756 Part VI: Measuring and Trading Volatility
Notice that the stock price is equal to the strike price (100). However, the deltas
are not at all equal. In fact, the delta of the call is 1.5 times that of the put (in absolute
value). One must buy three puts and two calls in order to have adelta-neutral posi
tion.
Most experienced option traders know that the delta of an at-the-money call is
somewhat higher than that of an at-the-money put. Consequently, they often esti
mate, without checking, that buying three puts and two calls produces adelta-neu
tral "straddle buy." However, consider asimilar situation, but with amuch higher
implied volatility- 110%, say.
AAA Common: 100; Implied Volatility: 110%
Option
AAA October 100 call
AAA October 1 00 put
Price
31.00
28.00
Delta
0.67
-0.33
The delta-neutral ratio here is two-to-one (67 divided by 33), not three-to-two
as in the earlier case - even though both stock prices are 100 and both sets of options
have six months remaining. This is abig difference in the delta-neutral ratio, espe
cially if one is trading alarge quantity of options. This shows how different levels of
implied volatility can alter one'sperception of what is aneutral position. It also points
out that one can'tnecessarily rely on his intuition; it is always best to check with amodel.
Carrying this thought astep further, one must be mindful of achange in implied
volatility if he wants to keep his position delta-neutral. If the implied volatility of AAA
options should drop significantly, the 2-to-lratio will no longer be neutral, even if the
stock is still trading at 100. Hence, atrader wishing to remain delta-neutral must
monitor not only changes in stock price, but changes in implied volatility as well. For
more complex strategies, one will also find the delta-neutral ratio changing due to achange in implied volatility.
The preceding examples summarize the major variables that might affect the
vega and also show how vega affects things other than itself, such as delta and, there
fore, delta neutrality. By the way, the vega of the underlying is zero; an increase in
implied volatility does not affect the price of the underlying instrument at all, in the
ory. In reality, if options get very expensive (i.e., implied volatility spikes up), that
usually brings traders into astock and so the stock price will change. But that'snot amathematical relationship, just amarket cause-and-effect relationship.

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Chapter 37: How Volatility Affects Popular Strategies
POSITION VEGA
757
As can be done with delta or with any other of the partial derivatives of the model,
one can compute aposition vega - the vega of an entire position. The position vega
is determined by multiplying the individual option vegas by the quantity of options
bought or sold. The "position vega" is merely the quantity of options held, times the
vega, times the shares per options ( which is normally 100).
Example: Using asimple call spread as an example, assume the following prices
exist:
Security Position Vega Position Vego
XYZ Stock No position
XYZ July 50 call Long 3 calls 0.098 +0.294
XYZ July 70 call Short 5 calls 0.076 -0.380
Net Position Vega: -0.086
This concept is very important to avolatility trader, for it tells him if he has con
structed aposition that is going to behave in the manner he expects. For example,
suppose that one identifies expensive options, and he figures that implied volatility
will decrease, eventually becoming more in line with its historical norms. Then he
would want to construct aposition with anegative position vega. Anegative position
vega indicates that the position will profit if implied volatility decreases. Conversely,
abuyer of volatility - one who identifies some underpriced situation - would want to
construct aposition with apositive position vega, for such aposition will profit if
implied volatility rises. In either case, other factors such as delta, time to expiration,
and so forth will have an effect on the position'sactual dollar profit, but the concept
of position vega is still important to avolatility trader. It does no good to identify
cheap options, for example, and then establish some strange spread with anegative
position vega. Such aconstruct would be at odds with one'sintended purpose - in
this case, buying cheap options.
OUTRIGHT OPTION PURCHASES AND SALES
Let us now begin to investigate the affects of implied volatility on various strategies,
beginning with the simplest strategy of all - the outright option purchase. It was
already shown that implied volatility affects the price of an individual call or put in a

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758 Part VI: Measuring and Trading Volatility
direct manner. That is, an increase in implied volatility will cause the option price to
rise, while adecrease in volatility will cause adecline in the option price. That piece
of information is the most important one of all, for it imparts what an option trader
needs to know: An explosion in implied volatility is aboon to an option owner, but
can be adevastating detriment to an option seller, especially anaked option seller.
Acouple of examples might demonstrate more clearly just how powerful the
effect of implied volatility is, even when there isn'tmuch time remaining in the life
of an option. One should understand the notion that an increase in implied volatility
can overcome days, even weeks, of time decay. This first example attempts to quan
tify that statement somewhat.
Example: Suppose that XYZ is trading at 100 and one is interested in analyzing a 3-
month call with striking price of 100. Furthermore, suppose that implied volatility is
currently at 20%. Given these assumptions, the Black-Scholes model tells us that the
call would be trading at aprice of 4.64.
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
3 months
20%
4.64
Now, suppose that amonth passes. If implied volatility remained the same
(20% ), the call would lose nearly apoint of value due to time decay. However, how
much would implied volatility have had to increase to completely counteract the
effect of that time decay? That is, after amonth has passed, what implied volatility
will yield acall price of 4.64? lt turns out to be just under 26%.
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
2 months
25.9%
4.64
What would happen after another month passes? There is, of course, some
implied volatility at which the call would still be worth 4.64, but is it so high as to be
unreasonable? Actually, it turns out that if implied volatility increases to about 38%,
the call will still be worth 4.64, even with only one month of life remaining:

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Chapter 37: How Volatility Affects Popular Strategies
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
1 month
38.1%
4.64
759
So, if implied volatility increases from 20% to 26% over the first month, then
this call option would still be trading at the same price - 4.64. That'snot an unusual
increase in implied volatility; increases of that magnitude, 20% to 26%, happen all
the time. For it to then increase from 26% to 38% over the next month is probably
less likely, but it is certainly not out of the question. There have been many times in
the past when just such an increase has been possible - during any of the August,
September, or October bear markets or mini-crashes, for example. Also, such an
increase in implied volatility might occur if there were takeover rumors in this stock,
or if the entire market became more volatile, as was the case in the latter half of the
1990s.
Perhaps this example was distorted by the fact that an implied volatility of 20%
is afairly low number to begin with. What would asimilar example look like if one
started out with amuch higher implied volatility - say, 80%?
Example: Making the same assumptions as in the previous example, but now setting
the implied volatility to amuch higher level of 80%, the Black-Scholes model now
says that the call would be worth aprice of 16.45:
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
3 months
80%
16.45
Again, one must ask the question: "If amonth passes, what implied volatility
would be necessary for the Black-Scholes model to yield aprice of 16.45?" In this
case, it turns out to be an implied volatility of just over 99%.
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
2 months
99.4%
16.45

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758 Part VI: Measuring and Trading Volatility
direct manner. That is, an increase in implied volatility will cause the option price to
rise, while adecrease in volatility will cause adecline in the option price. That piece
of information is the most important one of all, for it imparts what an option trader
needs to know: An explosion in implied volatility is aboon to an option owner, but
can be adevastating detriment to an option seller, especially anaked option seller.
Acouple of examples might demonstrate more clearly just how powerful the
effect of implied volatility is, even when there isn'tmuch time remaining in the life
of an option. One should understand the notion that an increase in implied volatility
can overcome days, even weeks, of time decay. This first example attempts to quan
tify that statement somewhat.
Example: Suppose that XYZ is trading at 100 and one is interested in analyzing a 3-
month call with striking price of 100. Furthermore, suppose that implied volatility is
currently at 20%. Given these assumptions, the Black-Scholes model tells us that the
call would be trading at aprice of 4.64.
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
3 months
20%
4.64
Now, suppose that amonth passes. If implied volatility remained the same
(20% ), the call would lose nearly apoint of value due to time decay. However, how
much would implied volatility have had to increase to completely counteract the
effect of that time decay? That is, after amonth has passed, what implied volatility
will yield acall price of 4.64? It turns out to be just under 26%.
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
2 months
25.9%
4.64
What would happen after another month passes? There is, of course, some
implied volatility at which the call would still be worth 4.64, but is it so high as to be
unreasonable? Actually, it turns out that if implied volatility increases to about 38%,
the call will still be worth 4.64, even with only one month of life remaining:

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Chapter 37: How Volatility Affects Popular Strategies
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
1 month
38.1%
4.64
759
So, if implied volatility increases from 20% to 26% over the first month, then
this call option would still be trading at the same price 4.64. That'snot an unusual
increase in implied volatility; increases of that magnitude, 20% to 26%, happen all
the time. For it to then increase from 26% to 38% over the next month is probably
less likely, but it is certainly not out of the question. There have been many times in
the past when just such an increase has been possible - during any of the August,
September, or October bear markets or mini-crashes, for example. Also, such an
increase in implied volatility might occur if there were takeover rumors in this stock,
or if the entire market became more volatile, as was the case in the latter half of the
1990s.
Perhaps this example was distorted by the fact that an implied volatility of 20%
is afairly low number to begin with. What would asimilar example look like if one
started out with amuch higher implied volatility say, 80%?
Example: Making the same assumptions as in the previous example, but now setting
the implied volatility to amuch higher level of 80%, the Black-Scholes model now
says that the call would be worth aprice of 16.45:
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
3 months
80%
16.45
Again, one must ask the question: "If amonth passes, what implied volatility
would be necessary for the Black-Scholes model to yield aprice of 16.45?" In this
case, it turns out to be an implied volatility of just over 99%.
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
2 months
99.4%
16.45

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760 Part VI: Measuring and Trading Volatility
Finally, to be able to completely compare this example with the previous one, it
is necessary to see what implied volatility would have to rise to in order to offset the
effect of yet another month'stime decay. It turns out to be over 140%:
Stock Price:
Strike Price:
Time Remaining:
Implied Volatility:
Theoretical Call Value:
100
100
1 month
140.9%
16.45
Table 37-4 summarizes the results of these examples, showing the levels to
which implied volatility would have to rise to maintain the call'svalue as time passes.
Are the volatility increases in the latter example less likely to occur than the
ones in the former example? Probably yes - certainly the last one, in which implied
volatility would have to increase from 80% to nearly 141 % in order to maintain the
call'svalue. However, in another sense, it may seem more reasonable: Note that the
increase in volatility from 20% to 26% is a 30% increase. That is, 20% times 1.30
equals 26%. That'swhat'srequired to maintain the call'svalue for the lower volatility
over the first month - an increase in the magnitude of implied volatility of 30%. At
the higher volatility, though, an increase in magnitude of only about 25% is required
(from 80% to 99%). Thus, in those terms, the two appear on more equal footing.
What makes the top line of Table 37-4 appear more likely than the bottom line
is merely the fact that an experienced option trader knows that many stocks have
implied volatilities that can fluctuate in the 20% to 40% range quite easily. However,
there are far fewer stocks that have implied volatilities in the higher range. In fact,
until the Internet stocks got hot in the latter portion of the 1990s, the only ones with
volatilities like those were very low-priced, extremely volatile stocks. Hence one'sexperience factor is lower with such high implied volatility stocks, but it doesn'tmean
that the volatility fluctuations appearing in Table 37-4 are impossible.
If the reader has access to asoftware program containing the Black-Scholes
model, he can experiment with other situations to see how powerful the effect of
implied volatility is. For example, without going into as much detail, if one takes the
case of a 12-month option whose initial implied volatility is 20%, all it takes to main-
TABLE 37-4
Initial Implied
Volatility
20%
80%
Volatility Leveled Required to Maintain Call Value ...
... After One Month ... After Two Months
26%
99%
38%
141%

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Chapter 37: How Volatility Affects Popular Strategies 761
tain the call'svalue over a 6-month time period is an increase in implied volatility to
27%. Taken from the viewpoint of the option seller, this is perhaps most enlighten
ing: If you sell aone-year (LEAPS) option and six months pass, during which time
implied volatility increases from 20% to 27% - certainly quite possible -you will have
made nothing! The call will still be selling for the same price, assuming the stock is
still selling for the same price.
Finally, it was mentioned earlier that implied volatility often explodes during amarket crash. In fact, one could determine just how much of an increase in implied
volatility would be necessary in amarket crash in order to maintain the call'svalue.
This is similar to the first example in this section, but now the stock price will be
allowed to decrease as well. Table 37-5, then, shows what implied volatility would be
required to maintain the call'sinitial value (aprice of 4.64), when the stock price falls.
The other factors remain the same: time remaining (3 months), striking price (100),
and interest rate (5% ). Again, this table shows instantaneous price changes. In real
life, aslightly higher implied volatility would be necessary, because each market crash
could take aday or two.
Thus, from Table 37-5, one could say that even if the underlying stock dropped
20 points (which is 20% in this case) in one day, yet implied volatility exploded from
20% to 67% at the same time, the call'svalue would be unchanged! Could such an
outrageous thing happen? It has: In the Crash of '87, the market plummeted 22% in
one day, while the Volatility Index ($VIX) theoretically rose from 36% to 150% in one
day. In fact, call buyers of some $OEX options actually broke even or made alittle
money due to the explosion in implied volatility, despite the fact that the worst mar
ket crash in history had occurred.
If nothing else, these examples should impart to the reader how important it is
to be aware of implied volatility at the time an option position is established. If you
are buying options, and you buy them when implied volatility is "low," you stand to
TABLE 37-5
Stock Price
100
95
90
85
80
75
70
Implied Volatility Necessary for Call to Maintain Value
20% (the initial parameters)
33%
44%
55%
67%
78%
89%

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762 Part VI: Measuring and Trading Volatility
benefit if implied volatility merely returns to "normal" levels while you hold the posi
tion. Of course, having the underlying increase in price is also important.
Conversely, an option seller should be keenly aware of implied volatility when
the option is initially sold - perhaps even more so than the buyer of an option. This
pertains equally well to naked option writers and to covered option writers. If implied
volatility is "too low" when the option writing position is established, then an increase
(or worse, an explosion) in implied volatility will be very detrimental to the position,
completely overcoming the effects of time decay. Hence, an option writer should not
just sell options because he thinks he is collecting time decay each day that passes.
That may be true, but an increase in implied volatility can completely domin.ate what
little time decay might exist, especially for alonger-term option.
In asimilar manner, adecrease in implied volatility can be just as important.
Thus, if the call buyer purchases options that are "too costly," ones in which implied
volatility is "too high," then he could lose money even if the underlying makes amod
est move in his favor.
In the next chapters, the topic of just how an option buyer or seller should
measure implied volatility to determine what is "too low" or "too high" will be dis
cussed. For now, suffice it to grasp the general concept that achange in implied
volatility can have substantial effects on an option'sprice far greater effects than the
passage of time can have.
In fact, all of this calls into question just exactly what time value premium is.
That part of an option'svalue that is not intrinsic value is really affected much more
by volatility than it is by time decay, yet it carries the term "time value premium."
TIME VALUE PREMIUM IS A MISNOMER
Many (perhaps novice) option traders seem to think of time as the main antagonist to
an option buyer. However, when one really thinks about it, he should realize that the
portion of an option that is not intrinsic value is really much more related to stock
price movement and/or volatility than anything else, at least in the short term. For
this reason, it might be beneficial to more closely analyze just what the "excess value"
portion of an option represents and why abuyer should not primarily think of it as
time value premium.
An option'sprice is composed of two parts: (1) intrinsic value, which is the "real"
part of the option'svalue - the distance by which the option is in-the-money, and (2)
"excess value" - often called time value premium. There are actually five factors that
affect the "excess value" portion of an option. Eventually, time will dominate them

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Chapter 37: How Volah'lity Affects Popular Strategies 763
all, but the longer the life of the option, the more the other factors influence the
"excess value."
The five factors influencing excess value are:
1. stock price movements,
2. changes in implied volatility,
3. the passage of time,
4. changes in the dividend (if any exist), and
5. changes in interest rates.
Each is stated in terms of amovement or change; that is, these are not static
things. In fact, to measure them one uses the "greeks": delta, vega, theta, (there is no
"greek" for dividend change), and rho. Typically, the effect of achange in dividend or
achange in interest rate is small (although alarge dividend change or an interest rate
change on avery long-term option can produce visible changes in the prices of
options).
If everything remains static, then time decay will eventually wipe out all of the
excess value of an option. That'swhy it'scalled time value premium. But things don'tever remain static, and on adaily basis, time decay is small, so it is the remaining two
factors that are most important.
Example: XYZ is trading at 82 in late November. The January 80 call is trading at 8.
Thus, the intrinsic value is 2 (82 minus 80) and the excess value is 6 (8 minus 2). If
the stock is still at 82 at January expiration, the option will of course only be worth 2,
and one will say that the 6 points of excess value that was lost was due to time decay.
But on that day in late November, the other factors are much more dominant.
On this particular day, the implied volatility of this option is just over 50%. One
can determine that the call'sgreeks are:
Delta: 0.60
Vega: 0.13
Theta: -0.06
This means, for example, that time decay is only 6 cents per day. It would
increase as time went by, but even with aday or so to go, theta would not increase
above about 20 cents unless volatility increased or the stock moved closer to the
strike price.
From the above figures, one can see - and this should be intuitively appealing that
the biggest factor influencing the price of the option is stock price movement (delta).

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764 Part VI: Measuring and Trading VolatiRty
It'salittle unfair to say that, because it'sconceivable (although unlikely) that volatil
ity could jump by alarge enough margin to become agreater factor than delta for
one day'smove in the option. Furthermore, since this option is composed mostly of
excess value, these more dominant forces influence the excess value more than time
decay does.
There is adirect relationship between vega and excess value. That is, if implied
volatility increases, the excess value portion of the option will increase and, if implied
volatility decreases, so will excess value.
The relationship between delta and excess value is not so straightforward. The
farther the stock moves away from the strike, the more this will have the effect of
shrinking the excess value. If the call is in-the-money (as in the above example), then
an increase in stock price will result in adecrease of excess value. That is, adeeply in
the-money option is composed primarily of intrinsic value, while excess value is quite
small. However, when the call is out-of-the-money, the effect is just the opposite:
Then, an increase in call price will result in an increase in excess value, because the
stock price increase is bringing the stock closer to the option'sstriking price.
For some readers, the following may help to conceptualize this concept. The
part of the delta that addresses excess value is this:
Out-of-the-money call: 100% of the delta affects the excess value.
In-the-money call: "1.00 minus delta" affects the excess value. (So, if acall is very
deeply in-the-money and has adelta of 0.95, then the delta only has 1.00 - 0.95,
or 0.05, room to increase. Hence it has little effect on what small amount of
excess value remains in this deeply in-the-money call.)
These relationships are not static, of course. Suppose, for example, that in the
same situation of the stock trading at 82 and the January 80 call trading at 8, there is
only week remaining until expiration! Then the implied volatility would be 155%
(high, but not unheard of in volatile times). The greeks would bear asignificantly dif
ferent relationship to each other in this case, though:
Delta: 0.59
Vega: 0.044
Theta: -0 .5 1
This very short-term option has about the same delta as its counterpart in the previ
ous example (the delta of an at-the-money option is generally slightly above 0.50).
Meanwhile, vega has shrunk. The effect of achange in volatility on such ashort-term
option is actually about athird of what it was in the previous example. However, time
decay in this example is huge, amounting to half apoint per day in this option.

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Chapter 37: How Volatility Affects Popular Strategies 765
So now one has the idea of how the excess value is affected by the "big three"
of stock price movement, change in implied volatility, and passage of time. How can
one use this to his advantage? First of all, one can see that an option'sexcess value
may be due much more to the potential volatility of the underlying stock, and there
fore to the option'simplied volatility, than to time.
As aresult of the above information regarding excess value, one shouldn'tthink
that he can easily go around selling what appear to be options with alot of excess
value and then expect time to bring in the profits for him. In fact, there may be alot
of volatility both actual and implied - keeping that excess value nearly intact for afairly long period of time. In fact, in the coming chapters on volatility estimation, it
will be shown that option buyers have amuch better chance of success than conven
tional wisdom has maintained.
VOLATILITY AND THE PUT OPTION
While it is obvious that an increase in implied volatility ½ill increase the price of aput
option, much as was shown for acall option in. the preceding discussion, there are
certain differences between aput and acall, so alittle review of the put option itself
may be useful. Aput option tends to lose its premium fairly quickly as it becomes an
in-the-money option. This is due to the realities of conversion arbitrage. In acon
version arbitrage, an arbitrageur or market-maker buys stock and buys the put, while
selling the call. If he carries the position to expiration, he will have to pay carrying
costs on the debit incurred to establish the position. Furthermore, he would earn any
dividends that might be paid while he holds the position. This information was pre
sented in aslightly different form in the chapter on arbitrage, but it is recounted
here:
In aperfect world, all option prices would be so accurate that there would be
no profit available from aconversion. That is, the following equation (1) would apply:
(1) Call price+ Strike price - Stock price - Put price+ Dividend- Carrying cost= 0
where carrying cost = strike price/ (1 + r)tt = time to expiration
r = interest rate
Now, it is also known that the time value premium of aput is the amount by which
its value exceeds intrinsic value. The intrinsic value of an in-the-money put option is
merely the difference between the strike price and the stock price. Hence, one can
write the following equation (2) for the time value premium (TVP) of an in-the
money put option:

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766 Part VI: Measuring and Trading Volatility
(2) Put TVP = Put price - Strike price + Stock price
The arbitrage equation, (1), can be rewritten as:
(3) Put price - Strike price+ Stock price= Call price+ Dividends - Carrying cost
and substituting equation (2) for the terms in equation (3), one arrives at:
( 4) Put TVP = Call price + Dividends - Carrying cost
In other words, the time value premium of an in-the-money put is the same as the
(out-of-the-money) call price, plus any dividends to be ea med until expiration, less
any carrying costs over that same time period.
Assuming that the dividend is small or zero (as it is for most stocks), one can see
that an in-the-money put would lose its time value premium whenever carrying costs
exceed the value of the out-of-the-money call. Since these carrying costs can be rel
atively large ( the carrying cost is the interest being paid on the entire debit of the
position - and that debit is approximately equal to the strike price), they can quickly
dominate the price of an out-of-the-money call. Hence, the time value premium of
an in-the-money put disappears rather quickly.
This is important information for put option buyers, because they must under
stand that aput won'tappreciate in value as much as one might expect, even when
the stock drops, since the put loses its time value premium quickly. It'seven more
important information for put sellers: Ashort put is at risk of assignment as soon as
there is no time value premium left in the put. Thus, aput can be assigned well in
advance of expiration even a LEAPS put!
Now, returning to the main topic of how implied volatility affects aposition, one
can ask himself how an increase or decrease in implied volatility would affect equa
tion ( 4) above. If implied volatility increases, the call price would increase, and if the
increase were great enough, might impart some time value premium to the put.
Hence, an increase in implied volatility also may increase the price of aput, but if the
put is too far in-the-nwney, amodest increase in implied volatility still won'tbudge
the put. That is, an increase in implied volatility would increase the value of the call,
but until it increases enough to be greater than the carrying costs, an in-the-money
put will remain at parity, and thus ashort put would still remain at risk of assignment.
STRADDLE OR STRANGLE BUYING AND SELLING
Since owning astraddle involves owning both aput and acall with the same terms,
it is fairly evident that an increase in implied volatility will be very beneficial for astraddle buyer. Asort of double benefit occurs if implied volatility rises, for it will

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Chapter 37: How Volatility Affects Popular Strategies 767
positively affect both the put and the call in along straddle. Thus, if astraddle buyer
is careful to buy straddles in situations in which implied volatility is "low," he can
make money in one of two ways. Either (1) the underlying price makes amove great
enough in magnitude to exceed the initial cost of the straddle, or (2) implied volatil
ity increases quickly enough to overcome the deleterious effects of time decay.
Conversely, astraddle seller risks just the opposite - potentially devastating loss
es if implied volatility should increase dramatically. However, the straddle seller can
register gains faster than just the rate of time decay would indicate if implied volatil
ity decreases. Thus, it is very important when selling options - and this applies to cov
ered options as well as to naked ones - to sell only when implied volatility is "high."
Astrangle is the same as astraddle, except that the call and put have different
striking prices. Typically, the call strike price is higher than the put strike price.
Naked option sellers often prefer selling strangles in which the options are well out
of-the-money, so that there is less chance of them having any intrinsic value when
they expire. Strangles behave much like straddles do with respect to changes in
implied volatility.
The concepts of straddle ownership will be discussed in much more detail in the
following chapters. Moreover, the general concept of option buying versus option
selling will receive agreat deal of attention.
CALL BULL SPREADS
In this section, the bull spread strategy will be examined to see how it is affected by
changes in implied volatility. Let'slook at acall bull spread and see how implied
volatility changes might affect the price of the spread if all else remains equal. Make
the following assumptions:
Assumption Set 1 :
Stock Price: 1 00
Time to Expiration: 4 months
Position: long Call Struck at 90
Short Call Struck at 110
Ask yourself this simple question: If the stock remains unchanged at 100, and implied
volatility increases dramatically, will the price of the 90-110 call bull spread grow or
shrink? Answer before reading on.
The truth is that, if implied volatility increases, the price of the spread will
shrink. Iwould suspect that this comes as something of asurprise to agood number
of readers. Table 37-6 contains some examples, generated from a Black-Scholes

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768
TABLE 37-6
Implied
Volatility
20%
30%
40%
50%
60%
70%
80%
Stock Price = I 00
Part VI: Measuring and Trading VolatHity
90-110 Call
Bull Spread
(Theoretical Value)
10.54
9.97
9.54
9.18
8.87
8.58
8.30
model, using the assumptions stated above, the most important of which is that the
stock is at 100 in all cases in this table.
One should be aware that it would probably be difficult to actually trade the
spread at the theoretical value, due to the bid-asked spread in the options.
Nevertheless, the impact of implied volatility is clear.
One can quantify the amount by which an option position will change for each
percentage point of increase in implied volatility. Recall that this measure is called
the vega of the option or option position. In acall bull spread, one would subtract the
vega of the call that is sold from that of the call that is bought in order to arrive at the
position vega of the call bull spread. Table 37-7 is areprint of Table 37-6, but now
including the vega.
Since these vegas are all negative, they indicate that the spread will shrink in
value if implied volatility rises and that the spread will expand in value if implied
TABLE 37-7
90-110 Call
Implied Bull Spread Position
Volatility (Theoretical Value) Vega
20% 10.54 -0.67
30% 9.97 -0.48
40% 9.54 -0.38
50% 9.18 -0.33
60% 8.87 -0.30
70% 8.58 -0.28
80% 8.30 -0.26

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Chapter 37: How VolatHity Afleds Popular Strategies 769
volatility decreases. Again, these statements may seem contrary to what one would
expect from abullish call position.
Of course, it'shighly unlikely that implied volatility would change much in the
course of just one day while the stock price remained unchanged. So, to get abet
ter handle on what to expect, one really to needs to look at what might happen at
some future time (say acouple of weeks hence) at various stock prices. The graph
in Figure 37-3 begins the investigation of these more complex scenarios.
The profit curve shown in Figure 37-3 makes certain assumptions: (1) The bull
spread assumes the details in Assumption Set 1, above; (2) the spread was bought
with an implied volatility of 20% and remained at that level when the profit picture
above was drawn; and (3) 30 days have passed since the spread was bought. Under
these assumptions, the profit graph shows that the bull spread conforms quite well to
what one would expect; that is, the shape of this curve is pretty much like that of abull spread at expiration, although if you look closely you'll see that it doesn'twiden
out to nearly its maximum gain or loss potential until the stock is well above llO or
below 90 the strike prices used in the spread.
Now observe what happens if one keeps all the other assumptions the same,
except one. In this case, assume implied volatility was 80% at purchase and remains
at 80% one month later. The comparison is shown in Figure 37-4. The 80% curve is
overlaid on top of the 20% curve shown earlier. The contrast is quite startling.
Instead of looking like abull spread, the profit curve that uses 80% implied volatili-
FIGURE 37-3.
Bull spread profit picture in 30 days, at 20% IV.
1000
500
<J)
<J)
0 0 ...J
?i:: 70 80 90 100 110 120 130 140
ea.
"' -500
Stock Price

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770 Part VI: Measuring and Trading Volatility
FIGURE 37-4.
Bull spread profit picture in 30 days.
1000
500
iv=80%
(/)
(/)
0
0 -' :;:,
70 801 90 e
~
a.
fl'>
-500
130 140
iv= 20%
-1000 Stock
ty is arather flat thing, sloping only slightly upward - and exhibiting far less risk and
reward potential than its lower implied volatility counterpart. This points out anoth
er important fact: For volatile stocks, one cannot expect a 4-rrwnth bull spread to
expand or contract much during the first rrwnth of life, even if the stock makes asub
stantial rrwve. Longer-term spreads have even less movement.
As acorollary, note that if implied volatility shrinks while the stock rises, the
profit outlook will improve. Graphically, using Figure 37-4, if one'sprofit picture
moves from the 80% curve to the 20% curve on the right-hand side of the chart, that
is apositive development. Of course, if the stock drops and the implied volatility
drops too, then one'slosses would be worse - witness the left-hand side of the graph
in Figure 37-4.
Agraph could be drawn that would incorporate other implied volatilities, but
that would be overkill. The profit graphs of the other spreads from Tables 37-6 or
37-7 would lie between the two curves shown in Figure 37-4.
If this discussion had looked at bull spreads as put credit spreads instead of call
debit spreads, perhaps these conclusions would not have seemed so unusual.
Experienced option traders already understand much of what has been shown here,
but less experienced traders may find this information to be different from what they
expected.
Some general facts can be drawn about the bull spread strategy. Perhaps the
most important one is that, if used on avolatile stock, you won'tget much expansion
in the spread even if the stock makes anice move upward in your favor. In fact, for

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Cbapter 37: How Volatility Affects Popular Strategies 771
high implied volatility situations, the bull spread won'texpand out to its maximum
price until expiration draws nigh. That can be frustrating and disappointing.
Often, the bull spread is established because the option trader feels the options
are "too expensive" and thus the spread strategy is away to cut down on the total
debit invested. However, the ultimate penalty paid is great. Consider the fact that,
if the stock rose from 100 to 130 in 30 days, any reasonable four-month call pur
chase (i.e., with astrike initially near the current stock price) would make anice
profit, while the bull spread barely ekes out a 5-point gain. To wit, the graph in
Figure 37-5 compares the purchase of the at-the-money call with astriking price of
100 and the 90-110 call bull spread, both having implied volatility of 80%. Quite
clearly, the call purchase dominates to agreat extent on an upward move. Of course,
the call purchase does worse on the downside, but since these are bullish strategies,
one would have to assume that the trader had apositive outlook for the stock when
the position was established. Hence, what happens on the downside is not primary
in his thinking.
The bull spread and the call purchase have opposite position vegas, too. That is,
arise in implied volatility will help the call purchase but will harm the bull spread
( and vice versa). Thus, the call purchase and the bull spread are not very similar posi
tions at all.
If one wants to use the bull spread to effectively reduce the cost of buying an
expensive at-the-money option, then at least make sure the striking prices are quite
FIGURE 37-5.
Call buy versus bull spread in 30 days; IV = 80%.
Cl)
~
2500
2000
1500
1000
e 500
Cl.
-500
-1000
Outright Call Buy
Bull Spread
---
140
Stock

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772 Part VI: Measuring and Trading Volatility
wide apart. That will allow for areasonable amount of price appreciation in the bull
spread if the underlying rises in price. Also, one might want to consider establishing
the bull spread with striking prices that are both out-of-the-money. Then, if the stock
rallies strongly, agreater percentage gain can be had by the spreader. Still, though,
the facts described above cannot be overcome; they can only possibly be mitigated
by such actions.
A FAMILIAR SCENARIO?
Often, one may be deluded into thinking that the two positions are more similar than
they are. For example, one does some sort of analysis - it does not matter if it'sfun
damental or technical - and comes to aconclusion that the stock ( or futures contract
or index) is ready for abullish move. Furthermore, he wants to use options to imple
ment his strategy. But, upon inspecting the actual market prices, he finds that the
options seem rather expensive. So, he thinks, "Why not use abull spread instead? It
costs less and it'sbullish, too."
Fairly quickly, the underlying moves higher - agood prediction by the trader,
and atimely one as well. If the move is aviolent one, especially in the futures mar
ket, implied volatility might increase as well. If you had bought calls, you'dbe ahappy
camper. But if you bought the bull spread, you are not only highly disappointed, but
you are now facing the prospect of having to hold the spread for several more weeks
(perhaps months) before your spread widens out to anything even approaching the
maximum profit potential.
Sound familiar? Every option trader has probably done himself in with this line
of thinking at one time or another. At least, now you know the reason why: High or
increasing implied volatility is not afriend of the bull spread, while it is afriendly ally
of the outright call purchase. Somewhat surprisingly, many option traders don'treal
ize the difference between these two strategies, which they probably consider to be
somewhat similar in nature.
So, be careful when using bull spreads. If you really think acall option is too
expensive and want to reduce its cost, ti:ythis strategy: Buy the call and simultane
ously sell acredit put spread (bull spread) using slightly out-of-the-money puts. This
strategy reduces the call'snet cost and maintains upside potential (although it
increases downside risk, but at least it is still afixed risk).
Example: With XYZ at 100, atrader is bullish and wants to buy the July 100 calls,
which expire in two months. However, upon inspection, he finds that they are trad
ing at 10 - an implied volatility of 59%. He knows that, historically, the implied
volatility of this stock'soptions range from approximately 40% to 60%, so these are

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Chapter 37: How Volatility Affects Popular Strategies 773
very expensive options. If he buys them now and implied volatility returns to its
median range near 50%, he will suffer from the decrease in implied volatility.
As apossible remedy, he considers selling an out-of-the-money put credit
spread at the same time that he buys the calls. The credit from this spread will act as
ameans of reducing the net cost of the calls. If he'sright and the stock goes up, all
will be well. However, the introduction of the put spread into the mix has introduced
some additional downside risk.
Suppose the following prices exist:
XYZ: 100
July 100 call: 10 (as stated above)
July 90 put: 5
July 80 put: 2
The entire bullish position would now consist of the following:
Buy 1 July 100 call at 1 0
Buy 1 July 80 put at 2
Sell 1 July 90 put at 5
Net expenditure: 7 point debit (plus commission)
Figure 37-6 shows the profitability, at expiration, of both the outright call pur
chase and the bullish position constructed above.
FIGURE 37-6.
Profitability at expiration.
2000
Bullish Spread //
/
1000
"' "' 0 ...J
87 :!::
Outright Call Purchase
e 0
C. 70 80 90
<Ft 100 /100 ,,....... ___
120
-1000
Stock
-2000

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774 Part VI: Measuring and Trading Volatility
First, one can see that the bullish spread position has atotal risk of 17 points, if
X'YZ is below 80 (the lower striking price of the put spread) at expiration. That, of
course, is more than the 10-point cost of the July 100 call by itself, but if one is using
atrading stop of any sort, he probably would not be at risk for the entire 17 points,
since he wouldn'thold on while the stock fell all the way to 80 and below. Note also
that the bullish spread position would have aloss of 10 points (the same as the call)
at aprice of 87 for the common at expiration. Hence, the combined position actual
ly has less risk than the outright call purchase as long as XYZ is 87 or higher at expi
ration. Since one is supposedly bullish initially when establishing this strategy, it
seems likely that he would figure the stock would be 87 or higher at expiration. ·
Figure 37-7 offers another comparison, that of the two positions after 30 days
have passed. Note that the spread position once again does better on the upside and
worse on the downside. The crossover point between the two curves is at about aprice of95. That is, ifXYZ is above 95 in 30 days, the bullish spread position will out
perform the call buy.
One final point should be made regarding the investment required. The out
right call purchase requires an investment of $1,000 - the cost of the long call. The
bullish spread position requires that $1,000, plus $700 for the spread (IO-point dif
ference in the strikes, less the 3-point credit received for selling the spread). That'satotal of $1,700, the risk of the bullish spread position. Hence, the rate of return might
favor the outright call purchase, depending on how far the stock rallies.
FIGURE 37-7.
Results of the two positions in 30 days.
Cl)
_g
:;::,
e
Cl..
~
1000
-1000
-2000-
Spread
95
110
Stock
/
Outright Call Purchase
120

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Chapter 37: How Volatility Affects Popular Strategies 775
Overall, the bullish spread position is an attractive alternative to an outright call
purchase, especially when the call is overpriced. The spread does risk agreater
amount of money if the underlying stock should collapse heavily. Still, if one is truly
bullish, and if one employs areasonably tight downside stop on his entire position,
this spread can perform better than the outright purchase of an overpriced call.
VERTICAL PUT SPREADS
Also of interest is the effect that implied volatility has on put spreads. One of the
more popular strategies involving puts is the sale of acredit spread - abull spread
with puts. Assume that astock is selling at 100, and one is going to sell aput with a
110 strike and buy aput with a 90 strike. That is aput credit (bull) spread. Also
assume that the options have four months of life remaining. (See Table 37-8.)
One would not rationally sell this credit spread if implied volatility were as low
as 20%, because at that low level of volatility, the in-the-money December llO put is
trading for 10 dollars - parity - and thus would immediately be at risk of early
assignment. But one can see that an increase in implied volatility increases the value
of the spread. Now, if one had sold this spread to begin with, he would thus be los
ing money when implied volatility increased. This was proven with call bull spreads,
too: They lose money when implied volatility increases. Conversely, of course, the
put credit spread makes money when implied volatility decreases.
What happens after thirty days have passed? Figure 37-8 shows just two cases
- implied volatility at 30% and implied volatility at 80%. One can surmise that other
levels of implied volatility between 30% and 80% would have profit graphs that lie
between the two shown in Figure 37-8.
TABLE 37·8
Implied
Volatility
20%
30%
40%
50%
60%
70%
80%
*Short option trading at parity
90-110
Put Bull Spread
(Theoretical Value)
9.15 er*
9.70 er
10.12 er
10.46 er
10.78 er
11.05 er
11.33 er

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174 Part VI: Measuring and Trading Volatility
First, one can see that the bullish spread position has atotal risk of 17 points, if
XYZ is below 80 (the lower striking price of the put spread) at expiration. That, of
course, is more than the 10-point cost of the July 100 call by itself, but if one is using
atrading stop of any sort, he probably would not be at risk for the entire 17 points,
since he wouldn'thold on while the stock fell all the way to 80 and below. Note also
that the bullish spread position would have aloss of 10 points (the same as the call)
at aprice of 87 for the common at expiration. Hence, the combined position actual
ly has less risk than the outright call purchase as long as XYZ is 87 or higher at expi
ration. Since one is supposedly bullish initially when establishing this strategy, it
seems likely that he would figure the stock would be 87 or higher at expiration.
Figure 37-7 offers another comparison, that of the two positions after 30 days
have passed. Note that the spread position once again does better on the upside and
worse on the downside. The crossover point between the two curves is at about aprice of 95. That is, ifXYZ is above 95 in 30 days, the bullish spread position will out
perform the call buy.
One final point should be made regarding the investment required. The out
right call purchase requires an investment of $1,000 - the cost of the long call. The
bullish spread position requires that $1,000, plus $700 for the spread (IO-point dif
ference in the strikes, less the 3-point credit received for selling the spread). That'satotal of $1,700, the risk of the bullish spread position. Hence, the rate of return might
favor the outright call purchase, depending on how far the stock rallies.
FIGURE 37-7.
Results of the two positions in 30 days.
1000
en
Ill
0
...J
O 70 :;::,
~ ... a.
w
-1000
-2000-
80
95
Bullish Spread
Stock
Spread
110
/
1/
Outright Call Purchase
120

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Chapter 37: How Volatility Affects Popular Strategies 775
Overall, the bullish spread position is an attractive alternative to an outright call
purchase, especially when the call is overpriced. The spread does risk agreater
amount of money if the underlying stock should collapse heavily. Still, if one is truly
bullish, and if one employs areasonably tight downside stop on his entire position,
this spread can perform better than the outright purchase of an overpriced call.
VERTICAL PUT SPREADS
Also of interest is the effect that implied volatility has on put spreads. One of the
more popular strategies involving puts is the sale of acredit spread - abull spread
with puts. Assume that astock is selling at 100, and one is going to sell aput with a
110 strike and buy aput with a 90 strike. That is aput credit (bull) spread. Also
assume that the options have four months of life remaining. (See Table 37-8.)
One would not rationally sell this credit spread if implied volatility were as low
as 20%, because at that low level of volatility, the in-the-money December llO put is
trading for 10 dollars - parity - and thus would immediately be at risk of early
assignment. But one can see that an increase in implied volatility increases the value
of the spread. Now, if one had sold this spread to begin with, he would thus be los
ing money when implied volatility increased. This was proven with call bull spreads,
too: They lose money when implied volatility increases. Conversely, of course, the
put credit spread makes money when implied volatility decreases.
What happens after thirty days have passed? Figure 37-8 shows just two cases
- implied volatility at 30% and implied volatility at 80%. One can surmise that other
levels of implied volatility between 30% and 80% would have profit graphs that lie
between the two shown in Figure 37-8.
TABLE 37-8
Implied
Volatility
20%
30%
40%
50%
60%
70%
80%
*Short option trading at parity
90-110
Put Bull Spread
(Theoretical Value)
9.15 er*
9.70 er
10. 12 er
10.46 er
10.78 er
11.05 er
11.33 er

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776
FIGURE 37-8.
Put credit (bull) spread profit in 30 days.
1000
500
- 1000 _ _,±8±
Assignment Risk Area
IV=30%
Stock
Part VI: Measuring and Trading Volatility
-~
IV= 30%
130 140
First, one can observe that abull put spread does not widen out to anywhere
near its maximum potential if implied volatility increases. The same thing was seen
with the call bull spread in the previous section. But aput bull spreader is caught in
another trap: If implied volatility falls and the stock falls too, the risk of early assign
ment materializes quicky. Note the shaded area at the lower left of the graph, extend
ing from aprice of about 94 on down. After thirty days (so there would be three
months life remaining at that point in time), if implied volatility is 30%, the 110 put
(the short put) would be trading at parity for stock prices of 94 and below. Thus, it
would be at risk of early assignment. If implied volatility were even lower, the puts
would be at parity for much higher stock prices.
Now, in and of itself, early assignment on an equity or futures put spread is not
necessarily aterrible thing. There will be arequest for additional margin (because
the stock has to be paid for or the futures contract margined), but the risk is still the
same in dollar terms. Of course, the request for extra margin could be backbreaking
for astock trader if he can'tafford to fully pay for the stock, and the early assignment
would probably incur additional commission costs, too. However, with cash-based
index options there is amore serious increase in risk after an early assignment,
because one is left with only the long side of the spread. If that option happens to
have substantial value, then there is considerable risk if the underlying should quick
ly move higher. In fact, by the time one unwinds the spread, he might actually end
up losing more than his original limited risk amount - all due to the early assignment.
(This could happen if the underlying first plunges in price, placing both options

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Chapter 37: How Volatility Affects Popular Strategies 777
deeply in-the-money, after which one gets assigned on the short put option, followed
by the underlying then dramatically rising in price.)
The lesson to be learned is this: If one is considering using bull spreads in which
at least one of the options is at- or in-the-rrwney, then acall bull spread is asuperior
choice over aput bull spread. Early assignment is not really aconsideration for most
call spreads.
In both cases, however, amore serious problem exists, and that is that the
spread does not widen out much even when the stock makes anice bullish move.
Thus, once again it is actually better to buy acall option in most cases than to use the
bull spread, because profits are larger and an increase in implied volatility is afavor
able thing for an outright call buyer.
Note that these effects are similar, but much less pronounced, for out-of-the
money put credit spreads. Still, it should be noted that an increase in implied volatil
ity will harm an out-of-the-money put credit spread, too. Hence, if the underlying
goes into arapid fall (crash, plunge), then implied volatility usually increases quickly
and dramatically. So an out-of-the-money credit spreader is hit with the double
whammy of expanding implied volatility and the fact that the underlying is fast
approaching the strike price of his options, . thereby expanding the price of the
spread.
PUT BEAR SPREADS
What about the put spread in abearish situation? In avertical put spread one buys
the put with the higher strike and sells the put with the lower strike to construct asimple put bear spread. Actually, asudden increase in implied volatility is of help to
the bear put spread. That is, the spread will widen out slightly. To verify this, look at
Table 37-8 again, only now imagine that one is buying the spread for adebit. Note
that the smallest debit is at the lower implied volatility- 9.15 debit with IV at 30%.
If implied volatility were to instantaneously jump to 80%, the spread would widen
out to 11.33 debit. Avery quick profit could be had. So there'sadifference right away
between adebit call bull spread (which loses money when implied volatility sudden
ly increases) and adebit put bear spread.
Unfortunately, the other major drawback - that the spread doesn'twiden out
much if the underlying makes afavorable move - is still true. Figure 37-9 shows abear put spread, 30 days hence, for two different implied volatilities. Once again, the
lower-volatility spread widens out more quickly, because both options tend to go to
parity in that case. In fact, one can see on the graph that there is early assignment
risk in the low-volatility case, below aprice of about 77 on the stock. That is not a

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778 Part VI: Measuring and Trading Volatility
FIGURE 37-9.
Bear put spread profit in 30 days.
1000 IV= 30%
Assignment
Risk
Area
(/)
(/)
0
...J
~ 0
e 70 80 110 120 130 140
a.
ff7
IV= 80%
-1000
'~
Stock
problem, though, since the spread would have widened to its maximum potential in
that case and could just be removed when the risk of early assignment materialized.
When implied volatility remains high, though, the spread doesn'twiden out
much, even when the stock drops alot after 30 days. Since it is common for implied
volatility to rise (even skyrocket) when the underlying drops quickly, the put bear
spread probably won'twiden out much. That may not be apsychologically pleasing
strategy, because one won'tmake the level of profits that he had hoped to when the
underlying fell quickly.
Once again, it seems that the outright purchase of an option is probably superi
or to aspread. In these cases, it is true with respect to puts, much as it was with call
options. Spreading often unnecessarily complicates atrader'soutlook.
CALENDAR SPREADS
In the earlier chapter on calendar spreads, it was mentioned that an increase in
implied volatility will cause acalendar spread to widen out. Both options will become
more expensive, of course, since the increase in implied volatility affects both of
them, but the absolute price change will be greatest in the long-term option.
Therefore, the calendar spread will widen. This may seem somewhat counterintu
itive, especially where highly volatile stocks are concerned, so some examples may
prove useful.

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Chapter 37: How Volatility Affects Popular Strategies 779
Example: Suppose that XYZ is trading at 100, and one is interested in acalendar
spread in which an August (5-month) call is bought and a May (2-month) call is sold.
For the purpose of this example, it will be assumed that these are both at-the-money
options. First, the vegas of the two options will be examined, assuming that implied
volatility is 40%:
Stock: 100
Implied Volatility:
40% Option
Sell May 100 call
Buy August 1 00 call
Theoretical Price
6.91
11.22
Vega
0.162
0.251
In theory, this spread should be worth 4.31 - the difference in the theoretical
values. Perhaps more important, it has volatility exposure of 0.089 - the difference
between the vega of the long call and that of the short call. Since vega is positive, this
means that an increase in implied volatility will be beneficial to the spread. In other
words, one can expect the spread to widen if implied volatility rises, and can expect
the spread to shrink if implied volatility declines.
The following table can also be constructed, showing the theoretical value of
the spread at various levels of implied volatility. This table makes the assumption that
very little time has passed ( only one week) before the implied volatility changes take
place. It also assumes that the stock is still at 100.
Stock Price: 100
One week ofter the spread hos been established:
Implied Volatility Theoretical Spread Value
20% 2.58
30% 3.52
40% 4.46
50% 5.40
60% 6.33
80% 8.16
100% 12.92
From the above data, it is quite obvious that implied volatility levels have ahuge
effect on the value of acalendar spread. The actual initial contribution of time decay
is rather small in comparison. For example, note that if volatility remains unchanged
at 40%, then the spread will have widened only slightly - to 4.46 from 4.31 - after
the passage of one week'stime. That is small in comparison to the changes dictated
by volatility expansion or contraction.

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780 Part VI: Measuring and Trading Volatility
Acommon mistake that calendar spreaders make is to think that such aspread
looks overly attractive on avery volatile stock. Consider the same stock as above, still
trading at 100, but for some reason implied volatility has skyrocketed to 80% (per
haps atakeover rumor is present).
Stock: JOO
Implied Volatility: 80%
Coll
May 100 call
June 100 call
Theoretical Value
12.55
16.81
On the surface, this seems like avery attractive spread. There are two months
of life remaining in the May options (and three months in the Junes) and the spread
is trading at 4.36. However, both options are completely composed of time value
premium, and most certainly the June 100 call would be worth far more than 4.36
when the May expires, if the stock is still near 100. The fact that many traders miss
when they think of the calendar spread this way is that the June call will only be
worth "far more than 4.36" if implied volatility holds up. If implied volatility for this
stock is normally something on the order of 40%, say, then it is probably not reason
able to expect that the 80% level will hold up. Just for comparison, note that if the
stock is at 100 at May expiration - the maximum profit potential for such acalendar
spread - the June 100 call, with implied volatility now at 40%, and with one month
of life remaining, would be worth only 4. 77. Thus the spread would only have made
aprofit of afew cents (4.36 to 4.77), and if the underlying stock were farther from
the strike price at expiration, there would probably be aloss rather than aprofit.
The point to be remembered is that acalendar spread is a "long volatility" play
(and areverse calendar spread is just the opposite). Evaluate the position'srisk with
an eye to what might happen to implied volatility, and not just to where the stock
price might go or how much time decay there might be in the position.
RATIO SPREADS AND BACKSPREADS
The previous descriptions in this chapter describe fairly fully and accurately what
the effect of volatility changes are. More complicated strategies are usually nothing
more than combinations of the strategies presented earlier, so it is easy to discern
the effect that changes in implied volatility would have; just combine the effects on
the simpler strategies. For example, aratio call write is really just the equivalent of
astraddle sale - astrategy whose volatility ramifications are fairly simple to under
stand.
Ratio spreads, on the other hand, might not be as intuitive to interpret, but they
are fairly simple nonetheless. Acall ratio spread is really just the combination of some

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Chapter 37: How Volatility Affects Popular Strategies 781
call bull spreads and some naked call options. For example, acall ratio spread might
consist of buying an XYZ July 100 call and selling two XYZ July 120 calls. If one were
to break it down into its components, this spread is really long one XYZ July 100-120
call bull spread, plus an additional naked July 120 call.
We already know that an increase in implied volatility is very detrimental to anaked call option. In addition, it was shown earlier than an increase in implied volatil
ity actually harms the value of an at-the-money call bull spread. So, for aratio call
spread, both components are harmed by an increase in implied volatility. Conversely,
adecrease in implied volatility would be beneficial to aratio spread, but where naked
options are concerned, one should be more mindful of his risk than of this reward.
It was also shown previously that acall bull spread does not widen out much if
the underlying stock makes aquick upward move. The spread won'twiden out to its
maximum profit potential until expiration draws nigh or the stock is well above the
upper strike in the spread. This scenario also does not bode well for the ratio call
spread. Suppose that the underlying stock suddenly jumps upward and implied
volatility increases at the same time. That combination is seen quite frequently, espe
cially if the stock were previously "dull" or if there is some sort of active corporate
(takeover) rumor. The call ratio spread will fare miserably under these conditions,
because the increase in stock price certainly harms the naked call position and the
bull spread is not widening out much to compensate for it. In addition, the increase
in implied volatility is working against both components.
The same sort of thing happens with put ratio spreads. They are really the com
bination of aput bear spread plus some additional naked put options. If the under
lying falls in price, while implied volatility increases - avery common occurrence in
all markets then the put ratio spread will fare poorly. In fact, implied volatility
sometimes explodes if the underlying falls very rapidly (crashes), so the ratio put
spreader should clearly assess his risk in this light.
In summary, atrader utilizing ratio spread strategies should clearly understand
and attempt to analyze the risks of an increase in implied volatility. This includes not
only assessing the vega risk of the spread, but also using aprobability calculator with
some inflated volatility estimates to see just what the chances are of the spread get
ting into real trouble.
BACKSPREADS
Acall backspread is merely the opposite of acall ratio spread. Thus, any of the earli
er commentary about how an increase in implied volatility is detrimental to aratio
spread can be reversed when discussing the backspread. An increase in implied
volatility will be beneficial to abackspread strategy, while adecrease in implied

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782 Part VI: Measuring and Trading Volatility
volatility will be slightly harmful to the spread. However, since risk is limited in abackspread, such adecrease in implied volatility would not have catastrophic conse
quences unless one had overcommitted his funds to one position.
SUMMARY
In general, one can always determine the exposure of his position to volatility by com
puting the vega of this position. However, it is also useful for astrategist to have some
general feeling for how implied volatility will affect his positions and strategies. Thus,
this chapter was designed to point out the most common effects that changes .in
implied volatility will have on the basic types of option strategies. Once one has afeeling for his exposure to volatility, he can then assess whether an adverse volatility
movement is likely. For example, if an increase in implied volatility would be harm
ful, and the strategist sees that current levels of implied volatility are quite low in
comparison to historical norms, then perhaps he should remove or adjust the posi
tion.
Volatility and the price of the underlying are the two major components
affecting profitability for most option positions. Time decay is only most pertinent
as expiration approaches. Yet, many traders concentrate greatly on potential price
movements of the underlying, often while ignoring what changes in implied volatil
ity could do. That is amistake, for the most knowledgeable option traders plan for
volatility risk at all times. Understanding and handling that risk can have apositive
effect on an option trader'sprofits.

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The Distribution of
Stock Prices
Much of the work that has been done in statistics and related areas regarding the
stock market has made the assumption that stock prices are distributed normally, or
more specifically, lognonnally. In actual practice, this is usually an incorrect assump
tion. For the option strategist, this means that some of the things one might believe
about certain option strategies having an advantage over certain other option strate
gies might be incorrect. In this chapter, anumber of facts concerning stock price dis
tribution will be brought to light, including how it might affect the option strategist.
MISCONCEPTIONS ABOUT VOLATILITY
Statistics are used to estimate stock price movement (and futures and indices as well)
in many areas of financial analysis. Many authors have written extensively about the
use of probabilities to aid in choosing viable option strategies. Stock mutual fund
managers often use volatility estimates to help them determine how risky their port
folios are. The uses are myriad. Unfortunately, almost all of these applications are
wrong! Perhaps wrong is too strong aword, but almost all estimates of stock price
movement are overly conservative. This can be very dangerous if one is using such
estimates for the purposes of, say, writing naked options or engaging in some other
such strategy in which volatile stock price movement is undesirable.
As areview for those not familiar with mathematical distributions, the lognor
mal distribution is what'scommonly used to describe stock prices because its shape
is intuitively similar to the way stocks behave - they can'tgo below zero, they can rise
to infinity, and most of the time they don'tgo much of anywhere. On top of that, the
distribution'sshape is based on the historical volatility of the underlying instrument.
In alognormal distribution (and normal distribution, too), stocks remain within 3
standard deviations of their current price 99. 7 4% of the time. Astandard deviation
783

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788 Part VI: Measuring and Trading VolatiDty
ally including gap moves. There are not always gap moves, though, over astudy of
this length. Sometimes, there will be amore gradual transition. Consider the fact that
one of the stocks in the study moved 5.8 sigma in the 30 days. There weren'tany huge
gaps during that time, but anyone who was short calls while the stock made its run
surely didn'tthink it was agradual advance.
So, what does this information mean to the average option trader? For one,
you should certainly think twice about selling stock options in apotentially volatile
market ( or any market, for that matter, since these large moves are not by any means
limited to the volatile market periods). This statement encompasses naked option
selling, but also includes many forms of option selling, because of the possibilities of
large moves by the underlying stocks.
For example, covered call writing is considered to be "conservative." However,
when the stock has the potential to make these big moves, it will either cause one to
give up large upside profits or to suffer large downside losses. ( Covered call writing
has limited profit potential and relatively large downside risk, as does its equivalent
strategy, naked put selling.) When these large stock moves occur on the upside, acov
ered writer is often disappointed that he gave up too much of the upside profit poten
tial. Conversely, if the stock drops quickly, and one is assigned on his naked put, he
often no longer has much appetite for acquiring the stock ( even though he said he
"wouldn'tmind" doing so when he sold the puts to begin with).
Even spreading has problems along these lines. For example, avertical spread
limits profits so that one can'tparticipate in these relatively frequent large stock
moves when they occur.
What can an option seller do? First, he must carefully analyze his position and
allow for much larger stock movements than one would expect under the lognormal
distribution. Also, he must be careful to sell options only when they are expensive in
terms of implied volatility, so that any decrease in implied will work in his favor.
Probably most judicious, though, is that an option seller should really concentrate on
indices (or perhaps certain futures contracts), because they are statistically much less
volatile than stocks. Hard as it is to believe, futures are less volatile than stocks
(although the leverage available in futures can make them ariskier investment overall).
Two 30-day studies, similar to those conducted on stocks, were run on option
able indices, covering the same time periods: 10/22/99 to 12/7/99 for one study and
7/1/93 to 8/17/93 for the other. The results are shown in Tables 38-5 and 38-6. This
may be asomewhat distorted picture, though, because many of these indices overlap
(there are four Internet indices, for example). The largest mover was the Morgan
Stanley High-Tech Index (5 standard deviations), but it should also be noted that
something that is considered fairly tame, such as the Russell 2000 ($RUT), also had
a 3-standard deviation move in one study. The first study showed that 37% of the

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Chapter 38: The Distribution of Stock Prices 795
Figure 38-3 perhaps shows even more starkly how the bull market has affected
things over the last six-plus years. There are over 1,600 data points for IBM (i.e., daily
readings) in Figure 38-3, yet the whole distribution is skewed to the right. It appar
ently was able to move up quite easily throughout this time period. In fact, the worst
move that occurred was one move of -2.5 standard deviations, while there were
about ten moves of +4.0 standard deviations or more.
For alonger-term look at how IBM behaves, consider the longer-term distribu
tion of IBM prices, going back to March 1987, as shown in Figure 38-4.
From Figure 38-4, it'sclear that this longer-term distribution conforms more
closely to the normal distribution in that it has asort of symmetrical look, as opposed
to Figure 38-3, which is clearly biased to the right (upside).
These two graphs have implications for the big picture study shown in Figure
38-1. The database used for this study had data for most stocks only going back to
1993 (IBM is one of the exceptions); but if the broad study of all stocks were run
using data all the way back to 1987, it is certain that the "actual" price distribution
would be more evenly centered, as opposed to its justification to the right (upside).
That'sbecause there would be more bearish periods in the longer study (1987, 1989,
and 1990 all had some rather nasty periods). Still, this doesn'tdetract from the basic
premise that stocks can move farther than the normal distribution would indicate.
WHAT THIS MEANS FOR OPTION TRADERS
The most obvious thing that an option trader can learn from these distributions and
studies is that buying options is probably alot more feasible than conventional wisdom
would have you believe. The old thinking that selling an option is "best" because it
wastes away every day is false. In reality, when you have sold an option, you are exposed
to adverse price movements and adverse movements in implied volatility all during the
life of the option. The likelihood of those occurring is great, and they generally have
more influence on the price of the aption in the short run than does time decay.
You might ask, "But doesn'tall the volatility in 1999 and 2000 just distort the
figures, making the big moves more likely than they ever were, and possibly ever will
be again?" The answer to that is aresounding, "Nol" The reason is that the current
20-day historical volatility was used on each day of the study in order to determine
how many standard deviations each stock moved. So, in 1999 and 2000, that histori
cal volatility was ahigh number and it therefore means that the stock would have had
to move avery long way to move four standard deviations. In 1993, however, when
the market was in the doldrums, historical volatility was low, and so amuch smaller

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Chapter 38: The Distribution of Stock Prices 195
Figure 38-3 perhaps shows even more starkly how the bull market has affected
things over the last six-plus years. There are over 1,600 data points for IBM (i.e., daily
readings) in Figure 38-3, yet the whole distribution is skewed to the right. It appar
ently was able to move up quite easily throughout this time period. In fact, the worst
move that occurred was one move of -2.5 standard deviations, while there were
about ten moves of +4.0 standard deviations or more.
For alonger-term look at how IBM behaves, consider the longer-term distribu
tion of IBM prices, going back to March 1987, as shown in Figure 38-4.
From Figure 38-4, it'sclear that this longer-term distribution conforms more
closely to the normal distribution in that it has asort of symmetrical look, as opposed
to Figure 38-3, which is clearly biased to the right (upside).
These two graphs have implications for the big picture study shown in Figure
38-1. The database used for this study had data for most stocks only going back to
1993 (IBM is one of the exceptions); but if the broad study of all stocks were run
using data all the way back to 1987, it is certain that the "actual" price distribution
would be more evenly centered, as opposed to its justification to the right (upside).
That'sbecause there would be more bearish periods in the longer study (1987, 1989,
and 1990 all had some rather nasty periods). Still, this doesn'tdetract from the basic
premise that stocks can move farther than the normal distribution would indicate.
WHAT THIS MEANS FOR OPTION TRADERS
The most obvious thing that an option trader can learn from these distributions and
studies is that buying options is probably alot more feasible than conventional wisdom
would have you believe. The old thinking that selling an option is "best" because it
wastes away every day is false. In reality, when you have sold an option, you are exposed
to adverse price movements and adverse movements in implied volatility all during the
life of the option. The likelihood of those occurring is great, and they generally have
more influence on the price of the option in the short run than does time decay.
You might ask, "But doesn'tall the volatility in 1999 and 2000 just distort the
figures, making the big moves more likely than they ever were, and possibly ever will
be again?" The answer to that is aresounding, "Nol" The reason is that the current
20-day historical volatility was used on each day of the study in order to determine
how many standard deviations each stock moved. So, in 1999 and 2000, that histori
cal volatility was ahigh number and it therefore means that the stock would have had
to move avery long way to move four standard deviations. In 1993, however, when
the market was in the doldrums, historical volatility was low, and so amuch smaller

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796 Part VI: Measuring and Trading Volatility
move was needed to register a 4-standard deviation move. To see aspecific example
of how this works in actual practice, look carefully at the chart of IBM in Figure 38-
4, the one that encompasses the crash of '87. Don'tyou think it'salittle strange that
the chart doesn'tshow any moves of greater than minus 4.0 standard deviations? The
reason is that IBM'shistorical volatility had already increased so much in the days
preceding the crash day itself, that when IBM fell on the day of the crash, its move
was less than minus 4.0 standard deviations. (Actually, its one-day move was greater
than -4 standard deviations, but the 30-day move - which is what the graphs in Figure
38-3 and 38-4 depict - was not.)
STOCK PRICE DISTRIBUTION SUMMARY
One can say with agreat deal of certainty that stocks do not conform to the normal
distribution. Actually, the normal distribution is adecent approximation of stock
price movement rrwst of the time, but it'sthese "outlying" results that can hurt any
one using it as abasis for anonvolatility strategy.
Scientists working on chaos theo:ry have been trying to get abetter handle on
this. An article in Scientific American magazine ("A Fractal Walk Down Wall Street,"
Februa:ry 1999 issue) met some criticism from followers of Elliot Wave theo:ry, in that
they claim the article'sauthor is purporting to have "invented" things that R. N.
Elliott discovered years ago. Idon'tknow about that, but Ido know that the article
addresses these same points in more detail. In the article, the author points out that
chaos theo:ry was applied to the prediction of earthquakes. Essentially, it concluded
that earthquakes can'tbe predicted. Is this therefore auseless analysis? No, says the
author. It means that humans should concentrate on building stronger buildings that
can withstand the earthquakes, for no one can predict when they may occur. Relating
this to the option market, this means that one should concentrate on building strate
gies that can withstand the chaotic movements that occasionally occur, since chaotic
stock price behavior can'tbe predicted either.
It is important that option traders, above all people, understand the risks of
making too conservative an estimate of stock price movement. These risks are espe
cially great for the writer of an option (and that includes covered writers and spread
ers, who may be giving away too much upside by writing acall against long stock or
long calls). By quantifying past stock price movements, as has been done in this chap
ter, my aim is to convince you that "conventional" assumptions are not good enough
for your analyses. This doesn'tmean that it'sokay to buy overpriced options just
because stocks can make large moves with agreater frequency than most option

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Chapter 38: The Distribution of Stock Prices 797
models predict; but it certainly means that the buyer of underpriced options stands
to benefit in acouple of ways. Conversely, an option seller must certainly concentrate
his efforts where options are expensive, and even then should be acutely aware that
he may experience larger-than-expected stock price movements while the option
position is in place.
So what does this mean for option strategies? On the surface, it means that if
one uses the normal (or lognormal) distribution for estimating the probability of astrategy'ssuccess, he may get abig move in the stock that he didn'toriginally view as
possible. If one were long straddles, that'sgreat. However, if he is short naked
options, then there could be anasty surprise in store. That'sone reason why extreme
caution should be used regarding selling naked options on stocks; they can make
moves of this sort too often. At least with indices, such moves are far less frequent,
although the Dow drop of over 550 points in October 1997 was amove of seven stan
dard deviations, and the crash of '87 was about a 16-standard deviation move - which
Professor Mark Rubenstein of the University of California at Berkeley says was some
thing that should occur about once in ten times the life of our current universe! That'saccording to lognormal distribution, of course, which we know understates things
somewhat, but it'sstill abig number under any distribution.
There are two approaches that one can take, then, regarding option strategies.
One is to invent another method for estimating stock price distributions. Suffice it to
say that that is not an easy task, or someone would have made it well-known already.
There have been many attempts, including some in which alarge history of stock
price movements is observed and then adistribution is fitted to them. The problem
with accounting for these occasional large price moves is that it is perhaps an even
more grievous error to overestimate the probabilities of such moves than to underes
timate them.
The second approach is to continue to use the normal distribution, because it'sfast and accessible in alot of places. Then, either rely on option buying strategies
( straddles, for example) where implied volatility appears to be low - knowing that you
have achance at better results than the statistics might indicate - or adjust your cal
culations mentally for these large potential movements if you are using option selling
strategies.
THE PRICING OF OPTIONS
The extreme movements of the fat tail distribution should be figured into the pricing
of an option, but they really are not, at least not by most models. The Black-Scholes

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798 Part VI: Measuring and Trading Volatility
model, for example, uses alognormal distribution. Personally, this author believes
that the Black-Scholes model is an excellent tool for analyzing options and option
strategies, but one must understand that it may not be affording enough probability
to large moves by the underlying.
Does this mean that most options are underpriced, since traders and market
makers are using the Black-Scholes model (or similar models) to price them?
Without getting too technical, the answer is that yes, some options - particularly out
of-the-money options - are probably underpriced. However, one must understand
that it is still arelatively rare occurrence to experience one of these big moves - ifs
just not as rare as the lognormal distribution would indicate. So, an out-of-the-money
option might be slightly underpriced, but often not enough to make any real differ
ence.
In fact, futures options in grains, gold, oil, and other markets that often experi
ence large and sudden rallies display adistinct volatility skew. That is, out-of-the-money
call options trade at significantly higher implied volatilities than do at-the-money
options. Ironically, there is far less chance of one of these hyper-standard-deviation
moves occurring in commodities than there is in stocks, at least if history is aguide. So,
the fact that some out-of-the-money futures options are expensive is probably an incor
rect overadjustment for the possibility of large moves.
THE PROBABILITY OF STOCK PRICE MOVEMENT
The distribution information introduced in this chapter can be incorporated into
somewhat rigorous methods of determining probabilities. That is, one can attempt to
assess the chances of astock, futures contract, or index moving by agiven distance,
and those chances can incorporate the fat tails or other non-lognormal behavior of
prices.
The software that calculates such probabilities is typically named a "probability
calculator." There are many such software programs available in the marketplace.
They range from free calculators to completely overpriced ones selling for more than
$1,000. In reality, high-level probability calculation software can be created by some
one with agood understanding of statistics, or aprogram can be purchased for arather nominal fee - perhaps $100 or so.
Before getting into these various methods of probability estimation, it should be
noted that all of them require the trader to input avolatility estimate. There are only
afew other inputs, usually the stock price, target price(s), and length of time of the
study. The volatility one inputs is, of course, an estimate of future volatility - some-

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Chapter 38: The Distribution of Stock Prices 799
thing that cannot be predicted with certainty. Nevertheless, any probability calcula
tor requires this input. So, one must understand that the results one obtains from any
of these probability calculators is an estimate of what might happen. It should not be
relied on as "gospel."
Additionally, probability calculators make asecond assumption: that the volatil
ity one inputs will remain constant over the entire length of the study. We know this
is incorrect, for volatility can change daily. However, there really isn'tagood way of
estimating how volatility might change in the course of the study, so we are pretty
much forced to live with this incorrect assumption as well.
There is no certain way to mitigate these volatility "problems" as far as the prob
ability calculator is concerned, but one helpful technique is to bias the volatility pro
jection against your objectives. That is, be overly conservative in your volatility pro
jections. If things tum out to be better than you estimated, fine. However, at least
you won'tbe overstating things initially. An example may help to demonstrate this
technique.
Example: Suppose that atrader is considering buying astraddle on XYZ. The five
month straddle is selling for aprice of 8, with the stock currently trading near 40. Aprobability calculator will help him to determine the chances that XYZ can rise to 48
or fall to 32 (the break-even points) prior to the options' expiration. However, the
probability calculator'sanswer will depend heavily on the volatility estimate that the
trader plugs into the probability calculator. Suppose that the following information is
know about the historical volatility of XYZ:
l 0-day historical volatility:
20-day historical volatility:
50-day historical volatility:
l 00-day historical volatility
22%
20%
28%
33%
Which volatility should the trader use? Should he choose the 100-day historical
volatility since this is afive-month straddle, which encompasses just about 100 trad
ing days until expiration? Should he use the 20-day historical volatility, since that is
the "generally accepted" measure that most traders refer to? Should he calculate ahistorical volatility based exactly on the number of days until expiration and use that?
To be most conservative, none of those answers is right, at least not for the right
reasons. Since one is buying options in this strategy, he should use the lowest of the
above historical volatility measures as his volatility estimate. By doing so, he is taking
aconservative approach. If the straddle buy looks good under this conservative
assumption, then he can feel fairly certain that he has not overstated the possibilities

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800 Part VI: Measuring and Trading Volatility
of success. If it turns out that volatility is higher during the life of the position, that
will be an added benefit to this position consisting of long options. So, in this exam
ple, he should use the 20-day historical volatility because it is the lowest of the four
choices that he has.
Similarly, if one is considering the sale of options or is taking aposition with anegative vega ( one that will be harmed if volatility increases), then he should use the
highest historical volatility when making his probability projections. By so doing, he
is again being conservative. If the strategy in question still looks good, even under an
assumption of high volatility, then he can figure that he won'tbe unpleasantly sur
prised by ahigher volatility during the life of the position.
There have been times when a 100-day lookback period was not sufficient for
determining historical volatility. That is, the underlying has been performing in an
erratic or unusual manner for over 100 days. In reality, its true nature is not described
by its movements over the past 100 days. Some might say that 100 days is not enough
time to determine the historical volatility in any case, although most of the time the
four volatility measures shown above will be asufficient guide for volatility.
When alonger lookback period is required, there is another method that can be
used: Go back in ahistorical database of prices for the underlying and compute the
20-day, 50-day, and l 00-day historic volatilities for all the time periods in the data
base, or at least during afairly large segment of the past prices. Then use the medi
an of those calculations for your volatility estimates.
Example: XYZ has been behaving erratically for several months, due to overall mar
ket volatility being high as well as to aseries of chaotic news events that have been
affecting XYZ. Atrader wants to trade XYZ'soptions, but needs agood estimate of
the "true" volatility potential of XYZ, for he thinks that the news events are out of the
way now. At the current time, the historical volatility readings are:
20-day historical: 130%
50-day historical l 00%
100-day historical 80%
However, when the trader looks farther back in XYZ'strading history, he sees
that it is not normally this volatile. Since he suspects that XYZ'srecent trading histo
ry is not typical of its true long-term performance, what volatility should he use in
either an option model or aprobability calculator?
Rather than just using the maximum or minimum of the above three numbers
(depending on whether one is buying or selling options), the trader decides to look

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