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Option
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.



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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




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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




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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.




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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



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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.


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