to as leveraged cash flows.

At-the-money

one year expiration

+ on a 2-year Treasury

O

“

Key:

Denotes actual payment or receipt of cash for a cash flow value that™s known at

time of initial trade (as with a purchase price, or a coupon or principal payment)

Denotes a reference to payment or receipt amount that is known at time of

initial trade, but with no exchange of cash taking place

Denotes that cash flow™s value cannot be known at time of initial trade, and that

an exchange of cash may or may not take place

Of course, any of the cash flows shown above might be sold prior to actual

maturity/expiration at a gain, loss, or breakeven.

FIGURE 2.23 Evaluating spot, forwards, and options on the basis of cash flow profiles.

TLFeBOOK

66 PRODUCTS, CASH FLOWS, AND CREDIT

APPENDIX

Volatility

Volatility is perhaps the single most elusive variable in the marketplace. There

are a variety of opinions about what constitutes the best methodology for

calculating volatility on a given asset, and in the end there really is no right

way to do it. A variety of texts have been published (and have yet to be pub-

lished) on the topic of option pricing methodologies. The aim here is merely

to flesh out a better understanding and appreciation for a rather fundamental

and variable.

Historical Volatility and Implied Volatility

While volatility typically is characterized as a price phenomenon, in the

world of bonds where yield is a key pricing variable, volatility may be quoted

in price terms or in yield terms. Generally speaking, volatility on bond futures

is expressed in price terms, while volatility on bond cash instruments is

expressed in yield terms. And consistent with the properties of duration,

whereby longer maturities/durations are associated with greater risk poten-

tial, a price volatility term structure tends to be upward sloping. This

upward slope is consistent with incrementally greater price risks as matu-

rity/duration extends. Conversely, in recognition of the inverse price/yield

relationship that exists when pricing bonds, a yield volatility term structure

tends to be downward sloping.

Essentially, volatility is measured with either historical methodology or

implied methodology. Historical volatility usually is calculated by taking a

series of daily data (perhaps three months™ worth of daily closing prices of

IBM stock, or maybe a year™s worth of daily closes of the on-the-run 10-

year Treasury™s yield) and then calculating a rolling series of annualized stan-

dard deviations.

The classic statistical definition of a standard deviation is that it mea-

sures variation around a mean. Mean is a reference to average, and to cal-

culate an average, we need to refer to a subset of data points. Using about

20 data points is a popular technique when daily observations are being

tapped, because there are about 20 business days in a month. The formula

for standard deviation is:

1xi x2 2

T

π

t 1B n 1

TLFeBOOK

67

Cash Flows

where summation

xi individual observations (i.e., daily or weekly)

“

x mean or average of all observations

n the number of observations

A standard deviation attempts to measure just how choppy a market is

by comparing how extreme individual observations can become relative to

their average.

Just as yields are expressed on an annualized basis, so too are volatili-

ties. And just as there is no hard-and-fast rule for the number of data points

that are used to calculate the mean, there is no industry convention for how

annualizing is calculated. There is simply a reasonable amount of latitude

that may be used to calculate. Many times the annualizing number is some-

thing close to 250, with the rationale that there are about 250 business days

in the year. And why might we care only about business days? Perhaps

because a variable cannot deviate from its mean if it is not trading, and the

markets typically are closed over weekends and holidays. Yet U.S. Treasuries

may be trading in Tokyo on what is Monday morning in Asia but on what

is Sunday night in the United States. Such is the life of a global market.

Even if a market is closed over a weekend, this does not mean that the

world stops and that market-moving news is somehow held back from being

announced until the following Monday. Indeed, important meetings of the

Group of Seven (G-7) or G-10 and others often occur over a weekend. Often

an expectation of a weekend G-7 pronouncement gets priced into the mar-

ket on the Friday ahead of the weekend, and this price behavior certainly is

captured by an implied volatility calculation. And if a market-moving event

transpires, then new prices at the market™s open on Monday morning cer-

tainly become reflected within a historical volatility calculation that gets

made on the following Tuesday. These are some of the considerations that

frame the debate around annualizing conventions.

An annualizing term to the historical volatility formula is

365

.

n

When combined with the formula for standard deviation, it provides

1xi x2 2 365

T

π .

t 1B n 1 n

TLFeBOOK

68 PRODUCTS, CASH FLOWS, AND CREDIT

The term “rolling” refers to the idea that we want to capture an evolv-

ing picture of volatility over time. We achieve this by employing a moving-

mean1 (or moving-average) calculation.

Implied Volatility

To calculate implied volatility, we simply take an option pricing formula of

our own choosing and plug in values for every variable in the equation except

for standard deviation. By solving for “x” where x is standard deviation,

we can calculate an implied volatility number. It is “implied” because it

comes directly from the price being quoted in the market and embodies the

market™s view on the option™s overall value.

Some investors feel that from time to time there actually may be more

packed into an implied volatility number than just a simple standard devi-

ation. That is, a standard deviation calculated as just described assumes that

the relevant underlying price series is normally distributed. What if the mar-

ket is on a marked trend upward or downward, without the kind of offset-

ting price dynamics consistent with a normally distributed pattern of

observations? In statistical terms, kurtosis is a measure of the extent that data

fall more closely around the mean of a series or more into the tails of a dis-

tribution profile relative to a normally distributed data set. For example, a

kurtosis value that is less than that of a standard normal distribution may

suggest that the distribution is wider around the mean and with a lower peak,

while a kurtosis value greater than a normal distribution may suggest a

higher peak with a narrower distribution around the mean and fatter tails.

Indeed, there is a variety of literature on the topic of fat-tail distributions

for various asset classes, and with important implications for pricing and

valuation. Here it is important to note that there is no kurtosis variable in

the formula for an option™s fair market value. The only variable in any stan-

dard option formula pertaining to the distribution of a price series (where

price may be price, yield, or an exchange rate) is standard deviation, and

standard deviation in a form consistent with normally distributed variables.

Having said this, two observations can be offered.

1

A moving-mean or moving average calculation simply means that series of averages

are calculated from one data set. For example, to calculate a 20-day moving average

with a data set of 100 daily prices, one average is calculated using days 1 to 20,

then a second average is calculated using days 2 to 21, then a third using 3 to 22,

and so on. If standard deviations are then calculated that correspond with these

moving averages, a rolling series of volatilities can be calculated. For pricing an

option with a 20-day expiration, the last volatility data point of a 20-day moving

average series would be an appropriate value to use as an input.

TLFeBOOK

69

Cash Flows

1. Any standard option-pricing model can be modified to allow for the

pricing of options where the underlying price series is not normally dis-

tributed.

2. When an implied volatility value is calculated, it may well embody more

value than what would be expected for an underlying price series that

is normally distributed; it may embody some kurtosis value.

Perhaps for obvious reasons, historical volatility often is referred to as

a backward-looking picture of market variation, while implied volatility is

thought of as a forward-looking measure of market variation. Which one is

right? Well, let us say that it is Monday morning and that on Friday a very

important piece of news about the economy is scheduled to be released”

maybe for the United States it is the monthly employment report”with the

potential to move the market in a big way one direction or the other. Let us

assume an investor was looking to buy a call option on the Dow Jones

Industrial Average for expiration on Friday afternoon. To get a good idea

of fair value for volatility, would the investor prefer to use a historical cal-

culation going back 20 days (historical volatility) or an indication of what

the market is pricing in today (implied volatility) as it looks ahead to Friday™s

event? A third possibility would involve looking at a series of historical

volatilities taken from the same key week of previous months to identify any

meaningful pattern. It is consistently this author™s preference to rely upon

implied volatility values.

To use historical volatility, a relevant question would be: How helpful

is a picture of past data for determining what will happen in the week ahead?

A more insightful use of historical volatility would be to look at data taken

from those weeks in prior months when employment data were released. But

if the goal of doing this is to learn from prior experience and derive a bet-

ter idea of fair value on volatility this particular week, perhaps implied

volatility already incorporates these experiences by reflecting the market-

clearing price where buyers and sellers agree to trade the option. Perhaps in

this regard we can employ the best of what historical and implied volatili-

ties each have to offer. Namely, we can take implied volatility as an indica-

tion of what the market is saying is an appropriate value for volatility now,

and for our own reality check we can evaluate just how consistent this

volatility value is when stacked up against historical experience. In this way,

perhaps we could use historical and implied volatilities in tandem to think

about relative value. And since we are buying or selling options with a squar-

ing off of our own views versus the market™s embedded views, other factors

may enter the picture when we are attempting to evaluate volatility values

and the best possible vehicles for expressing market views.

The debate on volatility is not going to be resolved on the basis of which

calculation methodology is right or which one is wrong. This is one of those

TLFeBOOK

70 PRODUCTS, CASH FLOWS, AND CREDIT

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.