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.

TLFeBOOK

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:

TLFeBOOK

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).

TLFeBOOK

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.

TLFeBOOK

59

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.

TLFeBOOK

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-