areas within finance that is more of the art than the math. Over the longer run, historical and implied volatility series tend to do a pretty good job of moving with a fairly tight correlation. This is to be expected. Yet often what are of most relevance for someone actively trading options are the very short- term opportunities where speed and precision are paramount, and where implied volatility might be most appropriate.
Many investors are biased to using those inputs that are most relevant for a scenario whereby they would have to engineer (or reverse-engineer) a product in the marketplace. For example, if attempting to value a callable bond (which is composed of a bullet bond and a short call option), the incli- nation would be to price the call at a level of volatility consistent with where an investor actually would have to go to the market and buy a call with the relevant features required. This true market price would then be used to get an idea of where the callable would trade as a synthetic bullet instrument having stripped out the short call with a long one, and the investor then could compare this new value to an actual bullet security trading in the market.
In the end, the investor might not actually synthetically create these prod- ucts in the market if only because of the extra time and effort required to do so (unless, of course, doing so offered especially attractive arbitrage opportunities). Rather, the idea would be to go through the machinations on paper to determine if relative values were in line and what the appro- priate strategy would be.
Since anything divided by zero is zero, we have
And since N(Ø) simply means that the role of the normal distribution function has no meaningful influence on the value of SandK, we now have
Note that SKrtis equivalent to FK.
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 CSKrt, the purpose of rtis nothing more than to adjust K (the strike price) to a present value. An equivalent statement would be CSrtK,whereSrtis 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 CMax (Ø,FK).
And if we continue this story and let both Ø and tØ, we have
or
A variable raised to the power of Ø is equal to 1, so
SK.
CS1K SrK.
CSrtK, CSKrt. CSN12KrtN12.
SNa log1S>Krt2
b KrtNalog1S>Krt2 b. KrtNalog1S>Krt2
1t 1
2 1tb 1t
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
S F
Options Forwards
C = SN(X) – Kr – tN(X–σ t )
With both σ = ∅ and t =∅, C = Srt K
= Sr∅K
=S K
Withσ equal to zero we have
SN log(S/Krt) KrtN log(S / Krt)
∅ ∅
=SN (∅)KrtN(∅)
=S Krt
=F K
FIGURE A2.1 Applying Black-Scholes to the interrelated values of spot, forwards, and options.
Credit
CHAPTER 3
Products
Issuers
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.
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
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
“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 insuranceto 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.”