to be the case at all.
As the concept of volatility is very important for option valuation, the
appendix to this chapter is devoted to the various ways volatility is calcu-
lated. In fact, a principal driver of why various option valuation models exist
is the objective of wanting to capture the dynamics of volatility in the best
possible way. Differences among the various options models that exist today
are found in existing texts on the subject.15



15
See, for example, John C. Hull, Options, Futures, and Other Derivatives (Saddle-
River. NJ: Prentice Hall, 1989).




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58 PRODUCTS, CASH FLOWS, AND CREDIT




Options
Bonds



Because bonds are priced both in terms of dollar price and yield, an overview
of various yield types is appropriate. Just as there are nominal yield spreads and
forward yield spreads, there are also option-adjusted spreads (OASs).
Refer again to the cash flow triangle and the notion of forwards building
on spots, and options, in turn, building on forwards. Recall that a spot spread
is defined as being the difference (in basis points) between two spot yield lev-
els (and being equivalent to a nominal yield spread when the spot curve is a
par bond curve) and that a forward spread is the difference (in bps) between
two forward yield levels derived from the entire relevant portion of respective
spot curves (and where the forward curve is equivalent to a spot curve when
the spot curve is flat). A nominal spread typically reflects a measure of one
security™s richness or cheapness relative to another. Thus, it can be of interest
to investors as a way of comparing one security against another. Similarly, a
forward spread also can be used by investors to compare two securities, par-
ticularly when it would be of interest to incorporate the information contained
within a more complete yield curve (as a forward yield in fact does).
An OAS can be a helpful valuation tool for investors when a security
has optionlike features. Chapter 4 examines such security types in detail.
Here the objective is to introduce an OAS and show how it can be of assis-
tance as a valuation tool for fixed income investors.
If a bond has an option embedded within it, a single security has charac-
teristics of a spot, a forward, and an option all at the same time. We would
expect to pay par for a coupon-bearing bond with an option embedded within
it if it is purchased at time of issue; this “pay-in-full at trade date” feature is
most certainly characteristic of spot. Yet the forward element of the bond is
a “deferred” feature that is characteristic of options. In short, an OAS is
intended to incorporate an explicit consideration of the option component
within a bond (if it has such a component) and to express this as a yield spread
value. The spread is expressed in basis points, as with all types of yield spreads.
Recall the formula for calculating a call option™s value for a bond, equity,
or currency.

Oc F K V.




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Cash Flows



Table 2.6 compares and contrasts how the formula would be modified
for calculating an OAS as opposed to a call option on a bond.
Consistent with earlier discussions on the interrelationships among spot, for-
wards, and options, if the value of volatility is zero (or if the par bond curve is
flat), then an OAS is the same thing as a forward spread. This is the case because
a zero volatility value is tantamount to asserting that just one forward curve is
of relevance: today™s forward curve. Readers who are familiar with the binom-
inal option model™s “tree” can think of the tree collapsing into a single branch
when the volatility value is zero; the single branch represents the single prevailing
path from today™s spot value to some later forward value. Sometimes investors
deliberately calculate a zero volatility spread (or ZV spread) to see where a given
security sits in relation to its nominal spread, whether the particular security is
embedded with any optionality or not. Simply put, a ZV spread is an OAS cal-
culated with the assumption of volatility being equal to zero. Similarly, if T
0 (i.e., there is immediate settlement), then volatility has no purpose, and the
OAS and forward spread are both equal to the nominal spread.
An OAS can be calculated for a Treasury bond where the Treasury bond
is also the benchmark security. For Treasuries with no optionality, calculating
an OAS is the same as calculating a ZV spread. For Treasuries with option-
ality, a true OAS is generated. To calculate an OAS for a non-Treasury secu-
rity (i.e., a security that is not regarded as risk free in a credit or liquidity
context), we have a choice; we can use a Treasury par bond curve as our ref-
erence curve for constructing a forward curve, or we can use a par bond curve
of the non-Treasury security of interest. Simply put, if we use a Treasury par
bond curve, the resulting OAS will embody measures of both the risk-free and
non“risk-free components of the future shape in the forward curve as well as
a measure of the embedded option™s value. Again, the term “risk free” refers
to considerations of credit risk and liquidity risk.

TABLE 2.6
Using Oc F K V to Calculate a Call Option on a Bond versus an OAS
(assuming the embedded option is a call option)

For a Bond For an OAS
• Oc is expressed as a dollar value OAS is expressed in basis points.
(or some other currency value).
• F is a forward price value. F is a forward yield value (which, via
bootstrapping, embodies a forward curve).
• K is a spot price reference value. K is expressed as a spot yield value
(typically equal to the coupon of the bond).
• V is the volatility price value. Same.




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60 PRODUCTS, CASH FLOWS, AND CREDIT



Conversely, if we use a non-Treasury par bond curve, the resulting OAS
embodies a measure of the non“risk-free component of the forward curve™s
future shape as well as a measure of the embedded option™s value. A fixed
income investor might very well desire both measures, and with the intent
of regularly following the unique information contained within each to
divine insight into the market™s evolution and possibilities. For example, an
investor might look at the historical ratio of the pure OAS embedded in a
Treasury instrument in relation to the OAS of a non-Treasury bond and cal-
culated with a non-Treasury par bond curve.
One very clear incentive for using a non-Treasury spot curve when gen-

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