802 Part VI: Measuring and Trading Volatility
Using 1,000 days of data:
Median 100-day historical volatility: 48%
Median 50-day historical volatility: 49%
Median 20-day historical volatility: 52%
Median 10-day historical volatility: 49%
If these were all the data that one had, then he would probably use avolatility esti
mate of 48% or so in his option models or probability calculators. Of course, this is
starkly different from the current levels of historical volatility (shown at the begin
ning of this example). So, one must be careful in assessing whether he expects the
stock to perform more in line with its longer-term (1,000 trading days) characteristics
or if there is some reason to think that the stock'sbehavior patterns have changed and
the higher, more recent volatilities should be used.
The pertinent volatilities to consider, then, in astrategy analysis are the medi
ans as well as the current figures. If the trader were going to be buying options in his
strategy, should he use the minimum of the volatilities shown, 48%? Probably.
However, if he'saseller of options, should he use the maximum, 130%? That might
be alittle too much of apenalty, but at least he would feel safe that if his volatility
selling position had apositive expected return with that high avolatility projection,
then it must truly be an attractive position.
In an analysis like that shown in this example, there is nothing magical about
using 1,000 trading days. Perhaps something like 600 trading days would be better.
The idea is to use enough trading days to bring in some historic data to counterbal
ance the recent, erratic behavior of the stock.
Among other things, this example also shows that volatilities are unstable, no
matter how much work and mathematics one puts into calculating them. Therefore,
they are at best afragile estimate of what might happen in the future. Still, it'sthe
best guess that one can make. The trader should realize, though, that when volatili
ties are this disparate when comparing recent and more distant activity, the results of
any mathematical projections using those volatilities should not be relied upon too
heavily. Those results will be just as tenuous as the volatility projections themselves.
Of course, in any case, the actual volatility that occurs while the position is in place
may be even more unfavorable than the one the trader used in his initial analysis. There
is nothing that one can do about that. But if you choose what appears to be asomewhat
unfavorable volatility, and the position still looks good under those assumptions, then it
is likely that the trader will be pleasantly surprised more often than not - that actual
volatility during the life of the position will tend to be more in his favor than not.

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804 Part VI: Measuring and Trading Volatility
If one is interested in computing the probability of the stock being above the
given price, the formula is
P (above) = 1 - P (below)
In the above formula, Vt = v✓twhere tis time to expiration in years and vis
annual volatility, as usual.
This formula is quite elementary for predictive purposes, and it is used by
many traders. This calculator can be found for free at the Web site www.option
strategist.com. Its main problem is that it gives the probability of the stock being
above or below the target price at the end of the time period, t. That'snot atotally
realistic way of approaching probability analysis. Most option traders are very con
cerned with what happens to their positions during the life of the option, not just at
expiration.
Example: suppose atrader is aseller of naked put options. He sells $OEX October
550 puts naked, with $OEX currently trading at 600. He would not normally just walk
away from this position until October expiration, because of the large risk involved
with the sale of anaked option. There are essentially three scenarios that can occur:
1. $OEX might never fall to 550 by expiration. In this case, he would have avery comfortable trade that was never in jeopardy, and the options would
expire worthless.
2. $OEX might fall below 550 and remain there until expiration. In this case,
he would surely have aloss unless $OEX were just atiny bit below 550.
3. $OEX might fall below 550 at some time between when the trade was estab
lished and when expiration occurred, but then subsequently rally back above
550 by the time expiration arrived.
An experienced option trader would almost certainly adjust if scenario 3 arose,
in order to prevent large losses from occurring. He might roll his naked puts down
and out to another strike, or he might just close them out altogether. However, it is
unlikely that he would do nothing.
The simple probability calculator formula shown above does not take into
account the trader'sthird scenario. Since it is only concerned with where the stock is
at expiration of the options, only scenarios 1 and 2 apply to it. Hence the usage of this
simple calculator is not really descriptive of what might happen to atrade during its
lifetime.
Let'sassign some numbers to the above trade, so that you might see the differ
ence. Suppose that the volatility estimate is 25%, there are 30 days until expiration,
and the prices are as stated in the previous example: $OEX is at 600, and the strik-

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Chapter 38: The Distribution of Stock Prices 805
ing price of the naked put being sold is 550. The resulting probabilities might be
something like this:
Scenario Actual Probability of Occurrence
1 . $OEX never falls below 550
2. $OEX falls below 550 and remains there
3. $OEX falls below 550 but rallies later
67%
19%
14%
The probabilities stated above are the "real" probabilities of the three various
scenarios occurring. However, if one were using the simple probability calculator
presented above, he would only have the following information:
Probability of $OEX being above 550 at expiration: 81 %
Probability of $OEX being below 550 at expiration: 19%
So, with the simple calculator, it looks like there'san 81 % chance of aworry-free
trade. Just sit back and relax and let the option expire worthless. However, in real
life - as shown by the previous set of probabilities, there'sonly a 67% chance of aworry-free trade. The difference - the other 14% - is the probability of the third
scenario occurring ($OEX falls below 550, but rallies back above it by expiration).
The simple probability calculator doesn'taccount for that scenario at all.
Hence, most serious traders don'tuse the simple model. Does that mean it'snot
useful at all? No, it is certainly viable as acomparative tool; for example, to compare
the chances of the $OEX put expiring worthless versus those of another put sale
being considered, perhaps something in astock option. However, better analyses can
be undertaken.
Before leaving the scenario of the simple probability calculator, one more point
should be made. It has been mentioned earlier in this book that the delta of an option
is actually afairly good estimate of the probability of the option being in-the-money
at its expiration date. Thus, the delta and the simple endpoint probability calculator
shown above attempt to convey the same information to atrader. In reality, because
of the fact that implied volatility might be different for various strikes (avolatility
skew), especially in index options, the delta of the option might not agree exactly with
the probability calculator. Even so, the delta is aquick and dirty way of estimating the
probability of the stock being above the strike price (in the case of call options) or
below the strike price (in the case of put options) at expiration.
THE nEVER" CALCULATOR
Having seen the frailties of the endpoint calculator, the next step is to try to design acalculator that can estimate the probability of the stock ever hitting the target price(s)

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806 Part VI: Measuring and 1iading Vo/atillty
at any time during the life of the probability study, usually the life of an option. It
turns out that there are acouple of ways to approach this problem. One is with a
Monte Carlo analysis, whereby one lets acomputer run alarge number of random
ly-generated scenarios (say, 100,000 or so) and counts the number of times the tar
get price is hit. A Monte Carlo analysis is acompletely valid way of estimating the
probability of an event, but it is asomewhat complicated approach.
In reality, there is away to create asingle formula that can estimate the "ever"
probability, although it is not any easy task either. In the following discussion, Iam
borrowing liberally from correspondence with Dr. Stewart Mayhew, Professor of
Mathematics at the University of Georgia. For proprietary reasons, the exact formu
la is not given here, but the following description should be sufficient for amathe
matics or statistics major to encode it. If one is not interested in implementing the
actual formula, the calculation can be obtained through programs sold by McMillan
Analysis Corp. at www.optionstrategist.com.
This discussion is quite technical, so readers not interested in the description of
the mathematics can skip the next paragraph and instead move ahead to the next sec
tion on Monte Carlo studies.
These are the steps necessary in determining the formula for the "ever" proba
bility of astock hitting an upside target at any time during its life. First, make the
assumption that stock prices behave randomly, and perform at the risk-free rate, r.
Mathematicians call random behavior "Brownian motion." There are anumber of
formulae available in statistics books regarding Brownian motion. If one is to esti
mate the probability of reaching amaximum (upside target) point, what is needed is
the known formula for the cumulative density function (CDP) for arunning maxi
mum of a Brownian motion. In that formula, it is necessary to use the lognormal
function to describe the upside target. Thus, instead of using the actual target price
in the CDF formula, one substitutes ln( qlp ), where qis the target price and pis the
current stock price.
The "ever" probability calculator provides much more useful information to atrader of options. Not only does anaked option seller have amuch more realistic esti
mate of the probability that he'sgoing to have to make an adjustment during the life
of an option, but the option buyer can find the information useful as well. For exam
ple, if one is buying an option at aprice of 10, say, then he could use the "ever" prob
ability calculator to estimate the chances of the stock trading 10 points above the
striking price at any time during the life of the option. That is, what are the chances
that the option is going to at least break even? The option buyer can, cf course, deter
mine other things too, such as the probability that the option doubles in price ( or
reaches some other return on investment, such as he might deem appropriate for his
analysis).

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808 Part VI: Measuring and Trading Volatility
of them resulted in the stock being unchanged. Also, only about 2,500 or them, or
1110th of one percent, resulted in amove of-4.0 standard deviations or more. Those
percentages, along with all of the others, would be built into the computer, so that
the total distribution accounts for 100% of all possible stock movements.
Then, we tell the computer to allow astock to move randomly in accordance
with whatever volatility the user has input. So, there would be afairly large proba
bility that it wouldn'tmove very far on agiven day, and avery small probability that
it would move three or more standard deviations. Of course, with the fat tail distri
bution, there would be alarger probability of amovement of three or more standard
deviations than there would be with the regular lognormal distribution. The Monte
Carlo simulation progresses through the given number of trading days, moving the
stock cumulatively as time passes. If the stock hits the break-even price, that partic
ular simulation can be terminated and the next one begun. At the end of all the tri
als (100,000 perhaps), the number in which the upside target was touched is divided
by the total number of trials to give the probability estimate.
Is it really worth all this extra trouble to evaluate these more complicated prob
ability distributions? It seems so. Consider the following example:
Example: Suppose that atrader is considering selling naked puts on XYZ stock,
which is currently trading at aprice of 80. He wants to sell the November 60 puts,
which expire in two months. Although XYZ is afairly volatile stock, he feels that he
wouldn'tmind owning it if it were put to him. However, he would like to see the puts
expire worthless. Suppose the following information is available to him via the vari
ous probability calculators:
Simple "end point" probability of XYZ < 60 at expiration: 10%
Probability that XYZ ever trades < 60 (using the lognormal distribution) 20%
Probability that XYZ ever trades < 60 (using the fat tail distribution): 22%
If the chances of the put never needing attention were truly only 10%, this trader
would probably sell the puts naked and feel quite comfortable that he had atrade
that he wouldn'thave to worry too much about later on. However, if the true proba
bility that the put will need attention is 22%, then he might not take the trade. Many
naked option sellers try to sell options that have only probabilities of 15% or less of
potentially becoming troublesome.
Hence, the choice of which probability calculation he uses can make adiffer
ence in whether or not atrade is established.
Other strategies lend themselves quite well to probability analysis as well.
Credit spreaders - sellers of out-of-the-money put spreads - usually attempt to quan
tify the probability of having to take defensive action. Any action to adjust or remove

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Chapter 38: The Distribution of Stock Prices 809
adeeply out-of-the-money put credit spread usually destroys most or all of its prof
itability, so an accurate initial assessment of the probabilities of having to make such
an adjustment is important.
Option buyers, too, would benefit from the use of amore accurate probability
estimate. This is especially true for neutral strategies, such as straddle or strangle
buying, when the trader is interested in the chances of the stock being able to move
far enough to hit one or the other of the straddle'sbreak-even points at some time
during the life of the straddle.
The Monte Carlo probability calculation can be expanded to include other sorts
of distributions. In the world of statistics, there are many distributions that define ran
dom patterns. The lognormal distribution is but one of them (although it is the one
that most closely follows stock prices movements in general). Also, there is aschool of
thought that says that each stock'sindividual price distribution patterns should be ana
lyzed when looking at strategies on that stock, as opposed to using ageneral stock
price distribution accumulated over the entire market. There is much debate about
that, because an individual stock'strading pattern can change abruptly just consider
any of the Internet stocks in the late 1990s and early 2000s. Thus, aprobability esti
mate based on asingle stock'sbehavior, even if that behavior extends back several
years, might be too unreliable astatistic upon which to base aprobability estimate.
In summary, then, one should use aprobability calculator before taking an
option position, even an outright option buy. Perhaps straight stock traders should
use aprobability calcutor as well. In doing so, though, one should be aware of the
limitations of the estimate: It is heavily biased by the volatility estimate that is input
and by the assumption of what distribution the underlying instrument will adhere to
during the life of the position. While neither of those limitations can be overcome
completely, one can mitigate the problems by using aconservative volatility estimate.
Also, he can look at the results of the probability calculation under several distribu
tions (perhaps lognormal, fat tail, and the distribution using only the past price
behavior of the underlying instrument in question) and see how they differ. In that
case, he would at least have afeeling for what could happen during the life of the
option position.
EXPECTED RETURN
The concept of expected return was described in the chapter on mathematical appli
cations. In short, expected return is aposition'sexpected profit divided by its invest
ment ( or expected investment if the investment varies with stock price, as in anaked
option position or afutures position). The crucial component, though, is expected
profit.

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810 Part VI: Measuring and Trading Volatility
Expected profit is computed by calculating the profitability of aposition at acertain stock price times the probability of the stock being at that price, and summing
that multiple over all possible stock prices. When the concept was first introduced,
the "probability of the stock being at that price" was given as what we now know is
the "endpoint" probability. In reality, amuch better measure of the expected profit
of aposition can be obtained by using one of the more advanced probability estima
tion models presented above.
In generalized expected return studies done using the fat tails Monte Carlo sim
ulation, certain general conclusions can be drawn about some strategies.
• Abull spread is an inferior strategy when the options are fairly priced, no matter
which distribution is assumed. This more or less agrees with observations that
have been made previously regarding the disappointments that traders often
encounter when using vertical spreads.
• While covered writing might seem superior to stock ownership under the log
normal distribution, the two are about equal under afat tail distribution.
• Most startling, though, is the fact that option buying strategies fare much, much
better under afat tail distribution than alognormal one. This most clearly
demonstrates the "power" of the fat tail distribution: Alimited-risk investment
with unlimited profit potential can be expected to perform very well if the fat tails
are allowed for.
Using the lognormal distribution more or less represents the conventional wisdom
regarding option strategies - the one that many brokers promote: "Don'tbuy options,
don'tmess with spreads, either buy stocks or do covered call writes." The fat tail dis
tribution column stands much of that advice on its head. In real life (as demonstrat
ed by the fat tail distribution), strategies with limited profit potential and unlimited
or large risk potential are inferior strategies.
One should be aware that the phrase "expected return" is used in many quasi
sophisticated option analyses (and even in analyses not using options). Many
investors accept these "returns" on blind faith, figuring that if they're generated by acomputer, they must be correct. In reality, they may be not be representative, even
for comparisons.
SUMMARY
This chapter has demonstrated that probability analysis is an inexact science, because
markets behave in ways that are very difficult to describe mathematically. However,
probability analysis is also necessary for the option strategist; without it he would be

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Chapter 38: The Distribution of Stock Prices 811
in the dark as to the likelihood of profitable outcomes for his strategy. Overall, in adiversified set of positions, the option strategist should use the fat tail distribution in
a Monte Carlo simulation to estimate probabilities. However, if that is not available,
he can use the normal or lognormal distribution with the proviso that he understands
it is not "gospel." He should require ve:ry stringent criteria on any strategies that are
antivolatility strategies, such as naked option writing of stock options, for there is agreater than normal chance of alarge move by the underlying, especially if the
underlying is stock.
The sophisticated trader may want to view his probabilities in the light of more
than one proposed distribution of prices. Of course, this type of analysis ( using sev
eral distributions) puts the onus on the investor to choose the distribution that he
wants to use in order to analyze his investment. However, such an approach should
be extremely illustrative in that he can compare returns from different strategies and
have areasonable expectation as to which ones might perform the best under differ
ent market conditions.

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CHAPTER 39
Volatility Trading Techniques
The previous three chapters laid the foundation for volatility trading. In this chapter,
the actual application of the technique will be described. It should be understood
that volatility trading is both an art and ascience. It'sascience to the extent that one
must be rigorous about determining historical volatility or implied volatility, calculat
ing probabilities, and so forth. However, given the vagaries of those measurements
that were described in some detail in the previous chapters, volatility trading is also
something of an art. Just as two fundamental analysts with the same information
regarding earnings, sales projection, and so on might have two different opinions
about astock'sfortunes, so also can two volatility traders disagree about the potential
for movement in astock.
However, volatility traders do agree on the approach. It is based on comparing
today'simplied volatility with what one expects volatility to do in the future. As noted
previously, one'sexpectations for volatility might be based on volatility charts, pat
terns of historical volatility and implied volatility, or something as complicated as a
GAR CH forecasting model. None of them guarantees success. However, we do know
that volatility tends to trade in arange in the long run. Therefore, the approach that
traders agree upon is this: If implied volatility is "low," buy it. If it's "high," sell it with
caution. So simple: Buy low, sell high (not necessarily in that order). The theory
behind volatility trading is that it'seasier to buy low and sell high (or at least to deter
mine what's "low" and "high") when one is speaking about volatility, than it is to do
the same thing when one is talking about stock prices.
Most of the time, implied volatility will not be significantly high or low on any
particular stock, futures contract, or index. Therefore, the volatility trader will have
little interest in most stocks on any given day. This is especially true of the big-cap
stocks, the ones whose options are most heavily traded. There are so many traders
812

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Chapter 39: Volatility Trading Techniques 813
watching the situation for those stocks that they will rarely let volatility get to the
extremes that one would consider "too high" or "too low." Yet, with the large num
ber of optionable stocks, futures, and indices that exist, there are always some that
are out of line, and that'swhere the independent volatility trader will concentrate
his efforts.
Once avolatility extreme has been uncovered, there are different methods of
trading it. Some traders - market-makers and short-term traders - are just looking
for very fleeting trades, and expect volatility to fall back into line quickly after an
aberrant move. Others prefer more of aposition traders' approach: attempting to
determine volatility extremes that are so far out of line with accepted norms that it
will probably take some time to move back into line. Obviously, the trader'sown sit
uation will dictate, to acertain extent, which strategy he pursues. Things such as
commission rates, capital requirements, and risk tolerance will determine whether
one is more of ashort-term trader or aposition trader. The techniques to be
described in this chapter apply to both methods, although the emphasis will be on
position trading.
TWO WAYS VOLATILITY PREDICTIONS CAN BE WRONG
When traders determine the implied volatility of the options on any particular under
lying instrument, they may generally be correct in their predictions; that is, implied
volatility will actually be afairly good estimate of forthcoming volatility. However,
when they're wrong, they can actually be wrong in two ways: either in the outright
prediction of volatility or in the path of their volatility predictions. Let'sdiscuss both.
When they're wrong about the absolute level of volatility, that merely means that
implied volatility is either "too low" or "too high." In retrospect, one could only make
that assessment, of course, after having seen what actual volatility turned out to be
over the life of the option. The second way they could be wrong is by making the
implied volatility on some of the options on aparticular underlying instrument much
cheaper or more expensive than other options on that same underlying instrument.
This is called avolatility skew and it is usually an incorrect prediction about the way
the underlying will perform during the life of the options.
The rest of this chapter will be divided into two main parts, then. The first part
will deal with volatility from the viewpoint of the absolute level of implied volatility
being "wrong" (which we'll call "trading the volatility prediction"), and the second
part will deal with trading the volatility skew.

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814 Part VI: Measuring and Trading Volatility
TRADING THE VOLATILITY PREDICTION
The volatility trader must have some way of determining when implied volatility is
sufficiently out of line that it warrants atrade. Then he must decide what trade to
establish. Furthermore, as with any strategy- especially option strategies - follow-up
action is important too. We will not be introducing any new strategies, per se, in this
chapter. Those strategies have already been laid out in the previous chapters of this
book. However, we will briefly review important points about those strategies and
their follow-up actions where it is appropriate.
First, one must try to find situations in which implied volatility is out of line.
That is not the end of the analysis, though. After that, one needs to do some proba
bility work and needs to see how the underlying has behaved in the past. Other fine
tuning measures are often useful, too. These will all be described in this chapter.
DETERMINING WHEN VOLATILITY IS OUT OF LINE
There is much disagreement among volatility traders regarding the best method to use
for determining if implied volatility is "out ofline." Most favor comparing implied with
historical volatility. However, it was shown two chapters ago that implied volatility is
not necessarily agood predictor of historical volatility. Yet this approach cannot be dis
carded; however it must be used judiciously. Another approach is to compare today'simplied volatility with where it has been in the past. This concept relies heavily on the
concept of the percentile of implied volatility. Finally, there is the approach of trying
to "read" the charts of implied and historical volatility. This is actually something akin
to what GARCH tries to do, but on ashort-term horizon. So the approaches are:
1. Compare implied volatility to its own past levels (percentile approach).
2. Compare implied volatility to historical volatility.
3. Interpret the chart of volatility.
In addition, we will examine two lesser-used methods: comparing current levels of
historical volatility to past measures of historical volatility, and finally, using only aprobability calculator and trading the situation that has the best probabilities of
success.
THE PERCENTILE APPROACH
In this author'sopinion, there is much merit in the percentile approach. When one says
that volatility tends to trade in arange, which is the basic premise behind volatility trad-

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Chapter 39: Volatility Trading Techniques 815
ing, he is generally talking about implied volatility. Thus, it makes sense to know where
implied volatility is within the range of the past readings of implied volatility. If volatil
ity is low with respect to where it usually trades, then we can say the options are cheap.
Conversely, if it'shigh with respect to those past values, then we can say the options are
expensive. These conclusions do not draw on historical volatility.
The percentile of implied volatility is generally used to describe just where the
current implied volatility reading lies with respect to its past values. The "implied
volatility" reading that is being used in this case is the composite reading - the one
that takes into account all the options on an underlying instrument, weighting them
by their distance in- or out-of-the-money (at-the-money gets more weight) and also
weighting them by their trading volume. This technique has been referred to many
times and was first described in Chapter 28 on mathematical applications. That com
posite implied volatility reading can be stored in adatabase for each underlying
instrument every day. Such databases are available for purchase from firms that spe
cialize in option data. Also, snapshots of such data are available to members of
www.optionstrategist.com.
In general, most underlying instruments would have acomposite implied
volatility reading somewhere near the 50th percentile on any given day. However, it
is not uncommon to see some underlyings with percentile readings near zero or 100%
on agiven day. These are the ones that would interest avolatility trader. Those with
readings in the 10th percentile or less, say, would be considered "cheap"; those in the
90th percentile or higher would be considered expensive.
In reality, the percentile of implied volatility is going to be affected by what the
broad market is doing. For example, during asevere market slide, implied volatilities
will increase across the board. Then, one may find alarge number of stocks whose
options are in the 90th percentile or higher. Conversely, there have been other times
when overall implied volatility has declined substantially: 1993, for example, and the
summer of 2001, for another. At those times, we often find agreat number of stocks
whose options reside in the 10th percentile of implied volatility or lower. The point
is that the distribution of percentile readings is adynamic thing, not something stat
ic like alognormal distribution. Yes, perhaps over along period of time and taking
into account agreat number of cases, we might find that the percentiles of implied
volatility are normally distributed, but not on any given day.
The trader has some discretion over this percentile calculation. Foremost, he
must decide how many days of past history he wants to use in determining the per
centiles. There are about 255 trading days in ayear. So, if he wanted atwo-year his
tory, he would record the percentile of today'scomposite implied volatility with
respect to the 510 daily readings over the past two years. This author typically uses

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816 Part VI: Measuring and Trading Volatility
600 days of implied volatility history for the purpose of determining percentiles, but
acase could be made for other lengths of time. The purpose is to use enough implied
volatility history to give one agood perspective. Then, areading of the 10th per
centile or the 90th percentile will truly be significant and would therefore be agood
starting point in determining whether the options are cheap or expensive.
In addition to the actual percentile, the trader should also be aware of the width
of the implied volatility distribution. This was discussed in an earlier chapter, but
essentially the concept is this: If the first percentile is an implied volatility of 40% and
the 100th percentile is an implied volatility of 45%, then that entire range is so nar
row as to be meaningless in terms of whether one could classify the options as cheap
or expensive.
The advantage of buying options in alow percentile of implied volatility is to
give oneself two ways to make money: one, via movement in the underlying (if astraddle were owned, for example), and two, by an increase in implied volatility. That
is, if the options were to return to the 50th percentile of implied volatility, the volatil
ity trader who has bought "cheap" options should expect to make money from that
movement as well. That can only happen if the 50th percentile and the 10th per
centile are sufficiently far apart to allow for an increase in the price of the option to
be meaningful. Perhaps agood rule of thumb is this: If the option rises from the cur
rent (low) percentile reading to the 50th percentile in amonth, will the increase in
implied volatility be equal to or greater than the time decay over that period?
Alternatively stated, with all other things being equal, will the option be trading at
the same or agreater price in amonth, if implied volatility rises to the 50th percentile
at the end of that time? If so, then the width of the range of implied volatilities is
great enough to produce the desired results.
The attractiveness to this method for determining if implied volatility is out of
line is that the trader is "forced" to buy options that are cheap ( or to sell options that
are expensive), on arelative basis. Even though historical volatility has not been taken
into consideration, it will be later on when the probability calculators are brought to
bear. There is no guarantee, of course, that implied volatility will move toward the
50th percentile while the position is in place, but if it does, that will certainly be an
aid to the position.
In effect this method is measuring what the option trading public is "thinking"
about volatility and comparing it with what they've thought in the past. Since the pub
lic is wrong (about prices as well as volatility) at major turning points, it is valid to want
to be long volatility when "everyone else" has pushed it down to depressed levels. The
converse may not necessarily be true: that we would want to be short volatility when
everyone else has pushed it up to extremely high levels. The caveat in that case is that

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Chapter 39: Volatility Trading Techniques 817
someone may have inside information that justifies expensive options. This is another
reason why selling volatility can be difficult: You may be dealing with far less infor
mation than those who are actually making the implied volatility high.
COMPARING IMPLIED AND HISTORICAL VOLATILITY
The most common way that traders determine which options are cheap or expensive
is by comparing the current composite implied volatility with various historical
volatility measures. However, just because this is the conventional wisdom does not
necessarily mean that this method is the preferred one for determining which options
are best for volatility trades. In this author'sopinion, it is inferior to the percentile
method (comparing implied to past measures of implied), but it does have its merits.
The theory behind using this method is that it is avirtual certainty that implied and
historical volatility will eventually converge with each other. So, if one establishes
volatility trading positions when they are far apart, there is supposedly an advantage
there.
However, this argument has plenty of holes in it. First of all, there is no guar
antee that the two will converge in atimely manner, for example, before the options
in the trader'sposition can become profitable. Historical and implied volatility often
remain fairly far apart for weeks at atime.
Second, even if the convergence does occur, it doesn'tnecessarily mean one will
make money. As an example, consider the case in which implied volatility is 40% and
historical volatility is 60%. That'squite adifference, so you'dwant to buy volatility.
Furthermore, suppose the two do converge. Does that mean you'll make money? No,
it does not. What if they converge and meet at 40%? Or, worse yet, at 30%? You'dmost certainly lose in those cases as the stock slowed down while your options lost
time value.
Another problem with this method is that implied volatility is not necessarily
low when it is bought, nor high when it is sold. Consider the example just cited. We
merely knew that implied volatility was 40% and that historical volatility was 60%. We
had no perspective on whether 40% was high, medium, or low. Thus, it is also nec
essary to see what the percentile of implied volatility is. If it turns out that 40% is arelatively high reading for implied volatility, as determined by looking at where
implied volatility has been over the past couple of years, then we would probably not
want to buy volatility in this situation, even though implied and historical volatility
have alarge discrepancy between them.
Many market-makers and floor traders use this approach. However, they are
often looking for an option that is briefly mispriced, figuring that volatilities will

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818 Part VI: Measuring and Trading Volatility
quickly revert back to where they were. But for aposition trader, the problems cited
above can be troublesome.
Having said that, if one looks to implement this method of trying to determine
when options are out of line, something along the following lines should be imple
mented. One should ensure that implied volatility is significantly different from all of
the pertinent historical volatilities. For example, one might require that implied
volatility is less than 80% of each of the 10-, 20-, 50-, and 100-day historical volatili
ty calculations. In addition, the current percentile of implied volatility should be
noted so that one has some relative basis for determining if all of the volatilities, his
torical and implied, are very high or very low. One would not want to buy options if
they were all in avery high percentile, nor sell them if they were all in avery low per
centile.
Often, avolatility chart showing both the implied and certain historical volatili
ties will be auseful aid in making these decisions. One can not only quickly tell if the
options are in ahigh or low percentile, but he may also be able to see what happened
at similar times in the past when implied and historical volatility deviated substan
tially.
Finally, one needs some measure to ensure that, if convergence between
implied and historical volatility does occur, he will be able to make money. So, for
example, if one is buying astraddle, he might require that if implied rises to meet his
torical (say, the lowest of the historicals) in amonth, he will actually make money.
One could use adifferent time frame, but be careful not to make it something unrea
sonable. For example, if implied volatility is currently 40% and historical is 60%, it is
highly unlikely that implied would rise to 60% in aday or two. Using this criterion
also ensures that the absolute difference between implied and historical volatility is
wide enough to allow for profits to be made. That is, if implied is 10% and historical
is 13%, that'sadifference of 30% in the two - ostensibly a "wide" divergence
between implied and historical. However, if implied rises to meet historical, it will
mean only an absolute increase of 3 percentage points in implied volatility - proba
bly not enough to produce aprofit, after costs, if any length of time passes.
If all of these criteria are satisfied, then one has successfully found "mispriced"
options using the implied versus historical method, and he can proceed to the next
step in the volatility analysis: using the probability calculator.
READING THE VOLATILITY CHART
Another technique that traders use in order to determine if options are mispriced is
to actually try to analyze the chart of volatility - typically implied volatility, but it
could be historical. This might seem to be asubjective approach, except that it is real-

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Chapter 39: Volatility Trading Techniques 819
ly not much different from the GARCH approach, which is considered to be highly
advanced. When one views the volatility chart, he is not looking for chart patterns like
technical analysts might do with stock charts: support, resistance, head-and-shoul
ders, flags, pennants, and so on. Rather, he is merely looking for the trend of volatil
ity to change.
This is avalid approach in the use of many indicators, particularly sentiment
indicators, that can go to extreme levels. By waiting for the trend to change, the user
is not subjecting himself to buying into the midst of adowntrend in volatility, nor sell
ing into the midst of asteep uptrend in volatility.
Example: Suppose avolatility trader has determined that the current level of implied
volatility for XYZ stock is in the 1st percentile of all past readings. Thus, the options
are as cheap as they've ever been. Perhaps, though, the overall market is experienc
ing avery dull period, or XYZ itself has been in aprolonged, tight trading range -
either of which might cause implied volatility to decline steadily and substantially.
Having found these cheap options, he wants to buy volatility. However, he has no
guarantee that implied volatility won'tcontinue to decline, even though it is already
as cheap as it'sever been.
If he follows the technique of waiting for areversal in the trend of implied
volatility, then he would keep an eye on XYZ'simplied volatility daily until it had at
least amodest increase, something to indicate that option buyers have become more
interested in XYZ'soptions. The chart in Figure 39-1 shows how this situation might
look.
There are anumber of items marked on the chart, so it will be described in
detail. There are two graphs in Figure 39-1: The top line is the implied volatility
graph, while on the bottom is the stock price chart. The implied volatility chart shows
that, near the first ofJune, it made new all-time lows near 28% (i.e., it was in the 0th
percentile of implied volatility). Hence, one might have bought volatility at that
point. However, it is obvious that implied volatility was in asteep downtrend at that
time, so the volatility trader who reads the charts might have decided to wait for apop in volatility before buying. This turned out to be ajudicious decision, because
the stock went nowhere for nearly another month and ahalf, all the while volatility
was dropping. At the right of the chart, implied volatility has dropped to nearly 20%.
The solid lines on the two graphs indicate the data that is known about the
implied volatility and price history of XYZ. The dotted lines indicate ascenario that
might unfold. If implied volatility were to jump ( and the stock price might jump, too),
then one might think that the trend of implied volatility was no longer down, and he
would then buy volatility.

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820 Part VI: Measuring and Trading VolatHity
The reason that this approach has merit is that one never knows how low volatil
ity can go, and more important, how high it can get. It was mentioned that the same
sort of approach works well for other sentiment indicators, the put-call ratio, in par
ticular. During the bull market of the 1990s, the equity-only put-call ratio generally
ranged between about 30 and 55. Thus, some traders became accustomed to buying
the market when the put-call ratio reached numbers exceeding 50 (high put-call
ratio numbers are bullish predictors for the market in general). However, when the
bull market ended, or at least faltered, the put-call ratios zoomed to heights near 70
or 75. Thus, those using astatic approach (that is, "Buy at 50 or higher") were buried
as they bought too early and had to suffer while the put-call ratios went to new all
time highs. Atrend reversal approach would have saved them. It is amore dynamic
procedure, and thus one would have let the put-call ratio continue to rise until it
peaked. Then the market could have been bought.
This is exactly what reading the volatility chart is about. Rather than relying on
past data to indicate where the absolute maxima and minima of movements might
occur, one rather notes that the volatility data is at extreme levels ( 1st percentile or
100th percentile) and then watches it until it reverses direction. This is especially
useful for options sellers, because it avoids stepping into the vortex of massive option
FIGURE 39-1.
Chart of the trend of implied volatility.
XYZ
,J'
········································ · ························ ....................... ··················· 50. 0
... · ····················· 40. 0
Implied Volatility I/vi Ar
···-····························· .. •·•······•·········-·············································-··vw•···················)·•······ 30. 0
All-Time Volatility Low -
, .... , ..... , .. ··················,·······~················
t :t·•····•····· ...•. ':·LJJSL5~?.
--;---··········•·:
o, b ::r Ff1 hj ::r
34.000
32.000
30. 000
28. 000
26.000
24.000
22.000
20. 000

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Chapter 39: VolatiDty Trading Techniques 821
buying, where the buyers perhaps have inside information about some forthcominf
corporate event such as atakeover. True, the options might be very expensive ( 10ot
percentile), but there is areason they are, and those with the inside information know
the reason, whereas the typical volatility trader might not. However, if the volatility
trader merely waits for adownturn in implied volatility readings before selling these
options, he will most likely avoid the majority of trouble because the options will
probably not lose implied volatility until news comes out or until the buyers give up
(perhaps figuring that the takeover rumor has died).
Volatility buyers don'tface the same problems with early entry that volatility
sellers do, but still it makes sense to wait for the trend of volatility to increase (as in
Figure 39-1) before trying to guess the bottom in volatility. Just as it is usually fool
hardy to buy astock that is in asevere downtrend, so it may be, too, with buying
volatility.
Aless useful approach would be to apply the same techniques to historical
volatility charts, for such charts say nothing about option prices. See the next section
for expansion on these thoughts.
COMPARING HISTORICAL VERSUS HISTORICAL
The above paragraphs summarize the three major ways that traders attempt to find
options that are out of line. Sometimes, another method is mentioned: comparing
current levels of historical volatility with past levels of the same measure, historical
volatility. This method will be described, but it is generally an inferior method
because such acomparison doesn'ttell us anything about the option prices. It would
do little good, for example, to find that current historical volatility is in avery low per
centile of historical volatilities, only to learn later that the options are expensive and
that perhaps implied volatility is even higher than historical volatility. One would nor
mally not want to buy options in that case, so the initial analysis of comparing histor
ical to historical is awasted effort.
Comparing current levels of historical volatility with past measures of historical
volatility is sort of abackward-looking approach, since historical volatility involves
strictly the use of past stock prices. There is no consideration of implied volatility in
this approach. Moreover, this method makes the tacit assumption that astock'svolatility characteristics do not change, that it will revert to some sort of "normal" past
price behavior in terms of volatility. In reality, this is not true at all. Nearly every stock
can be shown to have considerable changes in its historical volatility patterns over
time.

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822 Part VI: Measuring and Trading Volatility
Consider the historical volatilities of one of the wilder stocks of the tech stock
boom, Rambus (RMBS). Historical volatilities had ranged between 50% and ll0%,
from the listing of RMBS stock, through February 2000. At that time, the stock aver
aged aprice of about $20 per share.
Things changed mightily when RMBS stock began to rise at atremendous rate
in February 2000. At that time, the stock blasted to ll5, pulled back to 35, made anew high near 135, and then collapsed to aprice near 20. Hence, the stock itself had
completed awild round-trip over the two-year period. See Figures 39-2 and 39-3 for
the stock chart and the historical volatility chart of RMBS over the time period in
question.
As this happened, historical volatility skyrocketed. After February 2000, and
well into 2001, historical volatility was well above 120%. Thus it is clear that the
behavior patterns of Rambus changed greatly after February 2000. However, if one
had been comparing historical volatilities at any time after that, he would have erro
neously concluded that RMBS was about to slow down, that the historic volatilities
were too high in comparison with where they'dbeen in the past. If this had led one
to sell volatility on RMBS, it could have been avery expensive mistake.
While RMBS may be an extreme example, it is certainly not alone. Many other
stocks experienced similar changes in behavior. In this author'sopinion, such behav
ior debunks the usefulness of comparing historical volatility with past measures of
historical volatility as avalid way of selecting volatility trades.
FIGURE 39-2.
Historical volatilities of RMBS.
RMBS 19.000 17.250 18.875 20010410
............ ······· 60.0
· ·················· 50.0
...... ······· 40. 0
39 Mi·h··-:; s ·~ sa ~--f:i--·5--r;;····~··hs·s- Asa N □ s--r·;; r

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Chapter 39: Volatility Trading Techniques 823
FIGURE 39-3.
Stock chart of RMBS.
19. 000 17. 250 18. 875 20010410
30.000
22. 000
14. 000
' ' : :
! : : : ; 1 : : 1 r : ! : : : : ; : : : : : il : l : 6. ooo
99 Mil M :i Jii sb Nb J F Mil M :i Jil sb Nb J Fh 1
What this method may be best used for is to complement the other methods
described previously, in order to give the volatility trader some perspective on how
volatile he can expect the underlying instrument to be; but it obviously has to be
taken only as ageneral guideline.
CHECK THE FUNDAMENTALS
Once these mispriced options have been found, it is always imperative to check the
news to see if there is some fundamental reason behind it. For example, if the options
are extremely cheap and one then checks the news stories and finds that the under
lying stock has been the beneficiary of an all-cash tender offer, he would not buy
those options. The stock is not going to go anywhere, and in fact will disappear if the
deal goes through as planned.
Similarly, if the options appear to be very expensive, and one checks the news
and finds that the underlying has aproduct up for review before agovernmental
agency (FDA, for example), then the options should not be sold because the stock
may be about to undergo alarge gap move based on the outcome of FDA hearings.
There could be any number of similar corporate events that would make the options
very expensive. The seller of volatility should not try to intercede when such events
or rumors are occurring.

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824 Part VI: Measuring and Trading Volatllity
However, if there is no news that would seem to explain why the options are so
cheap or so expensive, then the volatility trader can continue on to the rest of his
analyses.
SELECTING THE STRATEGY TO USE
In general, when one wants to trade volatility, asimple approach is best, especially if
one is buying volatility. If there is avolatility skew involved, then there may be other
strategies that are superior, and they are discussed in the latter part of this chapter.
However, when one is interested in the straight trading of volatility because he thinks
implied volatility is out of line, then only afew strategies apply.
If volatility is too low, then either astraddle or astrangle should be purchased.
One would normally choose astraddle if the underlying instrument is currently trad
ing near an available striking price. However, if the underlying is currently trading
between two ~riking prices, then astrangle might be the better choice. In either case,
aposition trader would want to buy astraddle with several months of life remaining,
in order to improve his chances of making aprofit. There is no "best" time length to
use, so one should use aprobability calculator to aid in that decision. The use of prob
ability calculators will be discussed shortly.
Example: XYZ is trading at 39.60 and avolatility trader has determined that he wants
to buy volatility. With this information, he should attempt to buy astraddle with astriking of 40 for both the put and the call.
Suppose that the current date is in December, and the available expiration
months for XYZ are January, February, April, July, and October, plus LEAPS for
January of the next year. Then he would analyze each straddle (January 40, February
40, April 40, etc.) to see which is the best one to buy. It generally seems to work out
that the midrange straddles have the best probabilities of success, given the way that
option prices are usually structured. Of course, the actual prices of each straddle
would be considered when using the probability calculator. In this case, then, the July
40 or October 40 straddle would probably be the best choices from astatistical view
point for aposition trader.
If XYZ had been trading at aprice of 37.50, say, then the trader would proba
bly want to consider buying astrangle: buying acall with astriking price of 40 and aput with astriking price of 35. From the viewpoint of buying strangles, it does not
make sense to separate the strikes by more than one striking price unit - 5 points for
stock options, for example. This just makes the position more neutral to begin with.

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Chapter 39: Volatility Trading Techniques 825
Speaking of neutrality, one can also use the deltas of the options in question to
alter the ratio of puts to calls, making the position initially as neutral as possible. This
is the suggested approach, since the volatility buyer does not care whether the stock
goes up or down. He is merely interested in stock movement and/or an increase in
implied volatility.
Example: XYZ is again trading at 39.60, and the trader wants aneutral position. He
should use the deltas of the options to construct aneutral position. Consider the
October 40 straddle, for example. Assume the volatility used for the probability cal
culations is 40%. Under those conditions (and the ones assumed in the previous
example), the October 40 call has adelta of 0.60 and the October 40 put has adelta
of -0.40. Thus aratio of buying 2 calls and 3 puts is aneutral ratio. If the call is sell
ing for 6 and the put is selling for 5, then the break-even points for a 2-by-3 position
would be 53.5 on the upside and 31 on the downside. This information is summarized
as follows:
Delta of October 40 call:
Delta of October 40 put:
+0.60
-0.40
Delta-neutral ratio: buy 2 calls and 3 puts
Price of October 40 coll:
Price of October 40 put:
Net cost of 2-by-3 position: 27 points
Break-even points:
6.00
5.00
Upside = 40 + 27 /2 = 53.50
Downside = 40 - 27 /3 = 31 .00
So, the probability calculations would endeavor to determine what the chances are of
the stock ever trading at either 53.50 or 31.00 at any time prior to expiration. In fact,
since there are straddles available in several expiration months, the strategist would
want to analyze each of them in asimilar fashion. Table 39-1 shows how his choices
might look. If one were considering buying astrangle, similar calculations could be
made using the deltas of the put and the call, where the call strike is higher than the
put strike.
Another simple strategy that can be used when volatility is low is the calendar
spread, because it has apositive vega. That is, it can be expected to expand if implied
volatility increases. For most traders, though, the limited profit nature of the calen-

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826 Part VI: Measuring and Trading Volatility
TABLE 39-1
January February April July October January LEAP
Call price 1.25 2.25 3.50 5.00 6.00 7.15
Put price 1.50 2.35 3.35 4.35 5.00 5.55
Call delta 0.48 0.52 0.55 0.58 0.60 0.62
Put delta -0.52 -0.48 -0.45 -0.42 -0.40 -0.38
Neutral ~ 1-to-1 ~l-to-1 ~ 1-to-1 ~2-to-3 2-to-3 ~2-to-3
Debit 2.75 4.60 6.85 23.05 27.00 30.95
Upside break-even 42.75 44.60 46.85 51.57 53.50 55.47
Downside break-even 37.25 35.40 33.15 32.30 31.00 29.68
dar spread is too much of aburden"° either psychologically or in terms of commis
sions, and so this strategy is only modestly used by volatility traders. Some traders will
use the calendar spread if they don'tsee immediate prospects for an increase in
implied volatility. They perhaps buy acall calendar slightly out-of-the-money and also
buy aput calendar with slightly out-of-the-money puts. Then, if not much happens
over the short term, the options that were sold expire worthless, and the remaining
long straddle or strangle is even more attractive than ever. Of course, this strategy has
its drawback in that aquick move by the underlying may result in aloss, something
that would not have happened had asimple straddle or strangle been purchased.
SELLING VOLATILITY
If one were selling volatility (i.e., volatility is too high), his choices are more complex.
Virtually anyone who has ever sold volatility has had abad experience or two with
either exploding stock prices or exploding volatility. Some of the concerns regarding
the sale of volatility will be discussed at length later in this chapter. For now, the sim
pler strategies will be considered, in keeping with the discussion involving the cre
ation of avolatility trading position.
Simplistically, avolatility seller would generally have achoice between one of
two strategies (although there is amore complicated strategy that can be introduced
as well). The simplest strategy is just to sell both an out-of-the-money put and an out
of-the-money call. The striking prices chosen should be far enough away from the
current underlying price so that the probabilities of the position getting in trouble
(i.e., the probabilities that the underlying actually trades at the striking prices of the
naked options during the life of the position) are quite small. Just as the option buyer

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Chapter 39: Volatility Trading Techniques 827
above outlined several expiration months, then computed the break-even prices, so
should avolatility seller. Generally he will probably want to sell short-term options,
but all expiration months should be considered, at least initially. Also, he may want to
try different strike prices in order to get abalance between alow probability of the
stock reaching the striking price of the naked options and taking in enough premium
to make the trade worthwhile. To this author, the sale of naked options at small frac
tional prices does not appear attractive.
Of course, merely selling such aput and acall means the options are naked, and
that strategy is not suitable for all traders. The next best choice then, Isuppose, is acredit spread. The problem with acredit spread is that one is both selling expensive
options and also buying expensive options as protection. The ramifications of volatil
ity changes on the credit spread strategy were detailed two chapters previously, so
they won'tbe recounted here, except to say that if volatility decreases, the profits to
be realized by acredit spreader are quite small (perhaps not even enough to over
come the commission expense of removing the position), whereas anaked option
seller would benefit to agreater and more obvious extent.
The choice between naked writing and credit spreading should be made based
largely on the philosophy and psychological makeup of the trader himself. If one feels
uncomfortable with naked options, or if he doesn'thave the ability to watch the mar
ket pretty much all the time (or have someone watch it for him), or ifhe doesn'thave
the financial wherewithal to margin the positions and carry them until the stock hits
the break-even point, then naked writing is not for that trader.
Another factor that might affect the choice of strategy for the option seller is
what type of underlying instrument is being considered. Index options are by far the
best choices for naked option selling. Futures are next, and stocks are last. This is
because of the ways those various instruments behave; stocks have by far the great
est capability of making huge gap moves that are the bane of naked option selling. So,
if one has found expensive stock or futures options, that might lend more credence
to the credit spread strategy.
There is one other strategy that can be employed, upon occasion, when options
are expensive. It is called the volatility backspread, but its discussion will be deferred
until later in the chapter.
USING A PROBABILITY CALCULATOR
No matter which method is used to find options that are out of line, and no matter
which strategy is preferred by the trader, it is still necessary to use aprobability cal
culator to get ameaningful idea of whether or not the underlying has the ability to

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828 Part VI: Measuring and Trading Volatility
make the move to profitability (or not make the move into loss territory, if you're sell
ing options). This is where historical volatility plays abig part, for it is the input into
the probability calculator. In fact, no probability calculator will give reasonable pre
dictions without agood estimate of volatility. Please refer to the previous chapter for
amore in-depth discussion of probability calculators and stock price distributions.
The use of probability analysis also mitigates some of the problems inherent in
the method of selection that compares implied and historical volatilities. If the prob
abilities are good for success, then we might not care so much whether the options
are currently in alow percentile of implied volatility or not (although we still would
not want to buy volatility when the options were in ahigh percentile of implied
volatility and we would not want to sell options that are in alow percentile).
In using the probability calculator, one first selects astrategy (straddle buying,
for example, if options are cheap) and then calculates the break-even points as
demonstrated in the previous section. Then the probability calculator is used to
determine what the chances are of the underlying instrument ever trading at one or
the other of those break-even prices at any time during the life of the option position.
It was shown in the previous chapter that a Monte Carlo simulation using the fat tail
distribution is best used for this process.
An attractive volatility buying situation should have probabilities in excess of
80% of the underlying ever exceeding the break-even point, while an attractive
volatility selling situation should have probabilities of less than 25% of ever trading
at prices that would cause losses. The volatility seller can, of course, heavily influence
those probabilities by choosing options that are well out-of-the-money. As noted
above, the volatility seller should, in fact, calculate the probabilities on several dif
ferent striking prices, striving to find abalance between high probability of success
and the ability to take in enough premium to make the risk worthwhile.
Example: The OEX Index is trading at 650. Suppose that one has determined that
volatilities are too high and wants to analyze the sale of some naked options.
Furthermore, suppose that the choices have been narrowed down to selling the
September options, which expire in about five weeks. The main choices under con
sideration are those in Table 39-2. The option prices in this example, being index
options, reflect avolatility skew (to be discussed later) to make the example realistic.
The two right-hand columns should be rejected because the probabilities of
the stock hitting one or the other of the striking prices prior to expiration are too
high well in excess of the 25% guideline mentioned earlier. That leaves the
September 500-800 strangle or the September 550-750 strangle to consider. The
probabilities are best for the farthest out-of-the-money options (September 500-
800 strangle), but the options are selling at such small prices that they will not pro-

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Chapter 39: Volatility Trading Techniques 829
TABLE 39-2
September September September September
800 call 750 call 730 call 700 call
September September September September
Naked sale: 500 put 550 put 570 put 600 put
Call price 0.20 1.50 3.50 8.80
Put price 0.40 2.00 3.70 8.50
Probability of call strike 4% 17% 30% 44%
Probability of put strike 1% 11% 20% 40%
vide much of areturn even if they expire worthless. Remember that one is
required to establish the position with margin of at least 10% of the index price
for naked index options, which would be $6,500 in this case. In fact, it has been
recommended that one margin the position at the striking price itself (15% of 800,
or $12,000 in this case). So, taking in only $60, less commissions, for the sale of
the September 500-800 strangle doesn'tseem to provide enough of areward.
Thus, the best choice seems to be the September 550-750 strangle. One would be
making about $320 after commissions if the options expired worthless, and the
recommended margin would be 15% of 750 (the higher strike), or $11,250 - areturn of about 2.8% for one month. One cannot annualize these returns, for he
has no idea if the same option pricing structure will exist in five weeks, when these
options expire.
Other probabilities can be calculated as well. For example, suppose one has
decided to buy astraddle. He might want to know what the odds are not only of
breaking even, but also of making at least acertain percentage return- say 20%. One
could also calculate the probability of the stock moving 20% past the break-even
points. That distance - 20% - is areasonable figure to use because one would most
likely be taking some partial profits or adjusting his position if the stock did indeed
move that far.
USING STOCK PRICE HISTORY
All of the work done so far - determining which options are expensive, selecting astrategy, and calculating the probabilities of success - has been somewhat theoretical
in that we haven'tdone any "back testing" with regard to the volatility of the under
lying instruments. At this point, one should look at past prices to see if the stock has
been able to make large moves (whether or not such amove is desired).

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830 Part VI: Measuring and Trading Volatility
Example: Atrader is considering the purchase of the XYZ October 40 straddle for
11 points, with the stock at 39.60. The options are cheap and the probabilities of suc
cess appear to be good, according to the probability calculator. The question that now
needs to be asked and answered is this: "In the past, has this stock been able to move
11 points in 10 months (the time remaining in the straddle'slife)?" Or, more impor
tantly, since 11 divided by 39.60 is about 28%, "Has this stock been able to make
moves of 28% over 10 months, in the past?" The answers to these questions can be
readily obtained if stock price history data is available. One could even look at achart
of the stock and attempt to answer the questions himself without the aid of acom
puter, but computer analysis of the price history is more rigorous and is therefore
encouraged.
The answers can be expressed in the form of probabilities, much as the results
of the probability calculator are.
Suppose one determines that the stock has been able to move 11 points in 10
months 77% of the time in the past. That'sokay, but not great. However, when one
looks at the price chart ofXYZ, he sees that it traded at much lower prices - near $10
ashare - for along time before rising to its current levels. It would be very hard to
expect a $10 stock to move 11 points in 10 months. That'swhy the second figure, the
one involving the 28% move, is the more significant one. In this case, one might find
that XYZ has been able to move 28% in 10 months over 90% of the time in the past.
Now one has what appears to be adecent-looking straddle buy.
This analysis of past prices can be done in amore sophisticated manner. Rather than
just asking whether or not the stock has moved the required distance in the past, one
might want to see just how the stock'smovements "look." That is, there are acouple
of scenarios under which the past movements might look attractive, but upon closer
examination, one would not be so sanguine.
For example, what ifXYZ had repeatedly moved 28%, but never much more in
most of the IO-month periods that comprise its stock history? Then, one would be
less inclined to want to own this straddle.
Another scenario of past movements might be that XYZ had made moves that
one could not reasonably expect to be repeated. Perhaps there was ahuge gap down
on an earnings shortfall, or if it was an Internet stock around the tum of the millen
nium, it had ahuge move upward, followed by ahuge move downward. That would
be another nonrepeating type of move, because absent the Internet mania, the stock
might have been arather range-bound item both prior to and after the one huge,
round-trip move.

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832 Part VI: Measuring and Trading Volatility
moved two or three times that far with great frequency. Finally, there is acontinu
ity to the points on the histogram: There are some y-axis data points at almost all
points on the x-axis (between the minimum and maximum x-axis points). That is
good, because it shows that there has not been aclustering of movements by XYZ
that might have dominated past activity.
As for what is not a "good" histogram, we would not be so enamored of ahis
togram that showed ahuge cluster of points near and between the "-1" and 'T' points
on the X-axis. We want the stock to have shown an ability to move farther than just the
break-even distance, if possible. As an example, see Figure 39-5, which shows astock
whose movements rarely exceed the "-1" or "+l" points, and even when they do, they
don'texceed it by much. Most of these would be losing trades because, even though
the stock might have moved the required percentage, that was its maximum move
during the 10-month period, and there is no way that atrader would know to take
profits exactly at that time. The straddles described by the histogram in Figure 39-5
should not be bought, regardless of what the previous analyses might have shown.
Nor would it be desirable for the histogram to show alarge number of move
ments above the "+3" level on the histogram, with virtually nothing below that. Such
ahistogram would most likely be reflective of the spiky, Internet-type stock activity
that was referred to earlier as being unreasonable to expect that it might repeat itself.
In ageneral sense, one doesn'twant to see too many open spaces on the histogram's
X-axis; continuity is desired.
If the histogram is afavorable one, then the volatility analysis is complete. One
would have found mispriced options, with agood theoretical probability of profit,
whose past stock movements verify that such movements are feasible in the future.
ANOTHER APPROACH?
After having considered the descriptions of all of these analyses, one other approach
comes to mind: Use the past movements exclusively and ignore the other analyses
altogether. This sounds somewhat radical, but it is certainly avalid approach. It'smore like giving some rigor to the person who "knows" IBM can move 18 points and
who therefore wants to buy the straddle. If the histogram (study of past movements)
tells us that IBM does, indeed, have agood chance of moving 18 points, what do we
really care about the relationship of implied and historical volatility, or about the cur
rent percentiles of either type of volatility, or what atheoretical probability calcula
tor might say? In some sense, this is like comparing implied volatility (the price of the
straddle) with historical volatility (the history of stock price movements) in astrictly
practical sense, not using statistics.

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Chapter 39: Volatility Trading Techniques 833
In reality, one would have to be mindful of not buying overly expensive options ( or
selling overly cheap ones), because implied volatility cannot be ignored. However, the
price of the straddle itself, which is what determines the x-axis on the histogram, does
reflect option prices, and therefore implied volatility, in anontechnical sense. This author
suspects that alist of volatility trading candidates generated only by using past movements
would be arather long list. Therefore, as apractical matter, it may not be useful.
MORE THOUGHTS ON SELLING VOLATILITY
Earlier, it was promised that another, more complex volatility selling strategy would
be discussed. An option strategist is often faced with adifficult choice when it comes
to selling (overpriced) options in aneutral manner - in other words, "selling volatili
ty." Many traders don'tlike to sell naked options, especially naked equity options, yet
many forms of spreads designed to limit risk seem to force the strategist into adirec
tional (bullish or bearish) strategy that he doesn'treally want. This section addresses
the more daunting prospect of trying to sell volatility with protection in the equity
and futures option markets.
The quandary in trying to sell volatility is in trying to find aneutral strategy that
allows one to benefit from the sale of expensive options without paying too much for
ahedge - the offsetting purchase of equally expensive options. The simple strategy
that most traders first attempt is the credit spread. Theoretically, if implied volatility
were to fall during the time the credit spread position is in place, aprofit might be
realized. However, after commissions on four different options in and possibly out
(assuming one sold both out-of-the-money put and call spreads), there probably
wouldn'tbe any real profit left if the position were closed out early. In sum, there is
nothing really wrong with the credit spread strategy, but it just doesn'tseem like any
thing to get too excited about. What other strategy can be used that has limited risk
and would benefit from adecline in implied volatility? The highest-priced options are
the longer-term ones. If implied volatility is high, then if one can sell options such as
these and hedge them, that might be agood strategy.
The simplest strategy that has the desired traits is selling acalendar spread
that is, sell alonger-term option and hedge it by buying ashort-term option at the
same strike. True, both are expensive (and the near-term option might even have aslightly higher implied volatility than the longer-term one). But the longer-term one
trades with afar greater absolute price, so if both become cheaper, the longer-term
one can decline quite abit farther in points than the near-term one. That is, the vega
of the longer-term option is greater than the vega of the shorter-term one. When one
sells acalendar spread, it is called areverse calendar spread. The strategy was

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834 Part VI: Measuring and Trading Volatility
described in the chapter on reverse spreads. The reader might want to review that
chapter, not only for the description of the strategy, but also for the description of the
margin problems inherent in reverse spreads on stocks and indices.
One of the problems that most traders have with the reverse calendar spread is
that it doesn'tproduce very large profits. The spread can theoretically shrink to zero
after it is sold, but in reality it will not do so, for the longer-term option will retain
some amount of time value premium even if it is very deeply in- or out-of-the-money.
Hence the spread ·will never really shrink to zero.
Yet, there is another approach that can often provide larger profit potential and
still retain the potential to make money if implied volatility decreases. In some sense
it is amodification of the reverse calendar spread strategy that can create apoten
tially even more desirable position. The strategy, known as avolatility backspread,
involves selling along-term at-the-money option (just as in the reverse calendar
spread) and then buying agreater number of near-er term out-of-the-money options.
The position is generally constructed to be delta-neutral and it has anegative vega,
meaning that it will profit if implied volatility decreases.
Example: XYZ is trading at 115 in early June. Its options are very expensive. Atrad
er would like to construct avolatility backspread using the following two options:
Call Option
July 130 call:
October 120 call:
Price
2.50
13
Delta
0.26
0.53
Vega
0.10
0.27
Adelta-neutral position would be to buy 2 of the July 130 calls and sell one of
the October 120 calls. This would bring in acredit of 8 points. Also, it would have asmall negative position vega, since tvvo times the vega of the July calls minus one
times the vega of the October call is -0.07. That is, for each one percentage point
drop in implied volatility of XYZ options in general, this position would make $7 -
not alarge amount, but it is asmall position.
The profitability of the position is shown in Figure 39-6. This strategy has lim
ited risk because it does not involve naked options. In fact, if XYZ were to rally by agood distance, one could make large profits because of the extra long call.
Meanwhile, on the downside, if XYZ falls heavily, all the options would lose most of
their value and one would have aprofit approaching the amount of the initial credit
received. Furthermore, adecrease in implied volatility produces asmall profit as
well, although time decay may not be in the trader'sfavor, depending on exactly
which short-term options were bought. The biggest risk is that XYZ is exactly at 130
at July expiration, so any strategist employing this strategy should plan to close it out

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Chapter 39: Volatility Trading Techniques
FIGURE 39-6.
Volatility backspread neutral position.
Underlying Price
835
in advance of the near-term expiration. It should not be allowed to deteriorate to the
point of maximum loss.
Modifications to the strategy can be considered. One is to sell even longer-term
options and of course hedge them with the purchase of the near-term options. The
longer-term the option is, the bigger its vega will be, so adecrease in implied volatil
ity will cause the heftier-priced long-term option to decline more in price. This mod
ification is somewhat tempered, though, by the fact that when options get really
expensive, there is often atendency for the near-term options to be skewed. That is,
the near-term options will be trading with amuch higher implied volatility than will
the longer-term options. This is especially true for LEAPS options. For that reason,
one should make sure that he is not entering into asituation in which the shorter
term options could lose volatility while the longer-term ones more or less retain the
same implied volatility, as LEAPS options often do. This concept of differing volatil
ity between near- and long-term options was discussed in more detail in Chapter 36
on the basics of volatility trading. As asort of general rule, if one finds that the longer
term option has amuch lower implied volatility than the one you were going to buy,
this strategy is not recommended. As acorollary, then, it is unlikely that this strate
gy will work well with LEAPS options.
One other thing that you should analyze when looking for this type of trade is
whether it might be better to use the puts than the calls. For one thing, you can estab
lish aposition in which the heavy profitability is on the downside (as opposed to the
upside, as in the XYZ example above). Then, of course, having considered that, it
might actually behoove one to establish both the call spread and the put spread. If

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836 Part VI: Measuring and Trading Volatility
you do both, though, you create a "good news, bad news" situation. The good news
is that the maximum risk is reduced; for example, if XYZ goes exactly to 130 (the
worst point for the call spread), the companion put spread'scredit would reduce that
risk alittle. However, the bad news is that there is amuch wider range over which
there is not profit, since there are two spots where losses are more or less maximized
(at the strike price of the long calls and again at the strike price of the long puts).
Margin will be discussed only briefly, since it was addressed in the chapter on
reverse spreads. For both index and stock options, this strategy is considered to have
naked options - apreposterous assumption, since one can see from the profit graph
that the position is fully hedged until the near-term options expire. This raises the
capital requirement for nonmember traders. The margin anomaly is not aproblem
with futures options, however. For those options, one need only margin the differ
ence in the strikes, less any credit received, because that is the true risk of the posi
tion. In summary, the volatility trader who wants to sell volatility in equity and futures
options markets needs to be hedged, because gaps are prevalent and potentially very
costly. This strategy creates amore neutral, less price-dependent way to benefit if
implied volatility decreases, especially when compared with simple credit spreads.
SUMMARY: TRADING THE
VOLATILITY PREDICTION
Attempting to establish trades when implied volatility is out of line is atheoretically
attractive strategy. The process outlined above consisted of afew steps, employing
both statistical and theoretical analysis. In any case, though, probability calculators
must "say" that avolatility trade has good probabilities of success. It'smerely amat
ter of what criteria we apply to limit our choices before we run the probability analy
sis. So, it might be more useful to view volatility trading analysis in this light:
Step I: Use aselection criterion to limit the myriad of volatility trading choices. Any
of these could be used as the first criterion, but not all of them at once:
a. Require implied volatility to be at an extreme percentile.
b. Require historical and implied volatility to have alarge discrepancy
between them.
c. Interpret the chart of implied volatility to see if it has reversed trend.
Step 2: Use aprobability calculator to project whether the strategy can be expected
to be asuccess.
Step 3: Using past price histories, determine whether the underlying has been able
to create profitable trades in the past. (For example, if one is considering

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Chapter 39: Volatility Trading Techniques 837
buying astraddle, ask the question, "Has this stock been able to move far
enough, with great enough frequency, to make this straddle purchase prof
itable?") Use histograms to ensure that the past distribution of stock prices
is smooth, so that an aberrant, nonrepeatable move is not overly influenc
ing the results.
Each criterion from Step 1 would produce adifferent list of viable volatility
trading candidates on any given day. If aparticular candidate were to appear on more
than one of the lists, it might be the best situation of all.
TRADING THE VOLATILITY SKEW
In the early part of this chapter, it was mentioned that there are two ways in which
volatility predictions could be "wrong." The first was that implied volatility was out of
line. The second is that individual options on the same underlying instrument have
significantly different implied volatilities. This is called avolatility skew, and presents
trading opportunities in its own right.
DIFFERING IMPLIED VOLATILITIES ON THE SAME UNDERLYING SECURITY
The implied volatility of an option is the volatility that one would have to use as input
to the Black-Scholes model in order for the result of the model to be equal to the
current market price of the option. Each option will thus have its own implied volatil
ity. Generally, they will be fairly close to each other in value, although not exactly the
same. However, in some cases, there will be large enough discrepancies between the
individual implied volatilities to warrant the strategist'sattention. It is this latter con
dition of large discrepancies that will be addressed in this section.
Example: XYZ is trading at 45. The following option prices exist, along with their
implied volatilities:
Actual Implied
Option Price Volatility
January 45 call 2.75 41%
January 50 call 1.25 47%
January 55 call 0.63 53%
February 45 call 3.50 38%
February 50 call 4.00 45%

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838 Part VI: Measuring and Trading Volatility
Note that the implied volatilities of the individual options range from alow of
38% to ahigh of 53%. This is arather large discrepancy for options on the same
underlying security, but it is useful for exemplary purposes.
Aneutral strategy could be established by buying options with lower implied
volatilities and simultaneously selling ones with higher volatilities, such as buy 10
February 45 calls and sell 20 January 50 calls. Examples of neutral spreads will be
expanded upon in the next chapter, when more exact measures for determining how
many calls to buy and sell are discussed.
Before jumping into such aposition, the strategist should ask himself if there is
avalid reason why the different options have such different implied volatilities. As ageneralization, it might be fair to say that out-of-the-money options have slightly
higher implieds than at-the-money ones, and that longer-term options have lower
implieds than short-term ones. But there are many instances in which such is not the
case, so one must be careful not to overgeneralize.
Speculators often desire the lowest dollar-cost option available. Thus, in atakeover rumor situation, they would buy the out-of-the-moneys as opposed to the
higher-priced at- or in-the-moneys. If the out-of-the-moneys are extremely expen
sive because of atakeover rumor, then the strategist must be careful, because the
neutral strategy concept may lead him into selling naked calls. This is not to say
he should avoid the situation altogether; he may be able to structure aposition
with enough upside room to protect himself, or he may believe the rumors are
false.
Returning to the general topic of differing implied volatilities on the same
underlying stock, the strategist might ask how he is to determine if the discrepancies
between the individual options are significantly large to warrant attention. Amathe
matical approach is presented at the end of the next chapter in asection on advanced
mathematical concepts. Suffice it to say that there is away that the differences in the
various implieds can be reduced to asingle number - asort of "standard deviation of
the implieds" that is easy for the strategist to use. Alist of these numbers can be con
structed, comparing which stocks or futures might be candidates for this type of neu
tral spreading. On agiven day, the list is usually quite short - perhaps 20 stocks and
10 futures contracts will qualify.
The concept of the implied volatilities of various options on the same underly
ing stock remaining out of line with each other is one that needs more discussion. In
the following section, the idea of semipermanent distortion between the volatilities
of different striking prices is discussed.

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Chapter 39: Volatility Trading Techniques
VOLATILITY SKEWING
839
After the stock market crashed in 1987, index options experienced what has since
proven to be apermanent distortion: Out-of-the-money puts have remained more
expensive than out-of-the-money calls. Furthermore, out-of-the-money puts are
more expensive than at-the-money puts; out-of-the-money calls are cheaper than at
the-money calls. This distorted effect is due to several factors, but it is so deep-seat
ed that it has remained through all kinds of up and down markets since then. Other
markets, particularly futures markets, have also experienced along-lasting distortion
between the implied volatilities at various strikes.
The proper name given to this phenomenon is volatility skewing: the long-last
ing effect whereby options at different striking prices trade with differing implied
volatilities. It should be noted that the calls and puts at the same strike must trade
for the same implied volatility; otherwise, conversion or reversal arbitrage would
eliminate the difference. However, there is no true arbitrage between different strik
ing prices. Hence, arbitrage cannot eliminate volatility skewing.
Example: Volatility skewing exists in OEX index options. Assume the average volatil
ity of OEX and its options is 16%. With volatility skewing present, the implied volatil
ities at the various striking prices might look like this:
OEX: 580
Implied Volatility
Strike of Both Puts and Calls
550 22%
560 19%
570 17%
580 16%
590 15%
600 14%
610 13%
In this form of volatility skewing, the out-of-the-money puts are the most
expensive options; the out-of-the-money calls are the cheapest. This pattern of
implied volatilities is called areverse volatility skew or, alternatively, anegative
volatility skew.

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840 Part VI: Measuring and Trading Volatility
The causes of this effect stem from the stock market'spenchant to crash occa
sionally. Investors who want protection buy index puts; they don'tsell index futures
as much as they used to because of the failure of the portfolio insurance strategy dur
ing the 1987 crash. In addition, margin requirements for selling naked index puts
have increased, especially for market-makers, who are the main suppliers of naked
puts. Consequently, demand for index puts is high and supply is low. Therefore, out
of-the-money index puts are overly expensive.
This does not entirely explain why index calls are so cheap. Part of the reason
for that is that institutional traders can help finance the cost of those expensive index
puts by selling some out-of-the-money index calls. Such sales would essentially be
covered calls if the institution owned stocks, which it most certainly would. This strat
egy is called acollar.
This distortion in volatilities is not in accordance with the probability distribu
tion of stock prices. These distorted implied volatilities define adifferent probability
curve for stock movement. They seem to say that there is more chance of the market
dropping than there is of it rising. This is not true; in fact, just the opposite is true.
Refer to the reasons for using lognormal distribution to define stock price move
ments. Consequently, there are opportunities to profit from volatility skewing, if one
is able to hold the position until expiration.
It was shown in previous examples that one would attempt to sell the options
with higher implied volatilities and buy ones with lower implieds as ahedge. Hence,
for OEX traders, three strategies seem relevant:
1. Buy abear put spread in OEX.
Example: Buy 10 OEX June 560 puts
Sell IO OEX June 540 puts
2. Buy OEX puts and sell alarger number of out-of-the-money puts - aratio write
of put options.
Example: Buy 10 OEX June 560 puts
Sell 20 OEX June 550 puts
3. Sell OEX calls and buy alarger number of out-of-the-money calls - abackspread
of call options.
Example: Buy 20 OEX June 590 calls
Sell IO OEX June 580 calls
In all three cases, one is selling the higher implied volatility and buying options
with lower implied volatilities. The first strategy is asimple bear spread. While it will

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