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WHEN STANDARD DEVIATION IS ZERO
What happens when a standard deviation is zero in the context of the Black-
Scholes model? Starting with the standard Black-Scholes option pricing for-
mula for a call option, we have

s1t2
t
C SN1X2 Kr N1X

log 1S>Kr t 2 1
s 1t.
where X
s1t 2
If there were absolutely no uncertainty related to the future value of an
asset, then we have


1log1S>Kr t 2 2
SN a
1
1tb
C
1t 2




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71
Cash Flows



log1S>Kr t 2
Na
1
1tb 1t
t
Kr
1t 2

log1S>Kr t 2 log1S>Kr t 2
SN a b Na b.
t
Kr


Since anything divided by zero is zero, we have

SN 1 2 Kr tN1 2.
C

And since N(˜) simply means that the role of the normal distribution
function has no meaningful influence on the value of S and K, we now have

Kr t.
C S

Note that S Kr t is equivalent to F K.
Thus, in the extreme case where there is zero market volatility (or, equiv-
alently, where the future value of the underlying asset is known with cer-
tainty), the value of the call is driven primarily by the underlying asset™s
forward price. Specifically, it is the maximum of zero or the difference
between the forward price and the strike price.
Again, rewriting C S Kr t, the purpose of r t is nothing more than
to adjust K (the strike price) to a present value. An equivalent statement
would be C Sr t K, where Srt is the forward price of the underlying asset
(or simply F). The strike price, K, is a constant (our marker to determine
whether the option has intrinsic value), so when we let equal zero, the value
of the option boils down to the relationship between the value of the for-
ward and the strike price, or the maximum value between zero or F K
(sometimes expressed as C Max (˜, F K).
And if we continue this story and let both ˜ and t ˜, we have

Srt
C K,
or
Sr K.

A variable raised to the power of ˜ is equal to 1, so

C S 1 K

S K.




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72 PRODUCTS, CASH FLOWS, AND CREDIT



In the extreme case where there is zero market volatility and no time
value (or, equivalently, we want today™s value of the underlying asset), then
the value of the call is driven primarily by the underlying asset™s spot price.
Specifically, it is the maximum of zero or the difference between the spot
price and the strike price. Figure A2.1 places these relationships in the con-
text of our triangle.
In summary, the Achilles™ heel of an option is volatility; without it, an
option becomes a forward, and without volatility and time, an option
becomes spot.




Spot Forwards
S F



Options

C = SN(X) “ Kr “ tN(X“σ t )


With σ equal to zero we have
SN log(S/Kr t) Kr t N log(S / Kr t )
With both σ = … and t =…,
C = Sr t … …
K
… t N(…)
= Sr K = SN (…) Kr
= S Kr t
=S K
=F K

FIGURE A2.1 Applying Black-Scholes to the interrelated values of spot, forwards, and
options.




TLFeBOOK
3
CHAPTER

Credit

Products Cash flows


Issuers




This chapter builds on the concepts presented in Chapters 1 and 2. Their
importance is accented by their inclusion in the credit triangle. Simply put,
credit considerations might be thought of as embodying the likelihood of
issuers making good on the financial commitments (implied and explicit) that
they have made. The less confident we are that an entity will be able to make
good on its commitments, the more of a premium we are likely to require
to compensate us for the added risk we are being asked to bear.




Issuers




There are hundreds and upon thousands of issuers (entities that raise funds
by selling their debt or equity into the marketplace), and each with its own
unique credit risk profile. To analyze these various credit risks, larger
investors (e.g., large-scale fund managers) often have the benefit of an in-
house credit research department. Smaller investors (as with individuals) may
have to rely on what they can read in the financial press or pick up from
the Internet or personal contacts. But even for larger investors, the task of



73

TLFeBOOK
74 PRODUCTS, CASH FLOWS, AND CREDIT



following the credit risk of so many issuers can be daunting. Thankfully, rat-
ing agencies (organizations that sell company-specific research) exist to pro-
vide a report card of sorts on many types of issuers around the globe. The
most creditworthy of issuers carries a rating (a formally assigned opinion of
a company or entity) of triple A, while at the lower end of the so-called
investment grade ratings a security is labeled as BBB or Baa3. An issuer
with a rating below C or C1 is said to be in default.
Table 3.1 lists the various rating classifications provided by major rat-
ing agencies. Since it is difficult for one research analyst (or even a team of
analysts) to stay apprised of all the credit stories in the marketplace at any
time, analysts subscribe to the services of one or more of the rating agen-
cies to assess an issuer™s situation and outlook.
Because the rating agencies have been around for a while, databases have
evolved with a wealth of historical data on drift and default experiences.


TABLE 3.1 Credit Ratings across Rating Agencies
Moody™s S&P Fitch D&P

Aaa AAA AAA AAA Highest quality
Aa1 AA+ AA+ AA+
Aa2 AA AA AA High quality
Aa3 AA AA AA
A1 A+ A+ A+
A2 A A A Upper-medium quality
A3 A A A
Baa1 BBB+ BBB+ BBB+
Baa2 BBB BBB BBB Lower-medium quality
Baa3 BBB BBB BBB
Ba1 BB+ BB+ BB+
Ba2 BB BB BB Low quality
Ba3 BB BB BB
B1 B+ B+ B+
B2 B B B Highly speculative
B3 B B B
CCC+
Caa CCC CCC CCC Substantial risk
CCC
Ca CC CC
C C C Extremely speculative
C1
DDD Default
DD
D D




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75
Credit



“Drift” means an entity™s drifting from one rating classification to another
” from an original credit rating of, say, single A down to a double B.
“Default” simply means an entity™s going from a nondefault rating into a
default rating. Indeed, the rating agencies regularly generate probability dis-
tributions to allow investors to answer questions such as: What is the like-
lihood that based on historical experience a credit that is rated single A today
will be downgraded to a single B or upgraded to a double A? In this way
investors can begin to attempt to numerically quantify what credit risk is all
about. For example, so-called credit derivatives are instruments that may be
used to create or hedge an exposure to a given risk of upgrade or down-
grade, and the drift and default tables are often used to value these types of
products. Further, entities sell credit rating insurance to issuers, whereby a
bond can be marketed as a triple-A risk instead of a single-A risk because
the debenture comes with third-party protection against the risk of becom-
ing a weaker security. Typically insurers insist on the issuer taking certain
measures in exchange for the insurance, and these are discussed later in the
chapter under the heading of “Credit: Cash Flows.”


THE ELUSIVE NATURE OF CREDIT RISK
Despite whatever comfort we might have with better quantifying credit risks,
we must guard against any complacency that might accompany these quan-
titative advances because in many respects the world of credit risk is a world
of stories. That is, as much as we might attempt to quantify such a phe-
nomenon as the likelihood of an upgrade or downgrade, there are any num-
ber of imponderables with a given issuer that can turn a bad situation into
a favorable one or a favorable one into a disaster. Economic cycles, global
competitive forces, regulatory dynamics, the unique makeup and style of an
issuer™s management team, and the potential to take over or be taken over
” all of these considerations and others can combine to frustrate even the
most thorough analysis of an issuer™s financial statements. Credit risk is the
third and last point on the risk triangle because of its elusive nature to be
completely quantified.
What happens when a security is downgraded or upgraded by a rating
agency? If it is downgraded, this new piece of adverse information must be
reflected somehow in the security™s value. Sometimes a security is not imme-

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