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 a put won't appreciate in value as much as one might expect, even when
the stock drops, since the put loses its time value premium quickly. It's even more
important information for put sellers: A short put is at risk of assignment as soon as
there is no time value premium left in the put. Thus, a put can be assigned well in
advance of expiration even a LEAPS put!
Now, returning to the main topic of how implied volatility affects a position, 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 a put, but if the
put is too far in-the-nwney, a modest increase in implied volatility still won't budge
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 a short put would still remain at risk of assignment.
STRADDLE OR STRANGLE BUYING AND SELLING
Since owning a straddle involves owning both a put and a call with the same terms,
it is fairly evident that an increase in implied volatility will be very beneficial for a
straddle buyer. A sort 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 a long straddle. Thus, if a straddle 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 a move 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, a straddle 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."
A strangle is the same as a straddle, 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 a great 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's look at a call 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. I would suspect that this comes as something of a surprise to a good 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 a call 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 a reprint 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 a bullish call position.
Of course, it's highly unlikely that implied volatility would change much in the
course of just one day while the stock price remained unchanged. So, to get a bet
ter handle on what to expect, one really to needs to look at what might happen at
some future time (say a couple 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 a
bull spread at expiration, although if you look closely you'll see that it doesn't widen
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 a bull 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
e a.
"' -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 a rather 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 a sub
stantial rrwve. Longer-term spreads have even less movement.
As a corollary, note that if implied volatility shrinks while the stock rises, the
profit outlook will improve. Graphically, using Figure 37-4, if one's profit picture
moves from the 80% curve to the 20% curve on the right-hand side of the chart, that
is a positive development. Of course, if the stock drops and the implied volatility
drops too, then one's losses would be worse - witness the left-hand side of the graph
in Figure 37-4.
A graph 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 a volatile stock, you won't get much expansion
in the spread even if the stock makes a nice 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't expand 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 a way 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 a strike initially near the current stock price) would make a nice
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 a striking price of
100 and the 90-110 call bull spread, both having implied volatility of 80%. Quite
clearly, the call purchase dominates to a great 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 a positive 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,
a rise 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 a reasonable 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, a greater 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's fun
damental or technical - and comes to a conclusion that the stock ( or futures contract
or index) is ready for a bullish 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 a bull spread instead? It
costs less and it's bullish, too."
Fairly quickly, the underlying moves higher - a good prediction by the trader,
and a timely one as well. If the move is a violent one, especially in the futures mar
ket, implied volatility might increase as well. If you had bought calls, you'd be a happy
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 a friend of the bull spread, while it is a friendly ally
of the outright call purchase. Somewhat surprisingly, many option traders don't real
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 a call option is too
expensive and want to reduce its cost, ti:y this strategy: Buy the call and simultane
ously sell a credit put spread (bull spread) using slightly out-of-the-money puts. This
strategy reduces the call's net cost and maintains upside potential (although it
increases downside risk, but at least it is still a fixed risk).
Example: With XYZ at 100, a trader 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's options 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 a possible 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
a means of reducing the net cost of the calls. If he's right 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 a total 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
a trading stop of any sort, he probably would not be at risk for the entire 17 points,
since he wouldn't hold on while the stock fell all the way to 80 and below. Note also
that the bullish spread position would have a loss of 10 points (the same as the call)
at a price 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 a
price 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's a
total 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 a greater
amount of money if the underlying stock should collapse heavily. Still, if one is truly
bullish, and if one employs a reasonably 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 a credit spread - a bull spread
with puts. Assume that a stock is selling at 100, and one is going to sell a put with a
110 strike and buy a put with a 90 strike. That is a put 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 a total 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
a trading stop of any sort, he probably would not be at risk for the entire 17 points,
since he wouldn't hold on while the stock fell all the way to 80 and below. Note also
that the bullish spread position would have a loss of 10 points (the same as the call)
at a price 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 a
price 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's a
total 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 a greater
amount of money if the underlying stock should collapse heavily. Still, if one is truly
bullish, and if one employs a reasonably 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 a credit spread - a bull spread
with puts. Assume that a stock is selling at 100, and one is going to sell a put with a
110 strike and buy a put with a 90 strike. That is a put 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 a bull 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 a put 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 a price 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 a terrible thing. There will be a request 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 a stock trader if he can't afford 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 a more 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 a call bull spread is a superior
choice over a put bull spread. Early assignment is not really a consideration for most
call spreads.
In both cases, however, a more serious problem exists, and that is that the
spread does not widen out much even when the stock makes a nice bullish move.
Thus, once again it is actually better to buy a call option in most cases than to use the
bull spread, because profits are larger and an increase in implied volatility is a favor
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 a rapid 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 a bearish situation? In a vertical put spread one buys
the put with the higher strike and sells the put with the lower strike to construct a
simple put bear spread. Actually, a sudden 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 a debit. 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. A very quick profit could be had. So there's a difference right away
between a debit call bull spread (which loses money when implied volatility sudden
ly increases) and a debit put bear spread.
Unfortunately, the other major drawback - that the spread doesn't widen out
much if the underlying makes a favorable move - is still true. Figure 37-9 shows a
bear 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 a price 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't widen out
much, even when the stock drops a lot 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't widen out much. That may not be a psychologically pleasing
strategy, because one won't make 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 a spread. In these cases, it is true with respect to puts, much as it was with call
options. Spreading often unnecessarily complicates a trader's outlook.
CALENDAR SPREADS
In the earlier chapter on calendar spreads, it was mentioned that an increase in
implied volatility will cause a calendar 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 a calendar
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 a huge
effect on the value of a calendar 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's time. 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
A common mistake that calendar spreaders make is to think that such a spread
looks overly attractive on a very volatile stock. Consider the same stock as above, still
trading at 100, but for some reason implied volatility has skyrocketed to 80% (per
haps a takeover 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 a very 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 a calendar
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
a profit of a few cents (4.36 to 4.77), and if the underlying stock were farther from
the strike price at expiration, there would probably be a loss rather than a profit.
The point to be remembered is that a calendar spread is a "long volatility" play
(and a reverse calendar spread is just the opposite). Evaluate the position's risk 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, a ratio call write is really just the equivalent of
a straddle sale - a strategy 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. A call 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, a call 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 a
naked 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 a ratio call
spread, both components are harmed by an increase in implied volatility. Conversely,
a decrease in implied volatility would be beneficial to a ratio 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 a call bull spread does not widen out much if
the underlying stock makes a quick upward move. The spread won't widen 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 a put bear spread plus some additional naked put options. If the under
lying falls in price, while implied volatility increases - a very 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, a trader 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 a probability calculator with
some inflated volatility estimates to see just what the chances are of the spread get
ting into real trouble.
BACKSPREADS
A call backspread is merely the opposite of a call ratio spread. Thus, any of the earli
er commentary about how an increase in implied volatility is detrimental to a ratio
spread can be reversed when discussing the backspread. An increase in implied
volatility will be beneficial to a backspread strategy, while a decrease 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 a
backspread, such a decrease 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 a strategist 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 a
feeling 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 a mistake, for the most knowledgeable option traders plan for
volatility risk at all times. Understanding and handling that risk can have a positive
effect on an option trader's profits.

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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, a number 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 a word, 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 a review for those not familiar with mathematical distributions, the lognor
mal distribution is what's commonly used to describe stock prices because its shape
is intuitively similar to the way stocks behave - they can't go below zero, they can rise
to infinity, and most of the time they don't go much of anywhere. On top of that, the
distribution's shape is based on the historical volatility of the underlying instrument.
In a lognormal distribution (and normal distribution, too), stocks remain within 3
standard deviations of their current price 99. 7 4% of the time. A standard deviation
783

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Chapter 38: The Distribution of Stock Prices 187
Lest you think that this example was biased by the fact that it was taken during
a strong run in the NASDAQ market, here's another example, conducted with a dif
ferent set of data- using stock prices between June 1 and July 18, 1999 (also 30 trad
ing days in length). At that time, there were fewer large moves; about 250 stocks out
of 2,500 or so had moves of more than three standard deviations. However, that's still
one out of ten - way more than you've been led to expect if you believe in the nor
mal distribution. The results are shown in Table 38-3.
TABLE 38-3.
More stock price movements.
Total Stocks: 2,447 Dates: 6/1 /99-7 /18/99
Upside Moves:
Downside Moves:
3cr
104
54
4cr
28
19
Scr
13
7
>6cr
12
14
Total number of stocks moving >=3cr: 251 ( 10% of the stocks studied)
Total
157
94
Finally, one more example was conducted, using the least volatile period that
we had in our database - July of 1993. Those results are in Table 38-4.
TABLE 38-4.
Stock price movements during a nonvolatile period.
Total Stocks: 588 Dates: 7 /1 /93-8/17 /93
3cr 4cr Scr >6cr Total
Upside Moves: 14 5 1 1 21
Downside Moves: 28 5 3 4 40
Total number of stocks moving >=3cr: 61 ( 10% of the stocks studied)
At first glance, it appears that the number of large stock moves diminished dra
matically during this less volatile period in the market - until you realize that it still
represents 10% of the stocks in the study. There were just a lot fewer stocks with list
ed options in 1993 than there were in 1999, so the database is smaller (it tracks only
stocks with listed options). Once again, this means that there is a far greater chance
for large standard deviations moves - about one in ten - than the nearly zero percent
chance that the lognormal distribution would indicate.
VOLATILITY BUYER'S RULE!
The point of the previous discussion is that stocks move a lot farther than you might
expect. Moreover, when they make these moves, it tends to be with rapidity, gener-

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788 Part VI: Measuring and Trading VolatiDty
ally including gap moves. There are not always gap moves, though, over a study of
this length. Sometimes, there will be a more gradual transition. Consider the fact that
one of the stocks in the study moved 5.8 sigma in the 30 days. There weren't any huge
gaps during that time, but anyone who was short calls while the stock made its run
surely didn't think it was a gradual advance.
So, what does this information mean to the average option trader? For one,
you should certainly think twice about selling stock options in a potentially 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, a cov
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't mind" doing so when he sold the puts to begin with).
Even spreading has problems along these lines. For example, a vertical spread
limits profits so that one can't participate 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 a riskier 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 a somewhat 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
TABLE 38-5.
Index price movements.
Total Indices: 135
Upside Moves:
Downside Moves:
TABLE 38-6.
3cr
32
None
4cr
15
Scr
3
Index price movements, least volatile period.
Total Indices: 66
Upside Moves:
Downside Moves:
3cr
l
3
4cr
l
0
Scr
0
0
789
Dates: 10/22/99-12/7/99
>6cr
0
Total
50
Dates: 7/1/93-8/17/93
>6cr
0
0
Total
2
3
Total number of indices moving >=3cr: 5 (8% of the indices studied)
indices made oversized moves - probably a bias because of the strong Internet stock
market during that time period. The low-volatility period showed a more reasonable,
but still somewhat eye-opening, 8% making moves of greater than three standard
deviations. So, even selling index options isn't as safe as it's cracked up to be, when
they can make moves of this size, defying the "normal" probabilities.
Since that period in 1999 was rather volatile, and all on the upside, the same
study was conducted, once again using the least volatile period of July 1993.
In Table 38-6, the numbers are lower than they are for stocks, but still much
greater than one might expect according to the lognormal distribution.
These examples of stock price movement are interesting, but are not rigorous
ly complete enough to be able to substantiate the broad conclusion that stock prices
don't behave lognormally. Thus, a more complete study was conducted. The follow
ing section presents the results of this research.
THE DISTRIBUTION OF STOCK PRICES
The earlier examples pointed out that, at least in those specific instances, stock price
movements don't conform to the lognormal distribution, which is the distribution
used in many mathematical models that are intended to describe the behavior of
stock and option prices. This isn't new information to mathematicians; papers dating
back to the mid-1960s have pointed out that the lognormal distribution is flawed.
However, it isn't a terrible description of the way that stock prices behave, so many
applications have continued to use the lognormal distribution.

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794 Part VI: Measuring and Trading Volatility
FIGURE 38-3.
Stock price distribution, IBM, 7-year.
7
6
5
~ 4 :;:,
C: :::,
8 3
2
o----------------+---,.._,._""-¥-+-
-4.o -3.o -2.0 -1.0 o.o +1.0 +2.0 +3.o +4.o
Sigmas
dangerous naked puts and long stock on margin can be on days like this. No proba
bility calculator is going to give much likelihood to a day like this occurring, but it did
occur and it benefited those holding long puts greatly, while it seriously hurt others.
In addition to distributions for individual dates, distributions for individual stocks
were created for the time period in question. The graph for IBM, using data from the
same study as above (September 1993 to April 2000) is shown in Figure 38-3. In the
next graph, Figure 38-4, a longer price history of IBM is used to draw the distribution:
1987 to 2000. Both graphs depict 30-day movements in IBM.
FIGURE 38-4.
Actual stock price distribution, IBM, 13-year.
13
12
11
10
9
o a ,-
E 7
g 6
(.) 5
4
3
2
o -4.0 -3.0 -2.0 -1.0 0.0 + 1.0 +2.0 +3.0 +4.0
Sigmas

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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 a longer-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's clear that this longer-term distribution conforms more
closely to the normal distribution in that it has a sort 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's because there would be more bearish periods in the longer study (1987, 1989,
and 1990 all had some rather nasty periods). Still, this doesn't detract 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 a lot 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't all 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 a resounding, "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 a high number and it therefore means that the stock would have had
to move a very long way to move four standard deviations. In 1993, however, when
the market was in the doldrums, historical volatility was low, and so a much smaller

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794 Part VI: Measuring and Trading Volatility
FIGURE 38-3.
Stock price distribution, IBM, 7-year.
7
6
5
0 4
:g
::,
0 3 ()
2
0f-<--,.Jil,,J:.~------+---+---+----+--____;_,=!!:¥+-
-4.0 -3.0 -2.0 -1.0 0.0 + 1.0 +2.0 +3.0 +4.0
Sigmas
dangerous naked puts and long stock on margin can be on days like this. No proba
bility calculator is going to give much likelihood to a day like this occurring, but it did
occur and it benefited those holding long puts greatly, while it seriously hurt others.
In addition to distributions for individual dates, distributions for individual stocks
were created for the time period in question. The graph for IBM, using data from the
same study as above (September 1993 to April 2000) is shown in Figure 38-3. In the
next graph, Figure 38-4, a longer price history of IBM is used to draw the distribution:
1987 to 2000. Both graphs depict 30-day movements in IBM.
FIGURE 38-4.
Actual stock price distribution, IBM, 13-year.
13
12
11
10
9
0 8 ,...
E 7
5 6
() 5
4
3
2
1
o -4.0 -3.0 -2.0 -1.0 0.0 + 1.0 +2.0 +3.0 +4.0
Sigmas

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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 a longer-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's clear that this longer-term distribution conforms more
closely to the normal distribution in that it has a sort 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's because there would be more bearish periods in the longer study (1987, 1989,
and 1990 all had some rather nasty periods). Still, this doesn't detract 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 a lot 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't all 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 a resounding, "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 a high number and it therefore means that the stock would have had
to move a very long way to move four standard deviations. In 1993, however, when
the market was in the doldrums, historical volatility was low, and so a much smaller

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796 Part VI: Measuring and Trading Volatility
move was needed to register a 4-standard deviation move. To see a specific 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't you think it's a little strange that
the chart doesn't show any moves of greater than minus 4.0 standard deviations? The
reason is that IBM's historical 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 a great deal of certainty that stocks do not conform to the normal
distribution. Actually, the normal distribution is a decent approximation of stock
price movement rrwst of the time, but it's these "outlying" results that can hurt any
one using it as a basis for a nonvolatility strategy.
Scientists working on chaos theo:ry have been trying to get a better 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's author is purporting to have "invented" things that R. N.
Elliott discovered years ago. I don't know about that, but I do 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't be predicted. Is this therefore a useless 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't be 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 a call 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't mean that it's okay to buy overpriced options just
because stocks can make large moves with a greater frequency than most option

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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 a second 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't a good 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't be overstating things initially. An example may help to demonstrate this
technique.
Example: Suppose that a trader is considering buying a straddle on XYZ. The five
month straddle is selling for a price of 8, with the stock currently trading near 40. A
probability 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's answer 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 a five-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 a
historical 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
a conservative 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 a position with a
negative 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't be unpleasantly sur
prised by a higher 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 a sufficient guide for volatility.
When a longer lookback period is required, there is another method that can be
used: Go back in a historical 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 a fairly 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 a series of chaotic news events that have been
affecting XYZ. A trader wants to trade XYZ's options, but needs a good 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's trading history, he sees
that it is not normally this volatile. Since he suspects that XYZ's recent trading histo
ry is not typical of its true long-term performance, what volatility should he use in
either an option model or a probability 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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Chapter 38: The Distribution of Stock Prices 801
back over the last 1,000 trading days for XYZ. A 100-day historical volatility can be
computed, using 100 consecutive trading days of data, for 901 of those days (begin
ning with the 100th day and continuing through the l,000th day, which is presumably
the current trading day). Admittedly, these are not completely unique time periods;
there would only be ten non-overlapping (independent) consecutive 100-day periods
in 1,000 days of data. However, let's assume that the 901 periods are used. One can
then arrive at a distribution of 100-day historical volatilities. Suppose it looks some
thing like this:
Percentile 100-Day Historical
oth 34%
10th 37%
20th 43%
30th 45%
40th 46%
50th 48%
60th 51%
70th 58%
aoth 67%
90th 75%
1 ooth 81%
In other words, the 901 historical volatilities (100 days in each) are sorted and then
the percentiles are determined. The above table is just a snapshot of where the per
centiles lie. The range of those 901 volatilities is from 34% on the low side to 81 % on
the high side. Notice also that there is a very flat grouping from about the 20th per
centile to the 60th percentile: The 100-day historical volatility was between 43% and
51 % over that entire range. The median of the above figures is 48% - the 100-day
volatility at the 50th percentile.
Referring to the early part of this example, the current 100-day historical is
80%, a very high reading in comparison to what the measures were over the past
1,000 days, and certainly much higher than the median of 48%.
One could perform similar analyses on the 1,000 days of historical data to deter
mine where the 10-day, 20-day, and 50-day historical volatilities were over that time.
Those, too, could be sorted and arranged in percentile format, using the 50% per
centile (median) as a good estimate of volatility. After such computations, the trader
might then have this information:

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

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

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

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

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

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

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

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

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

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