Qapter 5: Naked Call Writing 133
THE UNCOVERED (NAKED) CALL OPTION
When one sells acall option without owning the underlying stock or any equivalent
security (convertible stock or bond or another call option), he is considered to have
written an uncovered call option. This strategy has limited profit potential and theo
retically unlimited loss. For this reason, this strategy is unsuitable for some investors.
This fact is not particularly attractive, but since there is no actual cash investment
required to write anaked call ( the position can be financed with collateral loan value
of marginable securities), the strategy can be operated as an adjunct to many other
investment strategies.
Asimple example will outline the basic profit and loss potential from naked
writing.
Example: XYZ is selling at 50 and a July 50 call is selling for 5. If one were to sell the
July 50 call naked - that is, without owning XYZ stock, or any security convertible into
XYZ, or another call option on XYZ - he could make, at most, 5 points of profit. This
profit would accrue if XYZ were at or anywhere below 50 at July expiration, as the
call would then expire worthless. If XYZ were to rise, however, the naked writer
could potentially lose large sums of money. Should the stock climb to 100, say, the
call would be at aprice of 50. If the writer then covered (bought back) the call for aprice of 50, he would have aloss of 45 points on the transaction. In theory, this loss
is unlimited, although in practice the loss is limited by time. The stock cannot rise an
infinite amount during the life of the call. Clearly, defensive strategies are important
in this approach, as one would never want to let aloss run as far as the one here.
Table 5-1 and Figure 5-1 (solid line) depict the results of this position at July expira
tion. Note that the break-even point in this example is 55. That is, if XYZ rose 10%,
or 5 points, at expiration, the naked writer would break even. He could buy the call
back at parity, 5 points, which is exactly what he sold it for. There is some room for
error to the upside. Anaked write will not necessarily lose money if the stock moves
up. It will only lose if the stock advances by more than the amount of the time value
premium that was in the call when it was originally written.
Naked writing is not the same as ashort sale of the underlying stock. While both
strategies have large potential risk, the short sale has much higher reward potential,
but the naked write will do better if the underlying stock remains relatively
unchanged. It is possible for the naked writer to make money in situations when the
short seller would have lost money. Using the example above, suppose one investor
had written the July 50 call naked for 5 points while another investor sold the stock
short at 50. If XYZ were at 52 at expiration, the naked writer could buy the call back
at parity, 2 points, for a 3-point profit. The short seller would have a 2-point loss.

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134
TABLE 5-1.
Position at July expiration.
XYZ Price at Call Price at
Expiration Expiration
30 0
40 0
50 0
55 5
60 10
70 20
80 30
FIGURE 5-1.
Uncovered (naked) call write.
+$500
C
0
~ ·15..
Xwcu
(/J
~ ...I
0
lt,
.... ......
", Naked Write
45 SO',
.... .. .... .. .. Short Sale ,,
.. ..
Stock Price at Expiration
..
Part II: Call Option Strategies
Profit on
Naked Write
+$ 500
+ 500
+ 500
0
500
- 1,500
- 2,500
.. .... ..
~
Moreover, the short seller pays out the dividends on the underlying stock, whereas
the naked call writer does not. The naked call will expire, of course, but the short sale
does not. This is asituation in which the naked write outperforms the short sale.
However, ifXYZ were to fall sharply- to 20, say- the naked writer could only make
5 points while the short seller would make 30 points. The dashed line in Figure 5-1
shows how the short sale of XYZ at 50 would compare with the naked write of the
July 50 call. Notice that the two strategies are equal at 45 at expiration; they both

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Cl,apter 5: Naked Call Writing 135
make a 5-point profit there. Above 45, the naked write does better; it has larger prof
its and smaller losses. Below 45, the short sale does better, and the farther the stock
falls, the better the short sale becomes in comparison. As will be seen later, one can
more closely simulate ashort sale by writing an in-the-money naked call.
INVESTMENT REQUIRED
The margin requirements for writing anaked call are 20% of the stock price plus the
call premium, less the amount by which the stock is below the striking price. If the
stock is below the striking price, the differential is subtracted from the requirement.
However, aminimum of 10% of the stock price is required for each call, even if the
C-'Omputation results in asmaller number. Table 5-2 gives four examples of how the ini
tial margin requirement would be computed for four different stock prices. The 20%
collateral figure is the minimum exchange requirement and may vary somewhat among
different brokerage houses. The call premium may be applied against the requirement.
In the first line of Table 5-2, if the XYZ July 50 call were selling for 7 points, the $700
call premium could be applied against the $1,800 margin requirement, reducing the
actual amount that the investor would have to put up as collateral to $1,100.
TABLE 5-2.
Initial collateral requirements for four stock prices.
Coll
Written
XYZ July 50
XYZ July 50
XYZ July 50
XYZ July 50
Stock Price When
Coll Written
55
50
46
40
*Requirement cannot be less than 10%.
Coll
Price
$700
400
200
100
20% of
Stock Price
$1,100
1,000
920
800
Out-of-the
Money
Differential
$ 0
0
400
- 1,000
Total Margin
Requirement
$1,800
1,400
720
400*
In addition to the basic requirements, abrokerage firm may require that for acustomer to participate in uncovered writing, he have aminimum equity in his
account. This equity requirement may range from as low as $2,000 to as high as
$100,000. Since naked call writing is ahigh-risk strategy, some brokerage firms
require that the customer be able to show both financial wherewithal and option

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136 Part II: Call Option Strategies
trading experience before the account can be approved for naked call writing. In
addition, some brokers require that amaintenance requirement be applied against
each option written naked. This requirement, sometimes called akicker, is usually
less than $250 per call and is generally used by the broker to ensure that, should the
customer fail to respond to an assignment notice against his naked call, the commis
sion costs for buying and selling the underlying stock would be defrayed.
Naked Option Positions Are Marked to the Market Daily. This
means that the collateral requirement for the position is recomputed daily, just as in
the short sale of stock. The same margin formula that was described above is applied
and, if the stock has risen far enough, the customer will be required to deposit addi
tional collateral or close the position. The need for such amark to market is obvious.
If the underlying stock should rise, the brokerage firm must ensure that the customer
has enough collateral to cover the eventuality of buying the stock in the open market
and selling it at the striking price if an assignment notice should be received against
the naked call. The mark to market works to the customer'sfavor if the stock falls in
price. Excess collateral is then released back into the customer'smargin account, and
may be used for other purposes.
It is important to realize that, in order to write anaked call, collateral is all that
is required. No cash need be "invested" if one owns securities with sufficient collat
eral loan value.
Example: An investor owns 100 shares of astock selling at $60 per share. This stock
is worth $6,000. If the loan rate on stock is 50% of $6,000, this investor has acollat
eral loan value equal to 50% of $6,000, or $3,000. This investor could write any of the
naked calls in Table 5-2 without adding cash or securities to his account. Moreover,
he would have satisfied aminimum equity requirement of at least $6,000, since his
stock is equity.
This aspect of naked call writing - using collateral value to finance the writing
- is attractive to many investors, since one is able to write calls and bring in premi
ums without disturbing his existing portfolio. Of course, if the stock underlying the
naked call should rise too far in price, additional collateral may be called for by the
broker because of the mark to market. Moreover, there is risk whether cash or col
lateral is used. If one buys in anaked call at aloss, he will then be spending cash, cre
ating adebit in his account.
Regardless of how one finances anaked option position, it is generally agood
idea to allow enough collateral so that the stock can move all the way to the point at
which one would cover the option or take follow-up action. For example, suppose a

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Gapter 5: Naked Call Writing 137
stock is trading at 50 and one sells an April 60 call naked, figuring that he will cover
the call if the stock rises to 60 ( that is, if the option becomes an in-the-money option).
He should set aside enough collateral to margin the position as if the stock were at
60 (even though the actual margin requirement will be smaller than that). If he
allows that extra collateral, then he will never be forced into amargin call at astock
price prior to (that is, below) where he wanted to take follow-up action. Simply stat
ed, let the market take you out of aposition, not amargin call.
THE PHILOSOPHY OF SELLING NAKED OPTIONS
The first and foremost question one must address when thinking about selling naked
options (or any strategy, for that matter) is: "Can Ipsychologically handle the thought
of naked options in my account?" Notice that the question does not have anything to
do with whether one has enough collateral or margin to sell calls (although that, too,
is important) nor does it ask how much money he will make. First, one must decide
if he can be comfortable with the risk of the strategy. Selling naked options means
that there is theoretically unlimited risk if the underlying instrument should make alarge, sudden, adverse move. It is one'sattitude regarding that fact alone that deter
mines whether he should consider selling naked options. If one feels that he won'tbe
able to sleep at night, then he should not sell naked options, regardless of any profit
projections that might seem attractive.
If one feels that the psychological suitability aspect is not aroadblock, then he
can consider whether he has the financial wherewithal to write naked options. On the
surface, naked option margin requirements are not large (although in equity and
index options, they are larger than they were prior to the crash of 1987).
In general, one would prefer to let the naked options expire worthless, if at all
possible, without disturbing them, unless the underlying instrument makes asignifi
cant adverse move. So, out-of-the-money options are the usual choice for naked sell
ing. Then, in order to reduce ( or almost eliminate) the chance of amargin call, one
should set aside the margin requirement as if the underlying had already rrwved to
the strike price of the option sold. By allowing margin as if the underlying were
already at the strike, one will almost never experience amargin call before the under
lying price trades up to the strike price, at which time it is best to close the position
or to roll the call to another strike.
Thus, for naked equity call options, allow as collateral 20% of the highest naked
strike price. In this author'sopinion, the biggest mistake atrader can make is to ini
tiate trades because of margin or taxes. Thus, by allowing the "maximum" margin,
one can make trading decisions based on what'shappening in the market, as opposed
to reacting to amargin call from his broker.

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138 Part II: Call Option Strategies
"Suitability" also means not risking nwre nwney than one can afford to lose. If
one allows the "maximum" margin, then he won'tbe risking alarge portion of his
equity unless he is unable to cover when the underlying trades through the strike
price of his naked option. Gaps in trading prices would be the culprits that could pre
vent one from covering. Gaps are common in stocks, less common in futures, and
almost nonexistent in indices. Hence, index options are the options of choice when it
comes to naked writing. Index options are discussed later in the book.
Finally, there is one more "rule" that anaked option writer must follow:
Someone has to be watching the position at all times. Disasters could occur if one
were to go on vacation and not pay attention to his naked options. Usually, one'sbro
ker can watch the position, even if the trader has to call him from his vacation site.
In sum, then, to write naked options, one needs to be prepared psychological
ly, have sufficient funds, be willing to accept the risk, be able to monitor the position
every day, sell options whose implied volatility is extremely high, and cover any naked
options that become in-the-money options.
RISK AND REWARD
One can adjust the apparent risks and rewards from naked call writing by his selec
tion of an in-the-money or out-of-the-money call. Writing an out-of-the-money call
naked, especially one quite deeply out-of-the-money, offers ahigh probability of
achieving asmall profit. Writing an in-the-money call naked has the most profit
potential, but it also has higher risks.
Example: XYZ is selling at 40 and the July 50 is selling for½. This call could be sold
naked. The probability that XYZ could rise to 50 by expiration has to be considered
small, especially if there is not alarge amount of time remaining in the life of the call.
In fact, the stock could rise 25%, or 10 points, by expiration to aprice of 50, and the
call would still expire worthless. Thus, this naked writer has agood chance of realiz
ing a $50 profit, less commissions. There could, of course, be substantial risk in terms
of potential profit versus potential loss if the stock rises substantially in price by expi
ration. Still, this apparent possibility of achieving additional limited income with ahigh probability of success has led many investors to use the collateral value of their
portfolios to sell deeply out-of-the-money naked calls.
For those employing this technique, afavored position is to have astock at or
just about 15 and then sell the near-term option with striking price 20 naked. This
option would sell for one-eighth or one-quarter, perhaps, although at times there
might not be any bid at all. At this price, the stock would have to rally nearly one-

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C.,,er 5: Naked Call Writing 139
third, or 33%, for the writer to lose money. Although there are not usually many
optionable stocks selling at or just above $10 per share, these same out-of-the-money
writers would also be attracted to selling acall with astriking price 15 when the stock
is at 10, because a 50% upward move by the stock would be required for aloss to be
realized.
This strategy of selling deeply out-of-the-money calls has its apparent attraction
in that the writer is assured of aprofit unless the underlying stock can rally rather
substantially before the call expires. The danger in this strategy is that one or two
losses, perhaps amounting to only acouple of points each, could wipe out many peri
ods of profits. The stock market does occasionally rally heavily in ashort period, as
witnessed repeatedly throughout history. Thus, the writer who is adopting this strat
egy cannot regard it as asure thing and certainly cannot afford to establish the writes
and forget them. Close monitoring is required in case the market begins to rally, and
by no means should losses be allowed to accumulate.
The opposite end of the spectrum in naked call writing is the writing of fairly
deeply in-the-money calls. Since an in-the-money call would not have much time
value premium in it, this writer does not have much leeway to the upside. If the
stock rallies at all, the writer of the deeply in-the-money naked call will normally
experience aloss. However, should the stock drop in price, this writer will make
larger dollar profits than will the writer of the out-of-the-money call. The sale of the
deeply in-the-money call simulates the profits that ashort seller could make, at least
until the stock drops close to the striking price, since the delta of adeeply in-the
money call is close to 1.
Example: XYZ is selling at 60 and the July 50 call is selling for 10½. IfXYZ rises, the
naked writer will lose money, because there is only ½ of apoint of time value pre
mium in the call. If XYZ falls, the writer will make profits on apoint-for-point basis
until the stock falls much closer to 50. That is, if XYZ dropped from 60 to 57, the call
price would fall by almost 3 points as well. Thus, for quick declines by the stock, the
deeply in-the-money write can provide profits nearly equal to those that the short
seller could accumulate. Notice that if XYZ falls all the way to 50, the profits on the
written call will be large, but will be accumulating at aslower rate as the time value
premium builds up with the stock near the striking price.
If one is looking to trade astock on the short side for just afew points of nwve
ment, he might use adeeply in-the-nwney naked write instead of shorting the stock.
His investment will be smaller - 20% of the stock price for the write as compared to
50% of the stock price for the short sale - and his return will thus be larger. (The
requirement for the in-the-money amount is offset by applying the call'spremium.)

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140 Part II: Call Option Strategies
The writer should take great caution in ascertaining that the call does have some time
premium in it. He does not want to receive an assignment notice on the written call.
It is easiest to find time premium in the more distant expiration series, so the writer
would normally be safest from assignment by writing the longest-term deep in-the
money call if he wants to make abearish trade in the stock.
Example: An investor thinks that XYZ could fall 3 or 4 points from its current price
of 60 in aquick downward move, and wants to capitalize on that move by writing anaked call. If the April 40 were the near-term call, he might have the choice of sell
ing the April 40 at 20, the July 40 at 20¼, or the October 40 at 20½. Since all three
calls will drop nearly point for point with the stock in amove to 56 or 57, he should
write the October 40, as it has the least risk of being assigned. Atrader utilizing this
strategy should limit his losses in much the same way ashort seller would, by cover
ing if the stock rallies, perhaps breaking through overhead technical resistance.
ROLLING FOR CREDITS
Most writers of naked calls prefer to use one of the two strategies described above.
The strategy of writing at-the-money calls, when the stock price is initially close to the
striking price of the written call, is not widely utilized. This is because the writer who
wants to limit risk will write an out-of-the-money call, whereas the writer who wants
to make larger, quick trading profits will write an in-the-money call. There is, how
ever, astrategy that is designed to utilize the at-the-money call. This strategy offers ahigh degree of eventual success, although there may be an accumulation of losses
before the success point is reached. It is astrategy that requires large collateral back
ing, and is therefore only for the largest investors. We call this strategy "rolling for
credits." The strategy is described here in full, although it can, at times, resemble a
Martingale strategy; that is, one that requires "doubling up" to succeed, and one that
can produce ruin if certain physical limits are reached. The classic Martingale strat
egy is this: Begin by betting one unit; if you lose, double your bet; if you win that bet,
you'll have netted aprofit of one unit (you lost one, but won two); if you lost the sec
ond bet, double your bet again. No matter how many times you lose, keep doubling
your bet each time. When you eventually win, you will profit by the amount of your
original bet (one unit). Unfortunately, such astrategy cannot be employed in real life.
For example, in agambling casino, after enough losses, one would bump up against
the table limit and would no longer be able to double his bet. Consequently, the strat
egy would be ruined just when it was at its worst point. While "rolling for credits"
doesn'texactly call for one to double the number of written calls each time, it does
require that one keep increasing his risk exposure in order to profit by the amount of
that original credit sold. In general, Martingale strategies should be avoided.

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Cl,apter 5: Naked Call Writing 141
In essence, the writer who is rollingf or credits sells the most time premium that
he can at any point in time. This would generally be the longest-term, at-the-money
call. If the stock declines, the writer makes the time premium that he sold. However,
if the stock rises in price, the writer rolls up for acredit. That is, when the stock
reaches the next higher striking price, the writer buys back the calls that were origi
nally sold and sells enough long-term calls at the higher strike to generate acredit.
In this way, no debits are incurred, although arealized loss is taken on the rolling up.
If the stock persists and rises to the next striking price, the process is repeated.
Eventually, the stock will stop rising - they always do - and the last set of written
options will expire worthless. At that time, the writer would make an overall profit
consisting of an amount equal to all the credits that he had taken in so far. In reality,
most of that credit will simply be the initial credit received. The "rolls" are done for
even money or asmall credit. In essence, the increased risk generated by continual
ly rolling up is all geared toward eventually capturing that initial credit. The similar
ity to the Martingale strategy is strongest in this regard: One continually increases his
risk, knowing that when he eventually wins (i.e., the last set of options expires worth
less), he profits by the amount of his original "bet."
There are really only two requirements for success in this strategy. The first is
that the underlying stock eventually fall back, that it does not rise indefinitely. This is
hardly arequirement; it is axiomatic that all stocks will eventually undergo acorrec
tion, so this is asimple requirement to satisfy. The second requirement is that the
investor have enough collateral backing to stay with the strategy even if the stock runs
up heavily against him.
This is amuch harder requirement to satisfy, and may in fact tum out to be
nearly impossible to satisfy. If the stock were to experience astraight-line upward
move, the number of calls written might grow so substantially that they would
require an unrealistically large amount of collateral (margin). At aminimum, this
strategy is applicable only for the largest investors. For such well-collateralized
investors, this strategy can be thought of as away to add income to aportfolio. That
is, alarge stock portfolio'sequity may be used to finance this strategy through its loan
value. There would be no margin interest charges, because all transactions are cred
it transactions. (No debits are created, as long as the Martingale "limits" are not
reached.) The securities portfolio would not have to be touched unless the strategy
were terminated before the last set of calls expired worthless.
This is where the danger comes in: If the stock upon which the calls are written
rises so fast that one completely uses up all of his collateral value to finance the naked
calls, and then one is required to roll again, the strategy could result in large losses.
For awhile, one could simply continue to roll the same number of calls up for deb
its, but eventually those debits would mount in size if the stock persisted in rising. At

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142 Part II: Call Option Strategies
this point, even if the stock did finally decline enough for the last set of calls to expire
worthless, the overall strategy might still have been operated at aloss.
Example: The basic strategy in the case of rising stock is shown in Table 5-3. Note
that each transaction is acredit and that all ( except the last) involve taking arealized
loss.
This example assumes that the stock rose so quickly that alonger-term call was
never available to roll into. That is, the October calls were always utilized. If there
were alonger-term call available (the January series, for example), the writer should
roll up and out as well. In this way, larger credits could be generated. The number of
calls written increased from 5 to 15 and the collateral required as backing for the
writing of the naked calls also increased heavily. Recall that the collateral require
ment is equal to 20% of the stock price plus the call premium, less the amount by
which the call is out-of-the-money. The premium may be used against the collateral
requirements. Using the stock and call prices of the example above, the investment
is computed in Table 5-4. While the number of written calls has tripled from 5 to 15,
the collateral requirement has more than quadrupled from $5,000 to $21,000. This is
why the investor must have ample collateral backing to utilize this strategy. The gen
eral philosophy of the large investors who do apply this strategy is that they hope to
eventually make aprofit and, since they are using the collateral value of large securi
ty positions already held, they are not investing any more money. The strategy does
not really "cost" these investors anything. All profits represent additional income and
do not in any way disturb the underlying security portfolio. Unfortunately, losses
taken due to aborting the strategy could seriously affect the portfolio. This is why the
investor must have sufficient collateral to carry through to completion.
The sophisticated strategist who implements this strategy will generally do
more rolling than that discussed in the simple example above. First, if the stock
drops, the calls will be rolled down to the next strike - for acredit - in order to con
stantly be selling the most time premium, which is always found in the longest-term
at-the-money call. Furthermore, the strategist may want to roll out to amore distant
expiration series whenever the opportunity presents itself. This rolling out, or for
ward, action is only taken when the stock is relatively unchanged from the initial
price and there is no need to roll up or down.
This strategy seems ve:ry attractive as long as one has enough collateral backing.
Should one use up all of his available collateral, the strategy could collapse, causing
substantial losses. It may not necessarily generate large rates of return in rising mar
kets, but in stable or declining markets the generation of additional income can be
quite substantial. Since the investor is not putting up any additional cash but is uti-

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144 Part II: Call Option Strategies
security, the strategist should diversify several moderately sized positions throughout
avariety of underlying stocks. If he does this, he will probably never have to exceed
the position limit of contracts short in any one security.
Even with as many precautions as one might take, there is no guarantee that
one would have the collateral available to withstand again of 1000% or more, such
as is occasionally seen with high-flying tech stocks or new IPOs. One would probably
be best served, if he really wants to operate this strategy, to stick with stocks that are
well capitalized (some of the biggest in the industry), so that they are less suscepti
ble to such violent upside moves. Even then, though, there is no guarantee that one
will not run out of collateral in asharply rising market, because it is impossible to esti
mate with complete certainty just how far any one stock might advance in aparticu
lar period of time.
TIME VALUE PREMIUM IS A MISNOMER
Once again, the topic of time value premium is discussed, as it was in Chapter 3.
Many novice option traders think that if they sell an out-of-the-money option
(whether covered or naked), all they have to do is sit back and wait to collect the pre
mium as time wears it away. However, alot of things can happen between the time
an option is sold and its expiration date. The stock can move agreat deal, or implied
volatility can skyrocket. Both are bad for the option seller and both completely coun
teract any benefit that time decay might be imparting. The option seller must con
sider what might happen during the life of the option, and not simply view it as astrategy to hold the option until expiration. Naked call writers, especially, should
operate with that thought in mind, but so should covered call writers, even though
most don't. What the covered writer gives away is the upside; and if he constantly
sells options without regard to the possibilities of volatility or stock price increases,
he will be doing himself adisservice.
So, while it is still proper to refer to the part of an option'sprice that is not
intrinsic value as "time value premium," the knowledgeable option trader under
stands that it is also more heavily influenced by volatility and stock price movement
than by time.
SUMMARY
In amajority of cases, naked call writing is applied as adeeply out-of-the-money
strategy in which the investor uses the collateral value of his security holdings to par
ticipate in astrategy that offers alarge probability of making avery limited profit. It
is apoor strategy, because one loss may wipe out many profits. The trader who

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Ratio Call Writing
Two basic types of call writing have been described in previous chapters: covered call
writing, in which one owns the underlying stock and sells acall; and naked call writ
ing. Ratio writing is acombination of these two types of positions.
THE RATIO WRITE
Simply stated, ratio call writing is the strategy in which one owns acertain number
of shares of the underlying stock and sells calls against more shares than he owns.
Thus, there is aratio of calls written to stock owned. The most common ratio is the
2:1 ratio, whereby one owns 100 shares of the underlying stock and sells 2 calls. Note
that this type of position involves writing anumber of naked call options as well as anumber of covered options. This resulting position has both downside risk, as does acovered write, and unlimited upside risk, as does anaked write. The ratio write gen
erally wilI provide much larger profits than either covered writing or naked writing if
the underlying stock remains relatively unchanged during the life of the calls.
However, the ratio write has two-sided risk, aquality absent from either covered or
naked writing.
Generally, when an investor establishes aratio write, he attempts to be neutral
in outlook regarding the underlying stock. This means that he writes the calls with
striking prices closest to the current stock price.
Example: Aratio write is established by buying 100 shares of XYZ at 49 and selling
two XYZ October 50 calls at 6 points each. If XYZ should decline in price and be
anywhere below 50 at October expiration, the calls will expire worthless and the
writer will make 12 points from the sale of the calls. Thus, even if XYZ drops 12
points to aprice of 37, the ratio writer will break even. The stock loss of 12 points
146

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Otapter 6: Ratio Call Writing 147
would be offset by a 12-point gain on the calls. As with any strategy in which calls are
sold, the maximum profit occurs at the striking price of the written calls at expiration.
In this example, if XYZ were at 50 at expiration, the calls would still expire worthless
for a 12-point gain and the writer would have a 1-point profit on his stock, which has
moved up from 49 to 50, for atotal gain of 13 points. This position therefore has
ample downside protection and arelatively large potential profit. Should XYZ rise
above 50 by expiration, the profit will decrease and eventually become aloss if the
stock rises too far. To see this, suppose XYZ is at 63 at October expiration. The calls
will be at 13 points each, representing a 7-point loss on each call, because they were
originally sold for 6 points apiece. However, there would be a 14-poirit gain on the
stock, which has risen from 49 to 63. The overall net is abreak-even situation at 63 -
a 14-point gain on the stock offset by 14 points ofloss on the options (7 points each).
Table 6-1 and Figure 6-1 summarize the profit and loss potential of this example at
October expiration. The shape of the graph resembles aroof with its peak located at
the striking price of the written calls, or 50. It is obvious that the position has both
large upside risk above 63 and large downside risk below 37. Therefore, it is imper
ative that the ratio writer plan to take follow-up action if the stock should move out
side these prices. Follow-up action is discussed later. If the stock remains within the
range 37 to 63, some profit will result before commission charges. This range
between the downside break-even point and the upside break-even point is called the
profit range.
This example represents essentially aneutral position, because the ratio writer
will make some profit unless the stock falls by more than 12 points or rises by more
than 14 points before the expiration of the calls in October. This is frequently an
attractive type of strategy to adopt because, normally, stocks do not move very far in
TABLE 6-1.
Profit and loss at October expiration.
XYZ Price at Stock Call Profit Total
Expiration Profit Price on Calls Profit
30 -$1,900 0 +$1,200 -$ 700
37 - 1,200 0 + 1,200 0
45 400 0 + 1,200 + 800
50 + 100 0 + 1,200 + 1,300
55 + 600 5 + 200 + 800
63 + 1,400 13 - 1,400 0
70 + 2,100 20 - 2,800 - 700

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148
FIGURE 6-1.
Ratio write (2: 1 ).
+$1,300
C
0
e ·5.
X
LU
al
rn rn
.3
0
-ea.
Part II: Call Option Strategies
Stock Price at Expiration
a 3- or 6-month time period. Consequently, this strategy has arather high probabili
ty of making alimited profit. The profit in this example would, of course, be reduced
by commission costs and margin interest charges if the stock is bought on margin.
Before discussing the specifics of ratio writing, such as investment required,
selection criteria, and follow-up action, it may be beneficial to counter two fairly
common objections to this strategy. The first objection, although not heard as fre
quently today as when listed options first began trading, is "Why bother to buy 100
shares of stock and sell 2 calls? You will be naked one call. Why not just sell one
naked call?" The ratio writing strategy and the naked writing strategy have very little
in common except that both have upside risk. The profit graph for naked writing
(Figure 5-1) bears no resemblance to the roof-shaped profit graph for aratio write
(Figure 6-1). Clearly, the two strategies are quite different in profit potential and in
many other respects as well.
The second objection to ratio writing for the conservative investor is slightly
more valid. The conservative investor may not feel comfortable with aposition that
has risk if the underlying stock moves up in price. This can be apsychological detri
ment to ratio writing: When stock prices are rising and everyone who owns stocks is
happy and making profits, the ratio writer is in danger of losing money. However, in
apurely strategic sense, one should be willing to assume some upside risk in
exchange for larger profits if the underlying stock does not rise heavily in price. The

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Chapter 6: Ratio Call Writing 149
covered writer has upside protection all the way to infinity; that is, he has no upside
risk at all. This cannot be the mathematically optimum situation, because stocks
never rise to infinity. Rather, the ratio writer is engaged in astrategy that makes its
profits in aprice range more in line with the way stocks actually behave. In fact, if
one were to try to set up the optimum strategy, he would want it to make its most
profits in line with the most probable outcomes for astock'smovement. Ratio writ
ing is such astrategy.
Figure 6-2 shows asimple probability curve for astock'smovement. It is most
likely that astock will remain relatively unchanged and there is very little chance that
it will rise or fall agreat distance. Now compare the results of the ratio writing strat
egy with the graph of probable stock outcomes. Notice that the ratio write and the
probability curve have their "peaks" in the same area; that is, the ratio write makes
its profits in the range of most likely stock prices, because there is only asmall chance
that any stock will increase or decrease by alarge amount in afixed period of time.
The large losses are at the edges of the graph, where the probability curve gets very
low, approaching zero probability. It should be noted that these graphs show the prof
it and probability at expiration. Prior to expiration, the break-even points are closer
to the original purchase price of the stock because there will still be some time value
premium remaining on the options that were sold.
FIGURE 6-2.
Stock price probability curve overlaid on profit graph of ratio
write.
+$1,300 Probability
Curve
Stock Price

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150 Part II: Call Option Strategies
INVESTMENT REQUIRED
The ratio writer has acombination of covered writes and naked writes. The margin
requirements for each of these strategies have been described previously, and the
requirements for aratio writing strategy are the sum of the requirements for anaked
write and acovered write. Ratio writing is normally done in amargin account,
although one could technically keep the stock in acash account.
Example: Ignoring commissions, the investment required can be computed as fol
lows: Buy 100 XYZ at 49 on 50% margin and sell 2 XYZ October 50 calls at 6 points
each (Table 6-2). The commissions for buying the stock and selling the calls would be
added to these requirements. Ashorter formula (Table 6-3) is actually more desirable
to use. It is merely acombination of the investment requirements listed in Table 6-2.
In addition to the basic requirement, there may be minimum equity require
ments and maintenance requirements, since naked calls are involved. As these vary
from one brokerage firm to another, it is best for the ratio writer to check with his
broker to determine the equity and maintenance requirements. Again, since naked
calls are involved in ratio writing, there will be amark to market of the position. If
the stock should rise in price, the investor will have to put up more collateral.
It is conceivable that the ratio writer would want to stay with his original posi
tion as long as the stock did not penetrate the upside break-even point of 63.
TABLE 6-2.
Investment required.
Covered writing portion (buy 100 XYZ and sell 1 call)
50% of stock price
Less premium received
Requirement for covered portion
Naked writing portion (sell 1 XYZ call)
20% of stock price
Less out-of-the-money amount
Plus call premium
Less premium received
Requirement for naked portion
Total requirement for ratio write
$2,450
600
$1,850
$ 980
100
+ 600
600
$ 880
$2,730

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Cl,apter 6: Ratio Call Writing
TABLE 6-3.
Initial investment required for aratio write.
70% of stock cost (XYZ = 49)
Plus naked call premiums
Less total premiums received
Plus or minus striking price differential
on naked calls
$3,430
+ 600
- 1,200
100
151
Total requirement $2,730 (plus commissions)
TABLE 6-4.
Collateral required with stock at upside break-even point of 63.
Covered writing requirement $1,850 (see Table 6-2)
20% of stock price (XYZ = 63) 1,260
Plus call premium
Less initial call premium received
Total requirement with XYZ at 63
1,400
600
$3,910
Therefore, he should allow for enough collateral to cover the eventuality of amove
to 63. Assuming the October 50 call is at 14 in this case, he would need $3,910 (see
Table 6-4). This is the requirement that the ratio writer should be concerned with,
not the initial collateral requirement, and he should therefore plan to invest $3,910
in this position, not $2,730 ( the initial requirement). Obviously, he only has to put up
$2,730, but from astrategic point of view, he should allow $3,910 for the position. If
the ratio writer does this with all his positions, he would not receive amargin call
even if all the stocks in his portfolio climbed to their upside break-even points.
SELECTION CRITERIA
To decide whether aratio write is adesirable position, the writer must first determine
the break-even points of the position. Once the break-even points are known, the
writer can then decide if the position has awide enough profit range to allow for
defensive action if it should become necessary. One simple way to determine if the
profit range is wide enough is to require that the next higher and lower striking prices
be within the profit range.

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152 Part II: Call Option Strategies
Example: The writer is buying 100 XYZ at 49 and selling 2 October 50 calls at 6
points apiece. It was seen, by inspection, that the break-even points in the position
are 37 on the downside and 63 on the upside. Amathematical formula allows one to
quickly compute the break-even points for a 2:1 ratio write.
Points of maximum profit = Strike price - Stock price + 2 x Call price
Downside break-even point = Strike price - Points of maximum profit
= Stock price - 2 x Call price
Upside break-even point = Strike price + Points of maximum profit
In this example, the points of maximum profit are 50 - 49 + 2 x 6, or 13. Thus,
the downside break-even point would be 37 (50 - 13) and the upside break-even
point would be 63 (50 + 13). These numbers agree with the figures determined ear
lier by analyzing the position.
This profit range is quite clearly wide enough to allow for defensive action
should the underlying stock rise to the next highest strikes of 55 or 60, or fall to the
next two lower strikes, at 45 and 40. In practice, aratio write is not automatically agood position merely because the profit range extends far enough. Theoretically,
one would want the profit range to be wide in relation to the volatility of the under
lying stock. If the range is wide in relation to the volatility and the break-even
points encompass the next higher and lower striking prices, adesirable position is
available. Volatile stocks are the best candidates for ratio writing, since their pre
miums will more easily satisfy both these conditions. Anonvolatile stock may, at
times, have relatively large premiums in its calls, but the resulting profit range may
still not be wide enough numerically to ensure that follow-up action could be taken.
Specific measures for determining volatility may be obtained from many data serv
ices and brokerage firms. Moreover, methods of computing volatility are present
ed later in the chapter on mathematical applications, and probabilities are further
addressed in the chapters on volatility trading.
Technical support and resistance levels are also important in establishing the
position. If both support and resistance lie within the profit range, there is abetter
chance that the stock will remain within the range. Aposition should not necessarily
be rejected if there is not support and resistance within the profit range, but the
writer is then subjecting himself to apossible undeterred move by the stock in one
direction or the other.
The ratio writer is generally aneutral strategist. He tries to take in the most
time premium that he can to earn the premium erosion while the stock remains rel
atively unchanged. If one is more bullish on aparticular stock, he can set up a 2:1
ratio write with out-of~the-money calls. This allows more room to the upside than to
the downside, and therefore makes the position slightly more bullish. Conversely, if

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Cl,apter 6: Ratio Call Writing 153
one is more bearish on the underlying stock, he can write in-the-money calls in a 2:1
ratio.
There is another way to produce aslightly more bullish or bearish ratio write.
This is to change the ratio of calls written to stock purchased. This method is also
used to construct aneutral profit range when the stock is not close to astriking price.
Example: An investor is slightly bearishly inclined in his outlook for the underlying
stock, so he might write more than two calls for each 100 shares of stock purchased.
His position might be to buy 100 XYZ at 49 and sell 3 XYZ October 50 calls at 6 points
each. This position breaks even at 31 on the downside, because if the stock dropped
by 18 points at expiration, the call profits would amount to 18 points and would pro
duce abreak-even situation. To the upside, the break-even point lies at 59½ for the
stock at expiration. Each call would be worth 9½ at expiration with the stock at 59½,
and each call would thus lose 3½ points, for atotal loss of 10½ points on the three
calls. However, XYZ would have risen from 49 to 59½ - a 10½-point gain - therefore
producing abreak-even situation. Again, aformula is available to aid in determining
the break-even point for any ratio.
Maximum profit= (Striking price - Stock price) x Round lots
purchased+ Number of calls written x Call price
D •dbak Striking Maximum profit owns1 ere -even = - ------~~----price Number of round lots purchased
U .dbak Striking Maximum profit psi ere -even = + price ( Calls written - Round lots purchased)
Note that in the case of a 2:1 ratio write, where the number of round lots purchased
equals 1 and the number of calls written equals 2, these formulae reduce to the ones
given earlier for the more common 2:1 ratio write. To verify that the formulae above
are correct, insert the numbers from the most recent example.
Example: Three XYZ October 50 calls at aprice of 6 were sold against the purchase
of 100 XYZ at 49. The number of round lots purchased is 1.
Maximum profit = (50 - 49) x 1 + 3 x 6 = 19
Downside break-even= 50-19/1 = 31
Upside break-even= 50 + 19/(3 1) = 59½
In the 2:1 ratio writing example given earlier, the break-even points were 37 and 63.
The 3:1 write has lower break-even points of 31 and 59½, reflecting the more bear
ish posture on the underlying stock.

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154 Part II: Call Option Strategies
Amore bullish write is constructed by buying 200 shares of the underlying stock
and writing three calls. To quickly verify that this ratio (3:2) is more bullish, again use
49 for the stock price and 6 for the call price, and now assume that two round lots
were purchased.
Maximum profit= (50-49) x 2 + 3 x 6 = 20
Downside break-even = 50 - 20/2 = 40
Upside break-even= 50 + 20/(3 - 2) = 70
Thus, this ratio of 3 calls against 200 shares of stock has break-even points of 40 and
70, reflecting amore bullish posture on the underlying stock.
A 2: 1 ratio may not necessarily be neutral. There is, in fact, amathematically
correct way of determining exactly what aneutral ratio should be. The neutral ratio
is determined by dividing the delta of the written call into 1. Assume that the delta of
the XYZ October 50 call in the previous example is .60. Then the neutral ratio is
1.0/.60, or 5 to 3. This means that one might buy 300 shares and sell 5 calls. Using
the formulae above, the details of this position can be observed:
Maximum profit= (50 -49) x 3 + 5 x 6 = 33
Downside break-even = 50 - 33/3 = 39
Upside break-even = 50 + 33/(5 --3) = 66½
According to the mathematics of the situation, then, this would be aneutral position
initially. It is often the case that a 5:3 ratio is approximately neutral for an at-the
money call.
By now, the reader should have recognized asimilarity between the ratio writ
ing strategy and the reverse hedge (or simulated straddle) strategy presented in
Chapter 4. The two strategies are the reverse of each other; in fact, this is how the
reverse hedge strategy acquired its name. The ratio write has aprofit graph that looks
like aroof, while the reverse hedge has aprofit graph that looks like atrough - the
roof upside down. In one strategy the investor buys stock and sells calls, while the
other strategy is just the opposite - the investor shorts stock and buys calls. Which
one is better? The answer depends on whether the calls are "cheap" or "expensive."
Even though ratio writing has limited profits and potentially large losses, the strate
gy will result in aprofit in alarge majority of cases, if held to expiration. However,
one may be forced to make adjustments to stock moves that occur prior to expiration.
The reverse hedge strategy, with its limited losses and potentially large profits, pro
vides profits only on large stock moves - aless frequent event. Thus, in stable mar
kets, the ratio writing strategy is generally superior. However, in times of depressed
option premiums, the reverse hedge strategy gains adistinct advantage. If calls are

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Chapter 6: Ratio Call Writing 155
underpriced, the advantage lies with the buyer of calls, and that situation is inherent
in the reverse hedge strategy.
The summaries stated in the above paragraph are rather simplistic ones, refer
ring mostly to what one can expect from the strategies if they are held until expira
tion, without adjustment. In actual trading situations, it is much more likely that one
would have to make adjustments to the ratio write along the way, thus disturbing or
perhaps even eliminating the profit range. Such travails do not befall the reverse
hedge (simulated straddle buy). Consequently, when one takes into consideration the
stock movements that can take place prior to expiration, the ratio write loses some of
its attractiveness and the reverse hedge gains some.
THE VARIABLE RATIO WRITE
In ratio writing, one generally likes to establish the position when the stock is trading
relatively close to the striking price of the written calls. However, it is sometimes the
case that the stock is nearly exactly between two striking prices and neither the in
the-money nor the out-of-the-money call offers aneutral profit range. If this is the
case, and one still wants to be in a 2:1 ratio of calls written to stock owned, he can
sometimes write one in-the-money call and one out-of-the-money call against each
100 shares of common. This strategy, often termed avariable ratio write or trape
zoidal hedge, serves to establish amore neutral profit range.
Example: Given the following prices: XYZ common, 65; XYZ October 60 call, 8; and
XYZ October 70 call, 3.
If one were to establish a 2:1 ratio write with only the October 60's, he would
have asomewhat bearish position. His profit range would be 49 to 71 at expiration.
Since the stock is already at 65, this means that he would be allowing room for 16
points of downside movement and only 6 points on the upside. This is certainly not
neutral. On the other hand, if he were to attempt to utilize only the October 70 calls
in his ratio write, he would have abullish position. This profit range for the October
70 ratio write would be 59 to 81 at expiration. In this case, the stock at 65 is too close
to the downside break-even point in comparison to its distance from the upside
break-even point.
Amore neutral position can be established by buying 100 XYZ and selling one
October 60 and one October 70. This position has aprofit range that is centered
about the current stock price. Moreover, the new position has both an upside and adownside risk, as does amore normal ratio write. However, now the maximum prof
it can be obtained anywhere between the two strikes at expiration. To see this, note

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156 Part II: Call Option Strategies
that if XYZ is anywhere between 60 and 70 at expiration, the stock will be called away
at 60 against the sale of the October 60 call, and the October 70 call will expire worth
less. It makes no difference whether the stock is at 61 or at 69; the same result will
occur. Table 6-5 and Figure 6-3 depict the results from this variable hedge at expira
tion. In the table, it is assumed that the option is bought back at parity to close the
position, but if the stock were called away, the results would be the same.
Note that the shape of Figure 6-3 is something like atrapezoid. This is the
source of the name "trapezoidal hedge," although the strategy is more commonly
known as avariable hedge or variable ratio write. The reader should observe that the
maximum profit is indeed obtained if the stock is anywhere between the two strikes
at eiqJiration. The maximum profit potential in this position, $600, is smaller than the
maximum profit potential available from writing only the October 60'sor only the
October 70's. However, there is avastly greater probability of realizing the maximum
profit in avariable ratio write than there is of realizing the maximum profit in anor
mal ratio write.
The break-even points for avariable ratio write can be computed most quickly
by first computing the maximum profit potential, which is equal to the time value
that the writer takes in. The break-even points are then computed directly by sub
tracting the points of maximum profit from the lower striking price to get the down
side break-even point and adding the points of maximum profit to the upper striking
price to arrive at the upside break-even point. This is asimilar procedure to that fol
lowed for anormal ratio write:
TABLE 6-5.
Results at expiration of variable hedge.
XYZ Price at XYZ October 60 October 70 Total
Expiration Profit Profit Profit Profit
45 -$2,000 +$ 800 +$ 300 -$900
50 - 1,500 + 800 + 300 - 400
54 - 1,100 + 800 + 300 0
60 500 + 800 + 300 + 600
65 0 + 300 + 300 + 600
70 + 500 - 200 + 300 + 600
76 + 1,100 - 800 300 0
80 + 1,500 -$1,200 700 - 400
85 + 2,000 -1,700 - 1,200 - 900

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Gopter 6: Ratio Call Writing
FIGURE 6-3.
Variable ratio write (trapezoidal hedge).
+$600
C:
i $
al
"' "' .3
5
;t:
e
0.
$0
Stock Price at Expiration
Points of maximum profit = Total option premiums + Lower
striking price - Stock price
Downside break-even point = Lower striking price - Points of
maximum profit
Upside break-even point = Higher striking price + Points of
maximum profit
157
Substituting the numbers from the example above will help to verify the formula.
The total points of option premium brought in were 11 (8 for the October 60 and 3
for the October 70). The stock price was 65, and the striking prices involved were 60
and 70.
Points of maximum profit = 11 + 60 - 65 = 6
Downside break-even point= 60- 6 = 54
Upside break-even point= 70 + 6 = 76
Thus, the break-even points as computed by the formula agree with Table 6-5 and
Figure 6-3. Nate that the formula applies only if the stock is initially between the two
striking prices and the ratio is 2:1. If the stock is above both striking prices, the for
mula is not correct. However, the writer should not be attempting to establish avari
able ratio write with two in-the-money calls.

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158 Part II: Call Option Strategies
FOLLOW-UP ACTION
Aside from closing the position completely, there are three reasonable approaches to
follow-up action in aratio writing situation. The first, and most popular, is to roll the
written calls up if the stock rises too far, or to roll down if the stock drops too far. Asecond method uses the delta of the written calls. The third follow-up method is to
utilize stops on the underlying stock to alter the ratio of the position as the stock
moves either up or down. In addition to these types of defensive follow-up action, the
investor must also have aplan in mind for taking profits as the written calls approach
expiration. These types of follow-up action are discussed separately.
ROLLING UP OR DOWN AS A DEFENSIVE ACTION
The reader should already be familiar with the definition of arolling action: The cur
rently written calls are bought back and calls at adifferent striking price are written.
The ratio writer can use rolling actions to his advantage to readjust his position if the
underlying stock moves to the edges of his profit range.
The reason one of the selection criteria for aratio write was the availability of
both the next higher and next lower striking prices was to facilitate the rolling actions
that might become necessary as afollow-up measure. Since an option has its great
est time premium when the stock price and the striking price are the same, one
would normally want to roll exactly at astriking price.
Example: Aratio writer bought 100 XYZ at 49 and sold two October 50 calls at 6
points each. Subsequently, the stock drops in price and the following prices exist:
XYZ, 40; XYZ October 50, l; and XYZ October 40, 4.
One would roll down to the October 40 calls by buying back the 2 October
50'sthat he is short and selling 2 October 40's. In so doing, he would reestablish asomewhat neutral position. His profit on the buy-back of the October 50 calls
would be 5 points each - they were originally sold for 6 - and he would realize a
10-point gain on the two calls. This 10-point gain effectively reduces his stock cost
from 49 to 39, so that he now has the equivalent of the following position: long 100
XYZ at 39 and short 2 XYZ October 40 calls at 4. This adjusted ratio write has aprofit range of 31 to 49 and is thus anew, neutral position with the stock currently
at 40. The investor is now in aposition to make profits if XYZ remains near this
level, or to take further defensive action if the stock experiences arelatively large
change in price again.
Defensive action to the upside - rolling up -works in much the same manner.

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Chapter 6: Ratio Call Writing 159
Example: The initial position again consists of buying 100 XYZ at 49 and selling two
October 50 calls at 6. If XYZ then rose to 60, the following prices might exist: XYZ,
60; XYZ October 50, 11; and XYZ October 60, 6.
The ratio writer could thus roll this position up to reestablish aneutral profit
range. If he bought back the two October 50 calls, he would take a 5-point loss on
each one for anet loss on the calls of 10 points. This would effectively raise his stock
cost by 10 points, to aprice of 59. The rolled-up position would then be long 100 XYZ
at 59 and short 2 October 60 calls at 6. This new, neutral position has aprofit range
of 47 to 73 at October expiration.
In both of the examples above, the writer could have closed out the ratio write
at avery small profit of about 1 point before commissions. This would not be advis
able, because of the relatively large stock commissions, unless he expects the stock to
continue to move dramatically. Either rolling up or rolling down gives the writer afairly wide new profit range to work with, and he could easily expect to make more
than 1 point of profit if the underlying stock stabilizes at all.
Having to take rolling defensive action immediately after the position is estab
lished is the most detrimental case. If the stock moves very quickly after having set
up the position, there will not be much time for time value premium erosion in the
written calls, and this will make for smaller profit ranges after the roll is done. It may
be useful to use technical support and resistance levels as keys for when to take
rolling action if these levels are near the break-even points and/or striking prices.
It should be noted that this method of defensive action - rolling at or near strik
ing prices - automatically means that one is buying back little or no time premium
and is selling the greatest amount of time premium currently available. That is, if the
stock rises, the call'spremium will consist mostly of intrinsic value and very little of
time premium value, since it is substantially in-the-money. Thus, the writer who rolls
up by buying back this in-the-money call is buying back mostly intrinsic value and is
selling acall at the next strike. This newly sold call consists mostly of time value. By
continually buying back "real" or intrinsic value and by selling "thin air" or time value,
the writer is taking the optimum neutral action at any given time.
If astock undergoes adramatic move in one direction or the other, the ratio
writer will not be able to keep pace with the dramatic movement by remaining in the
same ratio.
Example: If XYZ was originally at 49, but then undergoes afairly straight-line move
to 80 or 90, the ratio writer who maintains a 2:1 ratio will find himself in adeplorable
situation. He will have accumulated rather substantial losses on the calls and will not
be able to compensate for these losses by the gain in the underlying stock. Asimilar

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160 Part II: Call Option Strategies
situation could arise to the downside. If:X'YZ were to plunge from 49 to 20, the ratio
writer would make agood deal of profit from the calls by rolling down, but may still
have alarger loss in the stock itself than the call profits can compensate for.
Many ratio writers who are large enough to diversify their positions into anum
ber of stocks will continue to maintain 2:1 ratios on all their positions and will simply
close out aposition that has gotten out of hand by running dramatically to the upside
or to the downside. These traders believe that the chances of such adramatic move
occurring are small, and that they will take the infrequent losses in such cases in
order to be basically neutral on the other stocks in their portfolios.
There is, however, away to combat this sort of dramatic move. This is done by
altering the ratio of the covered write as the stock moves either up or down. For
example, as the underlying stock moves up dramatically in price, the ratio writer can
decrease the number of calls outstanding against his long stock each time he rolls.
Eventually, the ratio might decrease as far as 1:1, which is nothing more than acov
ered writing situation. As long as the stock continues to move in the same upward
direction, the ratio writer who is decreasing his ratio of calls outstanding will be giv
ing more and more weight to the stock gains in the ratio write and less and less weight
to the call losses. It is interesting to note that this decreasing ratio effect can also be
produced by buying extra shares of stock at each new striking price as the stock
moves up, and simultaneously keeping the number of outstanding calls written con
stant. In either case, the ratio of calls outstanding to stock owned is reduced.
When the stock moves down dramatically, asimilar action can be taken to
increase the number of calls written to stock owned. Normally, as the stock falls, one
would sell out some of his long stock and roll the calls down. Eventually, after the
stock falls far enough, he would be in anaked writing position. The idea is the same
here: As the stock falls, more weight is given to the call profits and less weight is given
to the stock losses that are accumulating.
This sort of strategy is more oriented to extremely large investors or to firm
traders, market-makers, and the like. Commissions will be exorbitant if frequent rolls
are to be made, and only those investors who pay very small commissions or who have
such alarge holding that their commissions are quite small on apercentage basis will
be able to profit substantially from such astrategy.
ADJUSTING WITH THE DELTA
The delta of the written calls can be used to determine the correct ratio to be used in
this ratio-adjusting defensive strategy. The basic idea is to use the call'sdelta to
remain as neutral as possible at all times.

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Cl,apter 6: Ratio Call Writing 161
Example: An investor initially sets up aneutral 5:3 ratio of XYZ October 50 calls to
XYZ stock, as was determined previously. The stock is at 49 and the delta is .60.
Furthermore, suppose the stock rises to 57 and the call now has adelta of .80. The
neutral ratio would currently be 1/.80 ( = 1.20) or 5:4. The ratio writer could thus buy
another 100 shares of the underlying stock.
Alternatively, he might buy in one of the short calls. In this particular example,
buying in one call would produce a 4:3 ratio, which is not absolutely correct. If he
had had alarger position initially, it would be easier to adjust to fractional ratios.
When the stock declines, it is necessary to increase the ratio. This can be accom
plished by either selling more calls or selling out some of the long stock. In theory,
these adjustments could be made constantly to keep the position neutral. In practice,
one would allow for afew points of movement by the underlying stock before adjust
ing. If the underlying stock rises too far, it may be logical for the neutral strategist to
adjust by rolling up. Similarly, he would roll down if the stock fell to or below the next
lower strike. The neutral ratio in that case is determined by using the delta of the
option into which he is rolling.
Example: With XYZ at 57, an investor is contemplating rolling up to the October 60'sfrom his present position of long 300 shares and short 5 XYZ October 50's. If the
October 60 has adelta of .40, the neutral ratio for the October 60'sis 2.5:l (1 + .40).
Since he is already long 300 shares of stock, he should now be short 7.5 calls (3 x 2.5).
Obviously, he would sell 7 or 8, probably depending on his short-term outlook for the
stock.
If one prefers to adopt an even more sophisticated approach, he can make
adjustments between striking prices by altering his stock position, and can make
adjustments by rolling up or down if the stock reaches anew striking price. For those
who prefer formulae, the following ones summarize this information:
1. When establishing anew position or when rolling up or down, at the next strike:
Nbfall t 11 Round lots held long um er ocsose =
Delta of call to be sold
Note: When establishing anew position, one must first decide how many shares
of the underlying stock he can buy before utilizing the formula; 1,000
shares would be aworkable amount.
2. When adjusting between strikes by buying or selling stock:

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162 Part II: Call Option Strategies
Number of
round lots = Current delta x Number of short calls - Round lots held long
to buy
Note: If anegative number results, stock should be sold, not bought.
These formulae can be verified by using the numbers from the examples above. For
example, when the delta of the October 50 was .80 with the stock at 57, it was seen
that buying 100 shares of stock would reestablish aneutral ratio.
Number of round lots to buy= .80 x 5 3 = 4- 3 = 1
Also, if the position was to be rolled up to the October 60 (delta = .40), it was seen
that 7.5 October 60'swould theoretically be sold:
Number of calls to sell = __l_ = 7.5 .40
There is amore general approach to this problem, one that can be applied to
any strategy, no matter how complicated. It involves computing whether the position
is net short or net long. The net position is reduced to an equivalent number of shares
of common stock and is commonly called the "equivalent stock position" (ESP). Here
is asimple formula for the equivalent stock position of any option position:
ESP = Option quantity x Delta x Shares per option
Example: Suppose that one is long 10 XYZ July 50 calls, which currently have adelta
of .45. The option is an option on 100 shares of XYZ. Thus, the ESP can be computed:
ESP = 10 x .45 x 100 = 450 shares
This is merely saying that owning 10 of these options is equivalent to owning 450
shares of the underlying common stock, XYZ. The reader should already understand
this, in that an option with adelta of .45 would appreciate by .45 points if the com
mon stock moved up 1 dollar. Thus, 10 options would appreciate by 4.5 points, or
$450. Obviously, 450 shares of common stock would also appreciate by $450 if they
moved up by one point.
Note that there are some options - those that result from astock split- that are
for more than 100 shares. The inclusion of the term "shares per option" in the for
mula accounts for the fact that such options are equivalent to adifferent amount of
stock than most options.
The ESP of an entire option and stock position can be computed, even if sev
eral different options are included in the position. The advantage of this simple cal-

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Chapter 6: Ratio Call Writing 163
culation is that an entire, possibly complex option position can be reduced to one
number. The ESP shows how the position will behave for short-term market move
ments.
Look again at the previous example of aratio write. The position was long 300
shares and short 5 options with acurrent delta of .80 after the stock had risen to 57.
The ESP of the 5 October 50'sis short 400 shares (5 x .80 x 100 shares per option).
The position is also long 300 shares of stock, so the total ESP of this ratio write is
short 100 shares.
This figure gives the strategist ameasure of perspective on his position. He now
knows that he has aposition that is the equivalent of being short 100 shares of XYZ.
Perhaps he is bearish on XYZ and therefore decides to do nothing. That would be
fine; at least he knows that his position is short.
Normally, however, the strategist would want to adjust his position. Again
returning to the previous example, he has several choices in reducing the ESP back
to neutral. An ESP of Ois considered to be aperfectly neutral position. Obviously,
one could buy 100 shares of XYZ to reduce the 100-share delta short. Or, given that
the delta of the October 50 call is .80, he could buy in 1.25 of these short calls (obvi
ously he could only buy l; fractional options cannot be purchased).
Later chapters include more discussions and examples using the ESP. It is avital concept that no strategist who is operating positions involving multiple options
should be without. The only requirement for calculating it is to know the delta of the
options in one'sposition. Those are easily obtainable from one'sbroker or from anumber of computer services, software programs, or Web sites.
For investors who do not have the funds or are not in aposition to utilize such
aratio adjusting strategy, there is aless time-consuming method of taking defensive
action in aratio write.
USING STOP ORDERS AS A DEFENSIVE STRATEGY
Aratio writer can use buy or sell stops on his stock position in order to automatical
ly and unemotionally adjust the ratio of his position. This type of defensive strategy
is not an aggressive one and will provide some profits unless awhipsaw occurs in the
underlying stock.
As an example of how the use of stop orders can aid the ratio writer, let us again
assume that the same basic position was established by buying XYZ at 49 and selling
two October 50 calls at 6 points each. This produces aprofit range of 37 to 63 at expi
ration. If the stock begins to move up too far or to fall too far, the ratio writer can
adjust the ratio of calls short to stock long automatically, through the use of stop
orders on his stock.

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164 Part II: Call Option Strategies
Example: An investor places a "good until canceled" stop order to buy 100 shares of
XYZ at 57 at the same time that he establishes the original position. If XYZ should
get to 57, the stop would be set off and he would then own 200 shares ofXYZ and be
short 2 calls. That is, he would have a 200-share covered write of XYZ October 50
calls.
To see how such an action affects his overall profit picture, note that his average
stock cost is now 53; he paid 49 for the first 100 shares and paid 57 for the second 100
shares bought via the stop order. Since he sold the calls at 6 each, he essentially has acovered write in which he bought stock at 53 and sold calls for 6 points. This does not
represent alot of profit potential, but it will ensure some profit unless the stock falls
back below the new break-even point. This new break-even point is 47 - the stock
cost, 53, less the 6 points received for the call. He will realize the maximum profit
potential from the covered write as long as the stock remains above 50 until expira
tion. Since the stock is already at 57, the probabilities are relatively strong that it will
remain above 50, and even stronger that it will remain above 47, until the expiration
date. If the buy stop order was placed just above atechnical resistance area, this prob
ability is even better.
Hence, the use of abuy stop order on the upside allows the ratio writer to auto
matically convert the ratio write into acovered write if the stock moves up too far.
Once the stop goes off, he has aposition that will make some profit as long as the
stock does not experience afairly substantial price reversal.
Downside protective action using asell stop order works in asimilar manner.
Example: The investor placed a "good until canceled" sell stop for 100 shares of
stock after establishing the original position. If this sell stop were placed at 41, for
example, the position would become anaked call writer'sposition if the stock fell to
41. At that time, the 100 shares of stock that he owned would be sold, at an 8-point
loss, but he would have the capability of making 12 points from the sale of his two
calls as long as the stock remained below 50 until expiration. In fact, his break-even
point after converting into the naked write would actually be 52 at expiration, since
at that price, the calls could be bought back for 2 points each, or 8 points total prof
it, to offset the 8-point loss on the stock. This action limits his profit potential, but
will allow him to make some profit as long as the stock does not experience astrong
price reversal and climb back above 52 by expiration.
There are several advantages for inexperienced ratio writers to using this
method of protection. First, the implementation of the protective strategies - buying
an extra 100 shares of stock if the stock moves up, or selling out the 100 shares that
are long if the stock moves down - is unemotional if the stop orders are placed at the

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166 Part II: Call Option Strategies
option investment, the writer who operates in large size will experience less of acommission charge, percentagewise. That is, the writer who is buying 500 shares
of stock and selling 10 calls to start with will be able to place his stop points far
ther out than the writer who is buying 100 shares of stock and selling 2 calls.
Technical analysis can be helpful in selecting the stop points as well. If there is
resistance overhead, the buy stop should be placed above that resistance. Similarly, if
there is support, the sell stop should be placed beneath the support point. Later,
when straddles are discussed, it will be seen that this type of strategy can be operat
ed at less of anet commission charge, since the purchase and sale of stock will not be
involved.
CLOSING OUT THE WRITE
The methods of follow-up action discussed above deal ,vith the eventuality of pre
venting losses. However, if all goes well, the ratio write will begin to accrue profits as
the stock remains relatively close to the original striking price. To retain these paper
profits that have accrued, it is necessary to move the protective action points closer
together.
Example: XYZ is at 51 after some time has passed, and the calls are at 3 points each.
The writer would, at this time, have an unrealized profit of $800 - $200 from the
stock purchase at 49, and $300 each on the two calls, which were originally sold at 6
points each. Recall that the maximum potential profit from the position, ifXYZ were
exactly at 50 at expiration, is $1,300. The writer would like to adjust the protective
points so that nearly all of the $800 paper profit might be retained while still allow
ing for the profit to grow to the $1,300 maximum.
At expiration, $800 profit would be realized ifXYZ were at 45 or at 55. This can
be verified by referring again to Table 6-1 and Figure 6-1. The 45 to 55 range is now
the area that the writer must be concerned with. The original profit range of 39 to 61
has become meaningless, since the position has performed well to this point in time.
If the writer is using the rolling method of protection, he would roll forward to the
next expiration series if the stock were to reach 45 or 55. If he is using the stop-out
method of protection, he could either close the position at 45 or 55 or he could roll
to the next expiration series and readjust his stop points. The neutral strategist using
deltas would determine the number of calls to roll forward to by using the delta of
the longer-term call.
By moving the protective action points closer together, the ratio writer can then
adjust his position while he still has aprofit; he is attempting to "lock in" his profit.
As even more time passes and expiration draws nearer, it may be possible to move

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Chapter 6: Ratio Call Writing 167
the protective points even closer together. Thus, as the position continues to improve
over time, the writer should be constantly "telescoping" his action points and finally
roll out to the next expiration series. This is generally the more prudent move,
because the commissions to sell stock to close the position and then buy another
stock to establish yet another position may prove to be prohibitive. In summary, then,
as aratio write nears expiration, the writer should be concerned with an ever-nar
rowing range within which his profits can grow but outside of which his profits could
dissipate if he does not take action.
COMMENTS ON DELTA-NEUTRAL TRADING
Since the concept of delta-neutral positions was introduced in this chapter, this is
an appropriate time to discuss them in ageneral way. Essentially, adelta-neutral
position is ahedged position in which at least two securities are used - two or more
different options, or at least one option plus the underlying. The deltas of the two
securities offset each other so that the position starts out with an "equivalent stock
position" (ESP) of 0. Another term for ESP is "position delta." Thus, in theory,
there is no price risk to begin with; the position is neutral with respect to price
movement of the underlying. That definition lasts for about ananosecond.
As soon as time passes, or the stock moves, or implied volatility changes, the
deltas change and therefore the position is no longer delta-neutral. Many people
seem to have the feeling that adelta-neutral position is somehow one in which it is
easy to make money without predicting the price direction of the underlying. That is
not true.
Delta-neutral trading is not "easy": Either (1) one assumes some price risk as
soon as the stock begins to move, or (2) one keeps constantly adjusting his deltas to
keep them neutral. Method 2 is not feasible for public traders because of commis
sions. It is even difficult for market-makers, who pay no commissions. Most public
practitioners of delta-neutral trading establish aneutral position, but then refrain
from adjusting it too often.
Acommon mistake that novice traders make with delta-neutral trading is to
short options in aneutral manner, figuring that they have little exposure to price
change because the position is delta-neutral. However, asizeable move by the under
lying (which often happens in ashort period of time) ruins the neutrality of the posi
tion and inevitably costs the trader alot of money. Asimple example: If one sells anaked straddle (i.e., he sells anaked put and anaked call with both having the same
striking price) with the stock initially just below the strike price, that'sadelta-ne~tral

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168 Part II: Call Option Strategies
position. However, the position has naked options on both sides, and therefore has
tremendous liability.
In practice, professionals watch more than just the delta; they also watch other
measures of the risk of aposition. Even then, price and volatility changes can cause
problems. Advanced risk concepts are addressed more fully in the chapter on
Advanced Concepts.
SUMMARY
Ratio writing is aviable, neutral strategy that can be employed with differing levels
of sophistication. The initial ratio of short calls to long stock can be selected simplis
tically by comparing one'sopinion for the underlying stock with projected break-even
points from the position. In amore sophisticated manner, the delta of the written
calls can be used to determine the ratio.
Since the strategy has potentially large losses either to the upside or the down
side, follow-up action is mandatory. This action can be taken by simple methods such
as rolling up or down in aconstant ratio, or by placing stop orders on the underlying
stock. Amore sophisticated technique involves using the delta of the option to either
adjust the stock position or roll to another call. By using the delta, atheoretically neu
tral position can be maintained at all times.
Ratio writing is arelatively sophisticated strategy that involves selling naked
calls. It is therefore not suitable for all investors. Its attractiveness lies in the fact that
vast quantities of time value premium are sold and the strategy is profitable for the
most probable price outcomes of the underlying stock. It has arelatively large prob
ability of making alimited profit, if the position can be held until expiration without
frequent adjustment.
AN INTRODUCTION TO CALL SPREAD STRATEGIES
Aspread is atransaction in which one simultaneously buys one option and sells
another option, with different terms, on the same underlying security. In acall
spread, the options are all calls. The basic idea behind spreading is that the strategist
is using the sale of one call to reduce the risk of buying another call. The short call in
aspread is considered covered, for margin purposes, only if the long call has an expi
ration date equal to or longer than the short call. Before delving into the individual
types of spreads, it may be beneficial to cover some general facts that pertain to most
spread situations.

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Chapter 6: Ratio Call Writing 169
All spreads fall into three broad categories: vertical, horizontal, or diagonal. Avertical spread is one in which the calls involved have the same expiration date but
different striking prices. An example might be to buy the XYZ October 30 and sell
the October 35 simultaneously. Ahorizontal spread is one in which the calls have the
same striking price but different expiration dates. This is ahorizontal spread: Sell the
XYZ January 35 and buy the XYZ April 35. Adiagonal spread is any combination of
vertical and horizontal and may involve calls that have different expiration dates as
well as different striking prices. These three names that classify the spreads can be
related to the way option prices are listed in any newspaper summary of closing
option prices. Avertical spread involves two options from the same column in anews
paper listing. Newspaper columns run vertically. Ahorizontal spread involves two
calls whose prices are listed in the same row in anewspaper listing; rows are hori
zontal. This relationship to the listing format in newspapers is not important, but it is
an easy way to remember what vertical spreads and horizontal spreads are. There are
many types of vertical and horizontal spreads, and several of them are discussed in
detail in later chapters.
SPREAD ORDER
The term "spread" designates not only atype of strategy, but atype of order as well.
All spread transactions in which both sides of the spread are opening (initial) trans
actions must be done in amargin account. This means that the customer must gen
erally maintain aminimum equity in the account, normally $2,000. Some brokerage
houses may also have amaintenance requirement, or "kicker."
It is possible to transact aspread in acash account, but one of the sides must be
aclosing transaction. In fact, many of the follow-up actions taken in the covered writ
ing strategy are actually spread transactions. Suppose acovered writer is currently
short one XYZ April call against 100 shares of the underlying stock. If he wants to roll
forward to the July 35 call, he will be buying back the April 35 and selling the July 35
simultaneously. This is aspread transaction, technically, since one call is being bought
and the other is being sold. However, in this transaction, the buy side is aclosing
transaction and the sell side is an opening transaction. This type of spread could be
done in acash account. Whenever acovered writer is rolling - up, down, or fmward
he should place the order as aspread order to facilitate abetter price execution.
The spreads discussed in the following chapters are predominantly spread
strategies, ones in which both sides of the spread are opening transactions. These are
designed to have their own profit and risk potentials, and are not merely follow-up
actions to some previously discussed strategy.

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170 Part II: Call Option Strategies
When aspread order is entered, the options being bought and sold must be
specified. Two other items must be specified as well: the price at which the spread is
to be executed, and whether that price is acredit or adebit. If the total price of the
spread results in acash inflow to the spread strategist, the spread is acredit spread.
This merely means that the sell side of the spread brings in ahigher price than is paid
for the buy side of the spread. If the reverse is true - that is, there is acash outflow
from the spread transaction - the spread is said to be adebit spread. This means that
the buy side of the spread costs more than is received from the sell side. It is also
common to refer to the purchased side of the spread as the long side and to refer to
the written side of the spread as the short side.
The price at which acertain spread can be executed is generally not the differ
ence between the last sale prices of the two options involved in the spread.
Example: An investor wants to buy an XYZ October 30 and simultaneously sell an
XYZ October 35 call. If the last sale price of the October 30 was 4 points and the last
sale price of the October 35 was 2 points, it does not necessarily mean that the spread
could be done for a 2-point debit (the difference in the last sale prices). In fact, the
only way to detennine the market price for aspread transaction is to know what the
bid and asked prices of the options involved are. Suppose the following quotes are
available on these two calls:
October 30 call
October 35 call
Bid
37/s
F/s
Asked
41/s
2
Lost Sole
4
2
Since the spread in question involves buying the October 30 call and selling the
October 35, the spreader will, at market, have to pay 41/sfor the October 30 ( the asked
or offering quote) and will receive only F/s (the bid quote) for the October 35. This
results in adebit of 2¼ points, significantly more than the 2-point difference in the
last sale prices. Of course, one is free to specify any price he wants for any type of
transaction. One might enter this spread order at a 21/s-point debit and could have areasonable chance of having the order filled if the floor broker can do better than the
bid side on the October 35 or better than the offering side on the October 30.
The point to be learned here is that one cannot assume that last sale prices are
indicative of the price at which aspread transaction can be executed. This makes
computer analysis of spread transactions via closing price data somewhat difficult.
Some computer data services offer (generally at ahigher cost) closing bid and asked
prices as well as closing sale prices. If astrategist is forced to operate with closing

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O,apter 6: Ratio Call Writing 171
prices only, however, he should attempt to build some screens into his output to allow
for the fact that last sale prices might not be indicative of the price at which the
spread can be executed. One simple method for screening is to look only at relative
ly liquid options - that is, those that have traded asubstantial number of contracts
during the previous trading day. If an option is experiencing agreat deal of trading
activity, there is amuch better chance that the current quote is "tight," meaning that
the bid and offering prices are quite close to the last sale price.
In the early days of listed options, it was somewhat common practice to "leg"
into aspread. That is, the strategist would place separate buy and sell orders for the
two transactions comprising his spread. As the listed markets have developed, adding
depth and liquidity, it is generally apoor idea to leg into aspread. If the floor broker
handling the transaction knows the entire transaction, he has amuch better chance
of "splitting aquote," buying on the bid, or selling on the offering. Splitting aquote
merely means executing at aprice that is between the current bid and asked prices.
For example, if the bid is 37/sand the offering is 41/s, atransaction at aprice of 4
would be "splitting the quote."
The public customer must be aware that spread transactions may involve sub
stantially higher commission costs, because there are twice as many calls involved in
any one transaction. Some brokers offer slightly lower rates for spread transactions,
but these are not nearly as low as spreads in commodity trading, for example.

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CHAPTER 7
Bull Spreads
The bull spread is one of the most popular forms of spreading. In this type of spread,
one buys acall at acertain striking price and sells acall at ahigher striking price.
Generally, both options have the same expiration date. This is avertical spread. Abull
spread tends to be profitable if the underlying stock rrwves up in price; hence, it is abullish position. The spread has both limited profit potential and limited risk.
Although both can be substantial percentagewise, the risk can never exceed the net
investment. In fact, abull spread requires asmaller dollar investment and therefore
has asmaller maximum dollar loss potential than does an outright call purchase of asimilar call.
Example: The following prices exist:
XYZ common, 32;
XYZ October 30 call, 3; and
XYZ October 35 call, 1.
Abull spread would be established by buying the October 30 call and simultaneous
ly selling the October 35 call. Assume that this could be done at the indicated 2-point
debit. Acall bull spread is always adebit transaction, since the call with the lower
striking price must always trade for more than acall with ahigher price, if both have
the same expiration date. Table 7-1 and Figure 7-1 depict the results of this transac
tion at expiration. The indicated call profits or losses would be realized if the calls
were liquidated at parity at expiration. Note that the spread has amaximum profit
and this profit is realized if the stock is anywhere above the higher striking price at
expiration. The maxipmm loss is realized if the stock is anywhere below the lower
strike at expiration, and is equal to the net investment, 2 points in this example.
172

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Chapter 7: Bull Spreads 173
Moreover, there is abreak-even point that always lies between the two striking prices
at expiration. In this example, the break-even point is 32. All bull spreads have prof
it graphs with the same shape as the one shown in Figure 7-1 when the expiration
dates are the same for both calls.
The investor who establishes this position is bullish on the underlying stock, but
is generally looking for away to hedge himself. If he were rampantly bullish, he
TABLE 7-1.
Results at expiration of bull spread.
XYZ Price of
Expiration
25
30
32
35
40
45
FIGURE 7-1.
Bull spread.
c: +$300
.Q
~
-~
w
October 30
Profit
-$ 300
- 300
100
+ 200
+ 700
+ 1,200
October 35
Profit
+$100
+ 100
+ 100
+ 100
- 400
- 900
,,
,,,'
;ff
,,,'
,,'
iii
~
,,,,'
$01---------'----J...__.... _ ___. _____ _
30 3:?,,' 35
0 ::: -$200
e 0..-$300
, ,
..------,,,,,'
Call Purchase
•-----------,'
Stock Price at Expiration
Total
Profit
-$200
- 200
0
+ 300
+ 300
+ 300

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