Chapter 27: Arbitrage 439
Example 1: XYZ is sold short at 60, and a January 50 call is bought for 10¼ points.
Assume that the prevailing interest rate is 1 % per month and that the position is
established one month prior to expiration. XYZ pays no dividend. The total credit
brought in from the trades is $4,975, so the arbitrageur will earn $49.75 in interest
over the course of 1 month. If the stock is above 50 at expiration, he will exercise his
call to buy stock at 50 and close the position. His loss on the security trades will be
$25 the amount of time value premium paid for the call option. (He makes 10
points by selling stock at 60 and buying at 50, but loses 10¼ points on the exercised
call.) His overall profit is thus $24.75.
Example 2: Areal-life example may point out the effect of interest rates even more
dramatically. In early 1979, IBM April 240 calls with about six weeks of life remain
ing were over 60 points in-the-money. IBM was not going to be ex-dividend in that
time. Normally, such adeeply in-the-money option would be trading at parity or even
adiscount when the time remaining to expiration is so short. However, these calls
were trading 3½ points over parity because of the prevailing high interest rates at the
time. IBM was at 300, the April 240 calls were trading at 63½, and the prevailing
interest rate was approximately 1 % per month. The credit from selling the stock and
buying the call was $23,700, so the arbitrageur earned $365.50 in interest for 1 ½
months, and lost $350 - the 3½ points of time value premium that he paid for the
call. This still left enough room for aprofit.
In Chapter 1, it was stated that interest rates affect option prices. The above
examples of the "interest play" strategy quite clearly show why. As interest rates rise,
the arbitrageur can afford to pay more for the long call in this strategy, thus causing
the call price to increase in times of high interest rates. If call prices are higher, so
will put prices be, as the relationships necessary for conversion and reversal arbitrage
are preserved. Similarly, if interest rates decline, the arbitrageur will make lower
bids, and call and put prices will be lower. They are active enough to give truth to the
theory that option prices are directly related to interest rates.
THE BOX SPREAD
An arbitrage consists of simultaneously buying and selling the same security or equiv
alent securities at different prices. For example, the reversal consists of selling aput
and simultaneously shorting stock and buying acall. The reader will recall that the
short stock/long call position was called asynthetic put. That is, shorting the stock
and buying acall is equivalent to buying aput. The reversal arbitrage therefore con
sists of selling a (listed) put and simultaneously buying a (synthetic) put. In asimilar

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440 Part IV: Additional Considerations
manner, the conversion is merely the purchase of a (listed) put and the simultaneous
sale of a (synthetic) put. Many equivalent strategies can be combined for arbitrage
purposes. One of the more common ones is the box spread.
Recall that it was shown that abull spread or abear spread could be construct
ed with either puts or calls. Thus, if one were to simultaneously buy a (call) bull
spread and buy a (put) bear spread, he could have an arbitrage. In essence, he is
merely buying and selling equivalent spreads. If the price differentials work out cor
rectly, arisk-free arbitrage may be possible.
Example: The following prices exist:
XYZ common, 55
XYZ January 50 call, 7
XYZ January 50 put, 1
XYZ January 60 call, 2
XYZ January 60 put, 5½
The arbitrageur could establish the box spread in this example by executing the
following transactions:
Buy acall bull spread:
Buy XYZ January 50 call
Sell XYZ January 60 call
Net call cost
Buy aput bear spread:
Buy XYZ January 60 put
Sell XYZ January 50 put
Net put cost
Total cost of position
7 debit
2 credit
51/2 debit
1 credit
5 debit
No matter where XYZ is at January expiration, this position will be worth 10 points.
The arbitrageur has locked in arisk-free profit of½ point, since he "bought" the box
spread for 9½ points and will be able to "sell" it for 10 points at expiration. To verify
this, evaluate the position at expiration, first with XYZ above 60, then with XYZ
between 50 and 60, and finally with XYZ below 50. If XYZ is above 60 at expiration,
the puts will expire worthless and the call bull spread will be at its maximum poten
tial of 10 points, the difference between the striking prices. Thus, the position can be
liquidated for 10 points if XYZ is above 60 at expiration. Now assume that XYZ is

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Chapter 27: Arbitrage 441
between 50 and 60 at expiration. In that case, the out-of-the-money, written options
would expire worthless-the January 60 call and the January 50 put. This would leave
along, in-the-money combination consisting of a January 50 call and a January 60
put. These two options must have atotal value of 10 points at expiration with XYZ
between 50 and 60. (For example, the arbitrageur could exercise his call to buy stock
at 50 and exercise his put to sell stock at 60.) Finally, assume that XYZ is below 50 at
expiration. The calls would expire worthless if that were true, but the remaining put
spread- actually abear spread in the puts -would be at its maximum potential of 10
points. Again, the box spread could be liquidated for 10 points.
The arbitrageur must pay acost to carry the position, however. In the prior
example, if interest rates were 6% and he had to hold the box for 3 months, it would
cost him an additional 14 cents (.06 x 9½ x 3112). This still leaves room for aprofit.
In essence, abull spread ( using calls) was purchased while abear spread ( using
puts) was bought. The box spread was described in these terms only to illustrate the
fact that the arbitrageur is buying and selling equivalent positions. The arbitrageur
who is utilizing the box spread should not think in terms of bull or bear spread, how
ever. Rather, he should be concerned with "buying" the entire box spread at acost of
less than the differential between the two striking prices. By "buying" the box spread,
it is meant that both the call spread portion and the put spread portion are debit
spreads. Whenever the arbitrageur observes that acall spread and aput spread using
the same strikes and that are both debit spreads can be bought for less than the dif
ference in the strikes plus carrying costs, he should execute the arbitrage.
Obviously, there is acompanion strategy to the one just described. It might
sometimes be possible for the arbitrageur to "sell" both spreads. That is, he would
establish acredit call spread and acredit put spread, using the same strikes. If this
credit were greater than the difference in the striking prices, arisk-free profit would
be locked in.
Example: Assume that adifferent set of prices exists:
XYZ common, 75
XYZ April 70 call, 8½
XYZ April 70 put, 1
XYZ April 80 call, 3
XYZ April 80 put, 6
By executing the following transactions, the box spread could be "sold":

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442
Sell acall (bear) spread:
Buy April 80 call
Sell April 70 call
Net credit on calls
Sell aput (bull) spread:
Buy April 70 put
Sell April 80 put
Net credit on puts
Total credit of position
3 debit
81/2 credit
1 debit
6 credit
Part IV: Additional Considerations
5 credit
10 1/2 credit
In this case, no matter where XYZ is at expiration, the position can be bought back
for 10 points. This means that the arbitrageur has locked in risk-free profit of¼
point. To verify this statement, first assume that XYZ is above 80 at April expiration.
The puts will expire worthless, and the call spread will have widened to 10 points -
the cost to buy it back. Alternatively, if XYZ were between 70 and 80 at April expira
tion, the long, out-of-the-money options would expire worthless and the in-the
money combination would cost 10 points to buy back. (For example, the arbitrageur
could let himself be put at 80, buying stock there, and called at 70, selling the stock
there - anet "cost" to liquidate of 10 points.) Finally, if XYZ were below 70 at expi
ration, the calls would expire worthless and the put spread would have widened to 10
points. It could then be closed out at acost of 10 points. In each case, the arbitrageur
is able to liquidate the box spread by buying it back at 10.
In this sale of abox spread, he would earn interest on the credit received while
he holds the position.
There is an additional factor in the profitability of the box spread. Since the sale
of abox generates acredit, the arbitrageur who sells abox will earn asmall amount
of money from that sale. Conversely, the purchaser of abox spread will have acharge
for carrying cost. Since profit margins may be small in abox arbitrage, these carrying
costs can have adefinite effect. As aresult, boxes may actually be sold for 5 points,
even though the striking prices are 5 points apart, and the arbitrageur can still make
money because of the interest earned.
These box spreads are not easy to find. If one does appear, the act of doing the
arbitrage will soon make the arbitrage impossible. In fact, this is true of any type of
arbitrage; it cannot be executed indefinitely because the mere act of arbitraging will
force the prices back into line. Occasionally, the arbitrageur will be able to find the
option quotes to his liking, especially in volatile markets, and can establish arisk-free

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Chapter 27: Arbitrage 443
arbitrage with the box spread. It can be evaluated at aglance. Only two questions
need to be answered:
1. If one were to establish adebit call spread and adebit put spread, using the same
strikes, would the total cost be less than the difference in the striking prices plus
carrying costs? If the answer is yes, an arbitrage exists.
2. Alternatively, if one were to sell both spreads - establishing acredit call spread
and acredit put spread - would the total credit received plus interest earned be
greater than the difference in the striking prices? If the answer is yes, an arbi
trage exists.
There are some risks to box arbitrage. Many of them are the same as those risks
faced by the arbitrageur doing conversions or reversals. First, there is risk that the
stock might close at either of the two strikes. This presents the arbitrageur with the
same dilemma regarding whether or not to exercise his long options, since he is not
sure whether he will be assigned. Additionally, early assignment may change the prof
itability: Assignment of ashort put will incur large carrying costs on the resulting long
stock; assignment of ashort call will inevitably come just before an ex-dividend date,
costing the arbitrageur the amount of the dividend.
There are not many opportunities to actually transact box arbitrage, but the fact
that such arbitrage exists can help to keep markets in line. For example, if an under
lying stock begins to move quickly and order flow increases dramatically, the special
ist or market-markers in that stock'soptions may be so inundated with orders that
they cannot be sure that their markets are correct. They can use the principles of box
arbitrage to keep prices in line. The most active options would be the ones at strikes
nearest to the current stock price. The specialist can quickly add up the markets of
the call and put at the nearest strike above the stock price and add to that the mar
kets of the options at the strike just below. The sum of the four should add up to aprice that surrounds the difference in the strikes. If the strikes are 5 points apart,
then the sum of the four markets should be something like 4½ bid, 5½ asked. If,
instead, the four markets add up to aprice that allows box arbitrage to be established,
then the specialist will adjust his markets.
VARIATIONS ON EQUIVALENCE ARBITRAGE
Other variations of arbitrage on equivalent positions are possible, although they are
relatively complicated and probably not worth the arbitrageur'stime to analyze. For
example, one could buy abutterfly spread with calls and simultaneously sell abut
terfly spread using puts. Alisted straddle could be sold and asynthetic straddle

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444 Part IV: Additional Considerations
could be bought - short stock and long 2 calls. Inversely, alisted straddle could be
bought against aratio write - long stock and short 2 calls. The only time the arbi
trageur should even consider anything like this is when there are more sizable mar
kets in certain of the puts and calls than there are in others. If this were the case, he
might be able to take an ordinary box spread, conversion, or reversal and add to it,
keeping the arbitrage intact by ensuring that he is, in fact, buying and selling equiv
alent positions.
THE EFFECTS OF ARBITRAGE
The arbitrage process serves auseful purpose in the listed options market, because it
may provide asecondary market where one might not otherwise exist. Normally,
public interest in an in-the-money option dwindles as the option becomes deeply in
the-money or when the time remaining until expiration is very short. There would be
few public buyers of these options. In fact, public selling pressure might increase,
because the public would rather liquidate in-the-money options held long than exer
cise them. The few public buyers of such options might be writers who are closing
out. However, if the writer is covered, especially where call options are concerned,
he might decide to be assigned rather than close out his option. This means that the
public seller is creating arather larger supply that is not offset by apublic demand.
The market created by the arbitrageur, especially in the basic put or call arbitrage,
essentially creates the demand. Without these arbitrageurs, there could conceivably
be no buyers at all for those options that are short-lived and in-the-money, after pub
lic writers have finished closing out their positions.
Equivalence arbitrage - conversion, reversals, and box spreads - helps to keep
the relative prices of puts and calls in line with each other and with the underlying
stock price. This creates amore efficient and rational market for the public to oper
ate in. The arbitrageur would help eliminate, for example, the case in which apublic
customer buys acall, sees the stock go up, but cannot find anyone to sell his call to
at higher prices. If the call were too cheap, arbitrageurs would do reversals, which
involve call purchases, and would therefore provide amarket to sell into.
Questions have been raised as to whether option trading affects stock prices,
especially at or just before an expiration. If the amount of arbitrage in acertain issue
becomes very large, it could appear to temporarily affect the price of the stock itself.
For example, take the call arbitrage. This involves the sale of stock in the market. The
corresponding stock purchase, via the call exercise, is not executed on the exchange.
Thus, as far as the stock market is concerned, there may appear to be an inordinate
amount of selling in the stock. If large numbers of basic call arbitrages are taking
place, they might thus hold the price of the stock down until the calls expire.

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Chapter 27: Arbitrage 445
The put arbitrage has an opposite effect. This arbitrage involves buying stock in
the market. The offsetting stock sale via the put exercise takes place off the exchange.
If alarge amount of put arbitrage is being done, there may appear to be an inordi
nate amount of buying in the stock. Such action might temporarily hold the stock
price up.
In avast majority of cases, however, the arbitrage has no visible effect on the
underlying stock price, because the amount of arbitrage being done is very small in
comparison to the total number of trades in agiven stock. Even if the open interest
in aparticular option is large, allowing for plenty of option volume by the arbi
trageurs, the actual act of doing the arbitrage will force the prices of the stock and
option back into line, thus destroying the arbitrage.
Rather elaborate studies, including doctoral theses, have been written that try
to prove or disprove the theory that option trading affects stock prices. Nothing has
been proven conclusively, and it may never be, because of the complexity of the task.
Logic would seem to dictate that arbitrage could temporarily affect astock'smove
ment if it has discount, in-the-money options shortly before expiration. However, one
would have to reasonably conclude that the size of these arbitrages could almost
never be large enough to overcome adirectional trend in the underlying stock itself.
Thus, in the absence of adefinite direction in the stock, arbitrage might help to per
petuate the inertia; but if there were truly apreponderance of investors wanting to
buy or sell the stock, these investors would totally dominate any arbitrage that might
be in progress.
RISK ARBITRAGE USING OPTIONS
Risk arbitrage is astrategy that is well described by its name. It is basically an arbi
trage - the same or equivalent securities are bought and sold. However, there is gen
erally risk because the arbitrage usually depends on afuture event occurring in
order for the arbitrage to be successful. One form of risk arbitrage was described
earlier concerning the speculation on the size of aspecial dividend that an underly
ing stock might pay. That arbitrage consisted of buying the stock and buying the put,
when the put' stime value premium is less than the amount of the projected special
dividend. The risk lies in the arbitrageur'sspeculation on the size of the anticipated
special dividend.
MERGERS
Risk arbitrage is an age-old type of arbitrage in the stock market. Generally, it con
cerns speculation on whether aproposed merger or acquisition will actually go
through as proposed.

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446 Part IV: Additional Considerations
Example: XYZ, which is selling for $50 per share, offers to buy out LMN and is offer
ing to swap one share of its (XYZ's) stock for every two shares of LMN. This would
mean that LMN should be worth $25 per share if the acquisition goes through as pro
posed. On the day the takeover is proposed, LMN stock would probably rise to about
$22 per share. It would not trade all the way up to 25 until the takeover was approved
by the shareholders of LMN stock. The arbitrageur who feels that this takeover will
be approved can take action. He would sell short XYZ and, for every share that he is
short, he would buy 2 shares of LMN stock. If the merger goes through, he will prof
it. The reason that he shorts XYZ as well as buying LMN is to protect himself in case
the market price of XYZ drops before the acquisition is approved. In essence, he has
sold XYZ and also bought the equivalent of XYZ (two shares of LMN will be equal to
one share of XYZ if the takeover goes through). This, then, is clearly an arbitrage.
However, it is arisk arbitrage because, if the stockholders of LMN reject the offer,
he will surely lose money. His profit potential is equal to the remaining differential
between the current market price of LMN (22) and the takeover price (25). If the
proposed acquisition goes through, the differential disappears, and the arbitrageur
has his profit.
The greatest risk in amerger is that it is canceled. If that happens, stock being
acquired (LMN) will fall in price, returning to its pre-takeover levels. In addition, the
acquiring stock (XYZ) will probably rise. Thus, the risk arbitrageur can lose money
on both sides of his trade. If either or both of the stocks involved in the proposed
takeover have options, the arbitrageur may be able to work options into his strategy.
In merger situations, since large moves can occur in both stocks ( they move in
concert), option purchases are the preferable option strategy. If the acquiring com
pany (XYZ) has in-the-money puts, then the purchase of those puts may be used
instead of selling XYZ short. The advantage is that if XYZ rallies dramatically during
the time it takes for the merger to take effect, then the arbitrageur'sprofits will be
increased.
Example: As above, assume that XYZ is at 50 and is acquiring LMN in a 2-for-lstock
deal. LMN is at 22. Suppose that XYZ rallies to 60 by the time the deal closes. This
would pull LMN up to aprice of 30. If one had been short 100 XYZ at 50 and long
200 LMN at 22, then his profit would be $600 - a $1,600 gain on the 200 long LMN
minus a $1,000 loss on the XYZ short sale.
Compare that result to asimilar strategy substituting along put for the short
XYZ stock. Assume that he buys 200 LMN as before, but now buys an XYZ put. If
one could buy an XYZ July 55 put with little time premium, say at 5½ points, then
he would have nearly the same dollars of profit if the merger should go through with
XYZ below 55.

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Chapter 27: Arbitrage 447
However, when XYZ rallies to 60, his profit increases. He would still make the
$1,600 on LMN as it rose from 22 to 30, but now would only lose $550 on the XYZ
put - atotal profit of $1,050 as compared to $600 with an all-stock position.
The disadvantage to substituting long puts for short stock is that the arbitrageur
does not receive credit for the short sale and, therefore, does not earn money at the
carrying rate. This might not be as large adisadvantage as it initially seems, however,
since it is often the case that it is very expensive - even impossible - to borrow the
acquiring stock in order to short it. If the stock borrow costs are very large or if no
stock can be located for borrowing, the purchase of an in-the-money put is aviable
alternative. The purchase of an in-the-money put is preferable to an at- or out-of-the
money put, because the amount of time value premium paid for the latter would take
too much of the profitability away from the arbitrage if XYZ stayed unchanged or
declined. This strategy may also save money if the merger falls apart and XYZ rises.
The loss on the long put may well be less than the loss would be on short XYZ stock.
Note also that one could sell the XYZ July 55 call short as well as buy the put.
This would, of course, be synthetic short stock and is apure substitute for shorting
the stock. The use of this synthetic short is recommended only when the arbitrageur
cannot borrow the acquiring stock. If this is his purpose, he should use the in-the
money put and out-of-the-money call, since if he were assigned on the call, he could
not borrow the stock to deliver it as ashort sale. The use of an out-of-the-money call
lessens the chance of eventual assignment.
The companion strategy is to buy an in-the-money call instead of buying the
company being acquired (LMN). This has advantages if the stock falls too far, either
because the merger falls apart or because the stocks in the merger decline too far.
Additionally, the cost of carrying the long LMN stock is eliminated, although that is
generally built into the cost of the long calls. The larger amount of time value pre
mium in calls as compared to puts makes this strategy often less attractive than that
of buying the puts as asubstitute for the short sale.
One might also consider selling options instead of buying them. Generally this
is an inferior strategy, but in certain instances it makes sense. The reason that option
sales are inferior is that they do not limit one'srisk in the risk arbitrage, but they cut
off the profit. For example, if one sells puts on the company being acquired (LMN),
he has abullish situation. However, if the company being acquired (XYZ) rallies too
far, there will be aloss, because the short puts will stop making money as soon as
LMN rises through the strike. This is especially disconcerting if atakeover bidding
war should develop for LMN. The arbitrageur who is long LMN will participate nice
ly as LMN rises heavily in price during the bidding war. However, the put seller will
not participate to nearly the same extent.

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448 Part IV: Additional Considerations
The sale of in-the-money calls as asubstitute for shorting the acquiring compa
ny (XYZ) can be beneficial at certain times. It is necessary to have aplus tick in order
to sell stock short. When many arbitrageurs are trying to sell astock short at the same
time, it may be difficult to sell such stock short. Morever, natural owners of XYZ may
see the arbitrageurs holding the price down and decide to sell their long stock rather
than suffer through apossible decline in the stock'sprice while the merger is in
progress. Additionally, buyers of XYZ will become very timid, lowering their bids for
the same reasons. All of this may add up to asituation in which it is very difficult to
sell the stock short, even if it can be borrowed. The sale of an in-the-money call can
overcome this difficulty. The call should be deeply in-the-money and not be too long
term, for the arbitrageur does not want to see XYZ decline below the strike of the
call. If that happened, he would no longer be hedged; the other side of the arbitrage
- the long LMN stock - would continue to decline, but he would not have any
remaining short against the long LMN.
LIMITS ON THE MERGER
There is another type of merger for stock that is more difficult to arbitrage, but
options may prove useful. In some merger situations, the acquiring company (XYZ)
promises to give the shareholders of the company being acquired (LMN) an amount
of stock equal to aset dollar price. This amount of stock would be paid even if the
acquiring company rose or fell moderately in price. If XYZ falls too far, however, it
cannot pay out an extraordinarily increased number of shares to LMN shareholders,
so XYZ puts alimit on the maximum number of shares that it will pay for each share
of LMN stock. Thus, the shareholders ofXYZ are guaranteed that there will be some
downside buffer in terms of dilution of their company in case XYZ declines, as is
often the case for an acquiring company. However, ifXYZ declines too far, then LMN
shareholders will receive less. In return for getting this downside guarantee, XYZ will
usually also stipulate that there is aminimum amount of shares that they will pay to
LMN shareholders, even if XYZ stock rises tremendously. Thus, if XYZ should rise
tremendously in price, then LMN shareholders will do even better than they had
anticipated. An example will demonstrate this type of merger accord.
Example: Assume that XYZ is at 50 and it intends to acquire LMN for astated price
of $25 per share, as in the previous example. However, instead of merely saying that
it will exchange two shares of LMN for one share of XYZ, the company says that it
wants the offer to be worth $25 per share to LMN shareholders as long as XYZ is
between 45 and 55. Given this information, we can determine the maximum and
minimum number of shares that LMN shareholders will receive: The maximum is

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450 Part IV: Additional Considerations
Problems arise if XYZ begins to fall below 45 well before the closing of the
merger, the lower "hook" in the merger. If it should remain below 45, then one
should set up the arbitrage as being short 0.556 shares ofXYZ for each share of LMN
that is held long. As long as XYZ remains below 45 until the merger closes, this is the
proper ratio. However, if, after establishing that ratio, XYZ rallies back above 45, the
arbitrageur can suffer damaging losses. XYZ may continue to rise in price, creating aloss on the short side. However, LMN will not follow it, because the merger is struc
tured so that LMN is worth 25 unless XYZ rises too far. Thus, the long side stops fol
lowing as the short side moves higher.
On the other hand, no such problem exists if XYZ rises too far from its original
price of 50, going above the upper "hook" of 55. In that case, the arbitrageur would
already be long the LMN and would not yet have shorted XYZ, since the merger was
not yet closing. LMN would merely follow XYZ higher after the latter had crossed 55.
This is not an uncommon dilemma. Recall that it was shown that the acquiring
stock will often fall in price immediately after amerger is announced. Thus, XYZ may
fall close to, or below, the lower "hook." Some arbitrageurs attempt to hedge them
selves by shorting alittle XYZ as it begins to fall near 45 and then completing the
short if it drops well below 45. The problem with handling the situation in this way
is that one ends up with an inexact ratio. Essentially, he is forcing himself to predict
the movements of XYZ.
If the acquiring stock drops below the lower "hook," there may be an opportu
nity to establish ahedge without these risks if that stock has listed options. The idea
is to buy puts on the acquiring company, and for those puts to have astriking price
nearly equal to the price of the lower "hook." The proper amount of the company
being acquired (LMN) is then purchased to complete the arbitrage. If the acquiring
company subsequently rallies back into the stated price range, the puts will not lose
money past the striking price and the problems described in the preceding paragraph
will have been overcome.
Example: Amerger is announced as described in the preceding example: XYZ is to
acquire LMN at astated value of $25 per share, with the stipulation that each share
of LMN will be worth at least 0.455 shares of XYZ and at most 0.556 shares. These
share ratios equate to prices of 45 and 55 on XYZ.
Suppose that XYZ drops immediately in price after the merger is announced,
and it falls to 40. Furthermore, suppose that the merger is expected to close some
time during July and that there are XYZ August 45 puts trading at 5½. This repre
sents only ½ point time value premium. The arbitrageur could then set up the arbi
trage by buying 10,000 LMN and buying 56 of those puts. Smaller investors might
buy 1,000 LMN and buy 6 puts. Either of these is in approximately the proper ratio
of 1 LMN to 0.556 XYZ.

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Chapter 27: Arbitrage
TENDER OFFERS
451
Another type of corporate takeover that falls under the broad category of risk arbi
trage is the tender offer. In atender offer, the acquiring company normally offers to
exchange cash for shares of the company to be acquired. Sometimes the off er is for
all of the shares of the company being acquired; sometimes it is for afractional por
tion of shares. In the latter case, it is important to know what is intended to be done
with the remaining shares. These might be exchanged for shares of the acquiring
company, or they might be exchanged for other securities (bonds, most likely), or
perhaps there is no plan for exchanging them at all. In some cases, acompany ten
ders for part of its own stock, so that it is in effect both the acquirer and the acquiree.
Thus, tender offers can be complicated to arbitrage properly. The use of options can
lessen the risks.
In the case in which the acquiring company is making acash tender for all the
shares (called an "any and all" offer), the main use of options is the purchase of puts
as protection. One would buy puts on the company being acquired at the same time
that he bought shares of that company. If the deal fell apart for some reason, the puts
could prevent adisastrous loss as the acquiring stock dropped. The arbitrageur must
be judicious in buying these puts. If they are too expensive or too far out-of-the
money, or if the acquiring company might not really drop very far if the deal falls
apart, then the purchase of puts is awaste. However, if there is substantial downside
risk, the put purchase may be useful.
Selling options in an "any and all" deal often seems like easy money, but there
may be risks. If the deal is completed, the company being acquired will disappear and
its options would be delisted. Therefore, it may often seem reasonable to sell out-of
the-money puts on the acquiring company. If the deal is completed, these expire
worthless at the closing of the merger. However, if the deal falls through, these puts
will soar in price and cause alarge loss. On the other hand, it may also seem like easy
money to sell naked calls with astriking price higher than the price being offered for
the stock. Again, if the deal goes through, these will be delisted and expire worthless.
The risk in this situation is that another company bids ahigher price for the compa
ny on which the calls were written. If this happens, there might suddenly be alarge
upward jump in price, and the written calls could suffer alarge loss.
Options can play amore meaningful role in the tender off er that is for only part
of the stock, especially when it is expected that the remaining stock might fall sub
stantially in price after the partial tender offer is completed. An example of apartial
tender offer might help to establish the scenario.
Example: XYZ proposes to buy back part of its own stock It has offered to pay $70
per share for half the company. There are no plans to do anything further. Based on

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452 Part IV: Additional Considerations
the fundamentals of the company, it is expected that the remaining stock will sell for
approximately $40 per share. Thus, the average share of XYZ is worth 55 if the ten
der offer is completed ( one-half can be sold at 70, and the other half will be worth
40). XYZ stock might sell for $52 or $53 per share until the tender is completed. On
the day after the tender offer expires, XYZ stock will drop immediately to the $40 per
share level.
There are two ways to make money in this situation. One is to buy XYZ at the
current price, say 52, and tender it. The remaining portion would be sold at the lower
price, say 40, when XYZ reopened after the tender expired. This method would yield
aprofit of $3 per share if exactly 50% of the shares are accepted at 70 in the tender
offer. In reality, aslightly higher percentage of shares is usually accepted, because afew people make mistakes and don'ttender. Thus, one'saverage net price tnight be
$56 per share, for a $4 profit from this method. The risk in this situation is that XYZ
opens substantially below 40 after the tender at 70 is completed.
Theoretically, the other way to trade this tender off er might be to sell XYZ short
at 52 and cover it at 40 when it reopens after the tender offer expires. Unfortunately,
this method cannot be effected because there will not be any XYZ stock to borrow in
order to sell it short. All owners will tender the stock rather than loan it to arbi
trageurs. Arbitrageurs understand this, and they also understand the risk they take if
they try to short stock at the last minute: They might be forced to buy back the stock
for cash, or they may be forced to give the equivalent of $70 per share for half the
stock to the person who bought the stock from them. For some reason, many indi
vidual investors believe that they can "get away" with this strategy. They short stock,
figuring that their brokerage firm will find some way to borrow it for them.
Unfortunately, this usually costs the customer alot of money.
The use of calls does not provide amore viable way of attempting to capitalize
on the drop of XYZ from 52 to 40. In-the-money call options on XYZ will normally
be selling at parity just before the tender offer expires. If one sells the call as asub
stitute for the short sale, he will probably receive an assignment notice on the day
after the tender offer expires, and therefore find himself with the same problems the
short seller has.
The only safe way to play for this drop is to buy puts on XYZ. These puts will be
very expensive. In fact, with XY"Lat 52 before the tender offer expires, if the con
sensus opinion is that XYZ will trade at 40 after the offer expires, then puts with a 50
strike will sell for at least $10. This large price reflects the expected drop in price of
XYZ. Thus, it is not beneficial to buy these puts as downside speculation unless one
expects the stock to drop farther than to the $40 level. There is, however, an oppor
tunity for arbitrage by buying XYZ stock and also buying the expensive puts.

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Chapter 27: Arbitrage 453
Before giving an example of that arbitrage, aword about short tendering is in
order. Short tendering is against the law. It comes about when one tenders stock into
atender offer when he does not really own that stock. There are complex definitions
regarding what constitutes ownership of stock during atender offer. One must be net
long all the stock that he tenders on the day the tender offer expires. Thus, he can
not tender the stock on the day before the offer expires, and then short the stock on
the next day ( even if he could borrow the stock). In addition, one must subtract the
number of shares covered by certain calls written against his position: Any calls with
astrike price less than the tender off er price must be subtracted. Thus, if he is long
1,000 shares and has written 10 in-the-money calls, he cannot tender any shares. The
novice and experienced investor alike must be aware of these definitions and should
not violate the short tender rules.
Let us now look at an arbitrage consisting of buying stock and buying the expen
sive puts.
Example: XYZ is at 52. As before, there is atender offer for half the stock at 70, with
no plans for the remainder. The July 55 puts sell for 15, and the July 50 puts sell for
10. It is common that both puts would be predicting the same price in the after-mar
ket: 40.
If one buys 200 shares ofXYZ at 52 and buys one July 50 put at 10, he has alocked
in profit as long as the tender offer is completed. He only buys one put because he
is assuming that 100 shares will be accepted by the company and only 100 shares will
be returned to him. Once the 100 shares have been returned, he can exercise the put
to close out his position.
The following table summarizes these results:
Initial purchase
Buy 200 XYZ at 52
Buy 1 July 50 put at 10
Total Cost
Closing sale
Sell 1 00 XYZ at 70 via tender
Sell 1 00 XYZ at 50 via put exercise
Total proceeds
Total profit: $600
$10,400 debit
1,000 debit
$11 ,400 debit
7,000 credit
5,000 credit
$12,000 credit
This strategy eliminates the risk ofloss ifXYZ opens substantially below 40 after
the tender offer. The downside price is locked in by the puts.

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454 Part IV: Additional Considerations
If more than 50% of XYZ should be accepted in the tender offer, then alarger
profit will result. Also, if XYZ should subsequently trade at ahigh enough price so
that the July 50 put has some time value premium, then alarger profit would result
as well. (The arbitrageur would not exercise the put, but would sell the stock and the
put separately in that case.)
Partial tender offers can be quite varied. The type described in the above exam
ple is called a "two-tier" offer because the tender offer price is substantially different
from the remaining price. In some partial tenders, the remainder of the stock is slat
ed for purchase at substantially the same price, perhaps through acash merger. The
above strategy would not be applicable in that case, since such an offer would more
closely resemble the "any and all" offer. In other types of partial tenders, debt secu
rities of the acquiring company may be issued after the partial cash tender. The net
price of these debt securities may be different from the tender offer price. If they
are, the above strategy might work.
In summary, then, one should look at tender offers carefully. One should be
careful not to take extraordinary option risk in an "any and all" tender. Conversely,
one should look to take advantage of any "two-tier" situation in apartial tender offer
by buying stock and buying puts.
PROFITABILITY
Since the potential profits in risk arbitrage situations may be quite large, perhaps 3
or 4 points per 100 shares, the public can participate in this strategy. Commission
charges will make the risk arbitrage less profitable for apublic customer than it
would be for an arbitrageur. The profit potential is often large enough, however, to
make this type of risk arbitrage viable even for the public customer.
In summary, the risk arbitrageur may be able to use options in his strategy,
either as areplacement for the actual stock position or as protection for the stock
position. Although the public cannot normally participate in arbitrage strategies
because of the small profit potential, risk arbitrages may often offer exceptions. The
profit potential can be large enough to overcome the commission burden for the
public customer.
PAIRS TRADING
Astock trading strategy that has gained some adherents in recent years is pairs trad
ing. Simplistically, this strategy involves trading pairs of stocks - one held long, the
other short. Thus, it is ahedged strategy. The two stocks' price movements are relat
ed historically. The pairs trader would establish the position when one stock was

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Chapter 27: Arbitrage 455
expensive with respect to the other one, historically. Then, when the stocks return to
their historical relationship, aprofit would result. In reality, some fairly complicated
computer programs search out the appropriate pairs.
The interest on the short sale offsets the cost of carry of the stock purchased.
Therefore, the pairs trader doesn'thave any expense except the possible differential
in dividend payout.
The bane of pairs trading is apossible escalation of the stock sold short without
any corresponding rise in price of the stock held long. Atakeover attempt might
cause this to happen. Of course, pairs traders will attempt to research the situation
to ensure that they don'toften sell short stocks that are perceived to be takeover can
didates.
Pairs traders can use options to potentially reduce their risk if there are in-the
money options on both stocks. One would buy an in-the-money put instead of selling
one stock short, and would buy an in-the-money call on the other stock instead of
buying the stock itself. In this option combination, traders are paying very little time
value premium, so their profit potential is approximately the same as with the pairs
trading strategy using stocks. ( One would, however, have adebit, since both options
are purchased; so there would be acost of carry in the option strategy.)
If the stocks return to their historical relationship, the option strategy will
reflect the same profit as the stock strategy, less any loss of time value premium. One
added advantage of the option strategy, however, is that if atakeover occurs, the put
has limited liability, and the trader'sloss would be less.
Another advantage of the option strategy is that if both stocks should experience
large moves, it could make money even if the pair doesn'treturn to historical norms.
This would happen, for example, if both stocks dropped agreat deal: The call has lim
ited loss, while the put' sprofits would continue to accrue. Similarly, to the upside, alarge move by both stocks would make the put worthless, but the call would keep
making money. In both cases, the option strategy could profit even if the pair of
stocks didn'tperform as predicted.
This type of strategy- buying in-the-money options as substitutes for both sides
of aspread or hedge strategy - is discussed in more detail in Chapter 31 on index
spreading and Chapter 35 on futures spreads.
FACILITATION (BLOCK POSITIONING)
Facilitation is the process whereby atrader seeks to aid in making markets for the
purchase or sale of large blocks of stock. This is not really an arbitrage, and its
description is thus deferred to Chapter 28.

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CHAPTER 28
Mathetnatical Applications
In previous chapters, many references have been made to the possibility of applying
mathematical techniques to option strategies. Those techniques are developed in this
chapter. Although the average investor - public, institutional, or floor trader - nor
mally has alimited grasp of advanced mathematics, the information in this chapter
should still prove useful. It will allow the investor to see what sorts of strategy deci
sions could be aided by the use of mathematics. It will allow the investor to evaluate
techniques of an information service. Additionally, if the investor is contemplating
hiring someone knowledgeable in mathematics to do work for him, the information
to be presented may be useful as afocal point for the work. The investor who does
have aknowledge of mathematics and also has access to acomputer will be able to
directly use the techniques in this chapter.
THE BLACK-SCHOLES MODEL
Since an option'sprice is the function of stock price, striking price, volatility, time to
expiration, and short-term interest rates, it is logical that aformula could be drawn
up to calculate option prices from these variables. Many models have been conceived
since listed options began trading in 1973. Many of these have been attempts to
improve on one of the first models introduced, the Black-Scholes model. This model
was introduced in early 1973, very near the time when listed options began trading.
It was made public at that time and, as aresult, gained arather large number of
adherents. The formula is rather easy to use in that the equations are short and the
number of variables is small.
The actual formula is:
456

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Chapter 28: Mathematical Applications
Theoretical option price= pN(d 1) se-rtN(d2)
pv2
ln(8 )+ (r +2 )twhere d1 = _ r.
V-4 td2 = d1 - v--ft
The variables are:
p = stock price
s = striking price
t = time remaining until expiration, expressed as apercent of ayear
r = current risk-free interest rate
v = volatility measured by annual standard deviation
ln = natural logarithm
N(x) = cumulative normal density function
457
An important by-product of the model is the exact calculation of the delta - that
is, the amount by which the option price can be expected to change for asmall
change in the stock price. The delta was described in Chapter 3 on call buying, and
is more formally known as the hedge ratio.
Delta= N(d1)
The formula is so simple to use that it can fit quite easily on most programmable cal
culators. In fact, some of these calculators can be observed on the exchange floors as
the more theoretical floor traders attempt to monitor the present value of option pre
miums. Of course, acomputer can handle the calculations easily and with great
speed. Alarge number of Black-Scholes computations can be performed in avery
short period of time.
The cumulative normal distribution function can be found in tabular form in
most statistical books. However, for computation purposes, it would be wasteful to
repeatedly look up values in atable. Since the normal curve is asmooth curve (it is
the "bell-shaped" curve used most commonly to describe population distributions),
the cumulative distribution can be approximated by aformula:
x = l-z(l.330274y 5 - l.821256y 4 + l.781478y 3 - .356538y 2 + .3193815y)
where y 1 and z = .3989423e-<r212
= 1 + .2316419lcrl
Then N(cr) = xif cr > 0 or N(cr) = 1- xif cr < 0

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458 Part IV: Additional Considerations
This approximation is quite accurate for option pricing purposes, since one is not
really interested in thousandths of apoint where option prices are concerned.
Example: Suppose that XYZ is trading at 45 and we are interested in evaluating the
July 50 call, which has 60 days remaining until expiration. Furthermore, assume that
the volatility of XYZ is 30% and that the risk-free interest rate is currently 10%. The
theoretical value calculation is shown in detail, in order that those readers who wish
to program the model will have something to compare their calculations against.
page:
Initially, determine t, d1, and d2, by referring to the formulae on the previous
t = 60/365 = .16438 years
d _ In (45/50) + (.1 + .3 x .3/2) x .16438
1-
.3 X ✓.16438
= -.10536 + (.145 X .16438) = __ 67025 .3 X .40544
d2 = -.67025 - .3 ✓.16438 = -.67025 - (.3 x .40544) = -.79189
Now calculate the cumulative normal distribution function for d1 and d2 by
referring to the above formulae:
dl = -.67025
l 1
y = l + (.2316419 I -.67025 I) = 1.15526 = ·86561
z = .3989423e--(-.67025 X -.67025)/2
= .3989423e-0·22462 = .31868
There are too many calculations involved in the computation of the fifth-order
polynomial to display them here. Only the result is given:
X = .74865
Since we are determining the cumulative normal distribution of anegative
number, the distribution is determined by subtracting xfrom l.
N(d1) = N(-.67O25) = l -x = l - .74865 = .25134
In asimilar manner, which requires computing new values for x, y, and z,
N(d2) = N(-.79179) = 1- .78579 = .21421

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Chapter 28: Mathematical Applications 459
Now, returning to the formula for theoretical option price, we can complete the
calculation of the July 50 call'stheoretical value, called value here for short:
value = 45 x N(d1) - 50 xe-·1 x ·16438 x N(d2)
= 45 X .25134 - 50 X .9837 X .21421
= .7746
Thus, the theoretical value of the July 50 call is just slightly over¼ of apoint.
Note that the delta of the call was calculated along the way as N(d1) and is equal to
just over .25. That is, the July 50 call will change price about¼ as fast as the stock
for asmall price change by the stock.
This example should answer many of the questions that readers of the first edi
tion have posed. The reader interested in amore in-depth description of the model,
possibly including the actual derivation, should refer to the article "Fact and Fantasy
in the Use of Options." 1 One of the less obvious relationships in the model is that call
option prices will increase (and put option prices will decrease) as the risk-free inter
est rate increases. It may also be observed that the model correctly preserves rela
tionships such as increased volatility, higher stock prices, or more time to expiration,
which all imply higher option prices.
CHARACTERISTICS Of THE MODEL
Several aspects of this model are worth further discussion. First, the reader will
notice that the model does not include dividends paid by the common stock. As has
been demonstrated, dividends act as anegative effect on call prices. Thus, direct
application of the model will tend to give inflated call prices, especially on stocks that
pay relatively large dividends. There are ways of handling this. Fisher Black, one of
the coauthors of the model, suggested the following method: Adjust the stock price
to be used in the formula by subtracting, from the current stock price, the present
worth of the dividends likely to be paid before maturity. Then calculate the option.
price. Second, assume that the option expires just prior to the last ex-dividend date
preceding actual option expiration. Again adjust the stock price and calculate the
option price. Use the higher of the two option prices calculated as the theoretical
price.
Another, less exact, method is to apply aweighting factor to call prices. The
weighting factor would be based on the dividend payment, with aheavier weight
being applied to call options on high-yielding stock. It should be pointed out that, in
1Fisher Black, Financial Analysts Journal, July-August 1975, pp. 36-70.

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460 Part IV: Additional Considerations
many of the applications that are going to be prescribed, it is not necessary to know
the exact theoretical price of the call. Therefore, the dividend "correction" might not
have to be applied for certain strategy decisions.
The model is based on alognormal distribution of stock prices. Even though the
normal distribution is part of the model, the inclusion of the exponential functions
makes the distribution lognormal. For those less familiar with statistics, anormal dis
tribution has abell-shaped curve. This is the most familiar mathematical distribution.
The problem with using anormal distribution is that it allows for negative stock
prices, an impossible occurrence. Therefore, the lognormal distribution is generally
used for stock prices, because it implies that the stock price can have arange only
between zero and infinity. Furthermore, the upward (bullish) bias of the lognormal
distribution appears to be logically correct, since astock can drop only 100% but can
rise in price by more than 100%. Many option pricing models that antedate the
Black-Scholes model have attempted to use empirical distributions. An empirical
distribution has adifferent shape than either the normal or the lognormal distribu
tion. Reasonable empirical distributions for stock prices do not differ tremendously
from the lognormal distribution, although they often assume that astock has agreater probability of remaining stable than does the lognormal distribution. Critics
of the Black-Scholes model claim that, largely because it uses the lognormal distri
bution, the model tends to overprice in-the-money calls and underprice out-of-the
money calls. This criticism is true in some cases, but does not materially subtract
from many applications of the model in strategy decisions. True, if one is going to buy
or sell calls solely on the basis of their computed value, this would create alarge prob
lem. However, if strategy decisions are to be made based on other factors that out
weigh the overpriced/underpriced criteria, small differentials will not matter.
The computation of volatility is always adifficult problem for mathematical
application. In the Black-Scholes model, volatility is defined as the annual standard
deviation of the stock price. This is the regular statistical definition of standard devi
ation:
where
P = average stock price of all P/s
Pi = daily stock price
n
~ (Pi -P)2
cr2 = _1=_1 __ _
n-1
v = a!P

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Chapter 28: Mathematical Applications 463
This is then the proper way to calculate historical volatility. Obviously, the
strategist can calculate 10-, 20-, and 50-day and annual volatilities if he wishes - or
any other number for that matter. In certain cases, one can discern valuable infor
mation about astock or future and its options by seeing how the various volatilities
compare with one another.
There is, in fact, away in which the strategist can let the market compute the
volatility for him. This is called using the implied volatility; that is, the volatility that
the market itself is implying. This concept makes the assumption that, for options
with striking prices close to the current stock price and for options with relatively
large trading volume, the market is fairly priced. This is something like an efficient
market hypothesis. If there is enough trading interest in an option that is close to the
money, that option will generally be fairly priced. Once this assumption has been
made, acorollary arises: If the actual price of an option is the fair price, it can be fixed
in the Black-Scholes equation while letting volatility be the unknown variable. The
volatility can be determined by iteration. In fact, this process of iterating to compute
the volatility can be done for each option on aparticular underlying stock This might
result in several different volatilities for the stock If one weights these various results
by volume of trading and by distance in- or out-of-the-money, asingle volatility can
be derived for the underlying stock This volatility is based on the closing price of all
the options on the underlying stock for that given day.
Example: XYZ is at 33 and the closing prices are given in Table 28-1. Each option
has adifferent implied volatility, as computed by determining what volatility in the
Black-Scholes model would result in the closing price for each option: That is, if .34
were used as the volatility, the model would give 4¼ as the price of the January 30
call. In order to rationally combine these volatilities, weighting factors must be
applied before avolatility for XYZ stock itself can be arrived at.
The weighting factors for volume are easy to compute. The factor for each
option is merely that option'sdaily volume divided by the total option volume on all
XYZ options (Table 28-2). The weighting functions for distance from the striking
price should probably not be linear. For example, if one option is 2 points out-of-the
money and another is 4 points out-of-the-money, the former option should not nec
essarily get twice as much weight as the latter. Once an option is too far in- or out-of
the-money, it should not be given much or any weight at all, regardless of its trading
volume. Any parabolic function of the following form should suffice:
{
(x - a)2 if xis less than a
Weighting factor = -;;,r-
= 0 if xis greater than a

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464 Part IV: Additional Considerations
TABLE 28-1.
Implied volatilities, closing price, and volume.
Option
Option Price Volume
January 30 41/2
January 35 11/2
April 35 21/2
April 40 11/2
TABLE 28-2.
Volume weighting factors.
Option
January 30
January 35
April 35
April 40
Volume
50
90
55
5
50
90
55
~
200
Implied
Volatility
.34
.28
.30
.38
Volume Weighting Factor
.25 (50/200)
.45 (90/200)
.275 (55/200)
.025 ( 5/200)
where xis the percentage distance between stock price and strike price and ais the
maximum percentage distance at which the modeler wants to give any weight at all
to the option'simplied volatility.
Example: An investor decides that he wants to discard options from the weighting
criterion that have striking prices more than 25% from the current stock price. The
variable, a, would then be equal to .25. The weighting factors, with XYZ at 33, could
thus be computed as shown in Table 28-3. To combine the weighting factors for both
volume and distance from strike, the two factors are multiplied by the implied volatil
ity for that option. These products are summed up for all the options in question.
This sum is then divided by the products of the weighting factors, summed over all
the options in question. As aformula, this would read:
Implied _ I,(Volume factor x Distance factor x Implied volatility)
volatility - I,(Volume factor x Distance factor)
In our example, this would give an implied volatility for XYZ stock of 29.8%
(Table 28-4). Note that the implied volatility, .298, is not equal to any of the individ-

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Chapter 28: Mathematical Applications
TABLE 28-3.
Distance weighting factors.
465
Option
Distonce
from
Stock Price
Distance
Weighting Factor
January 30
January 35
April 35
April 40
TABLE 28-4.
Option'simplied volatility.
.091 (3/33)
.061 (2/33)
.061 (2/33)
.212 (7 /33)
.41
.57
.57
.02
Volume Distance Option's Implied
Option Factor Factor Volotility
January 30 .25 .41 .34
January 35 .45 .57 .28
April 35 .275 .57 .30
April40 .025 .02 .38
Implied = .25 x .41 x .34 + .45 x .57 x .28 + .275 x .57 x .30 + .025 x .02 x .38
volatility. .25 x .41 + .45 x .57 + .275 x .57 + .025 x .02
= .298
ual option'simplied volatilities. Rather, it is acomposite figure that gives the most
weight to the heavily traded, near-the-money options, and very little weight to the
lightly-traded (5 contracts), deeply out-of-the-money April 40 call. This implied
volatility is still aform of standard deviation, and can thus be used whenever astan
dard deviation volatility is called for.
This method of computing volatility is quite accurate and proves to be sensitive
to changes in the volatility of astock. For example, as markets become bullish or
bearish (generating large rallies or declines), most stocks will react in avolatile man
ner as well. Option premiums expand rather quickly, and this method of implied
volatility is able to pick up the change quickly. One last bit of fine-tuning needs to be
done before the final volatility of the stock is arrived at. On aday-to-day basis, the
implied volatility for astock - especially one whose options are not too active may
fluctuate more than the strategist would like. Asmoothing effect can be obtained by

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466 Part IV: Additional Considerations
taking amoving average of the last 20 or 30 days' implied volatilities. An alternative
that does not require the saving of many previous days' worth of data is to use amomentum calculation on the implied volatility. For example, today'sfinal volatility
might be computed by adding 5% of today'simplied volatility to 95% of yesterday'sfinal volatility. This method requires saving only one previous piece of data - yester
day'sfinal volatility - and still preserves a "smoothing" effect.
Once this implied volatility has been computed, it can then be used in the
Black-Scholes model ( or any other model) as the volatility variable. Thus one could
compute the theoretical value of each option according to the Black-Scholes formu
la, utilizing the implied volatility for the stock. Since the implied volatility for the
stock will most likely be somewhat different from the implied volatility of this par
ticular option, there will be adiscrepancy between the option'sactual closing price
and the theoretical price as computed by the model. This differential represents the
amount by which the option is theoretically overpriced or underpriced, compared to
other options on that same stock.
EXPECTED RETURN
Certain investors will enter positions only when the historical percentages are on
their side. When one enters into atransaction, he normally has abelief as to the pos
sibility of making aprofit. For example, when he buys stock he may think that there
is a "good chance" that there will be arally or that earnings will increase. The investor
may consciously or unconsciously evaluate the probabilities, but invariably, an invest
ment is made based on apositive expectation of profit. Since options have fixed
terms, they lend themselves to amore rigorous computation of expected profit than
the aforementioned intuitive appraisal. This more rigorous approach consists of com
puting the expected return. The expected retum is nothing more than the retum that
the position should yield over alarge number of cases.
Asimple example may help to explain the concept. The crucial variable in com
puting expected return is to outline what the chances are of the stock being at acer
tain price at some future time.
Example: XYZ is selling at 33, and an investor is interested in determining where
XYZ will be in 6 months. Assume that there is a 20% chance of XYZ being below 30
in 6 months, and that there is a 40% chance that XYZ will be above 35 in 6 months.
Finally, assume that XYZ has an equal 10% chance of being at 31, 32, 33, or 34 in 6
months. All other prices are ignored for simplification. Table 28-5 summarizes these
assumptions.

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Chapter 28: Mathematical Applications
TABLE 28-5.
Calculation of expected returns.
Price of XYZ in 6 Months
Below 30
31
32
33
34
Above 35
467
Chance of XYZ Being at That Price ·
20%
10%
10%
10%
10%
40%
100%
Since the percentages total 100%, all the outcomes have theoretically been
allowed for. Now suppose a February 30 call is trading at 4 and a February 35 call is
trading at 2 points. Abull spread could be established by buying the February 30 and
selling the February 35. This position would cost 2 points - that is, it is a 2-point
debit. The spreader could make 3 points if XYZ were above 35 at expiration for areturn of 150%, or he could lose 100% if XYZ were below 30 at expiration. The
expected return for this spread can be computed by multiplying the outcome at expi
ration for each price by the probability of being at that price, and then summing the
results. For example, if XYZ is below 30 at expiration, the spreader loses $200. It was
assumed that there is a 20% chance of XYZ being below 30 at expiration, so the
expected loss is 20% times $200, or $40. Table 28-6 shows the computation of the
expected results at all the prices. The total expected profit is $100. This means that
the expected return (profit divided by investment) is 50% ($100/$200). This appears
to be an attractive spread, because the spreader could "expect" to make 50% of his
money, less commissions.
What has really been calculated in this example is merely the return that one
would expect to make in the long run if he invested in the same position many times
throughout history. Saying that aparticular position has an expected return of 8 or
9% is no different from saying that common stocks return 8 or 9% in the long run.
Of course, in bull markets stock would do much better, and in bear markets much
worse. In asimilar manner, this example bull spread with an expected return of 50%
may do as well as the maximum profit or as poorly as losing 100% in any one case. It
is the total return on many cases that has the expected return of 50%. Mathematical
theory holds that, if one constantly invests in positions with positive expected returns,
he should have abetter chance of making rrwney.

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468 Part IV: Additional Considerations
TABLE 28-6.
Computation of expected profit.
Chance of Being Profit at Expected
XYZ Price at at That Price That Price Profit:
Expiration (A) (B) (A) x (8)
Below 30 20% -$200 -$ 40
31 10% - 100 - 10
32 10% 0 0
33 10% + 100 + 10
34 10% + 200 + 20
Above 35 40% + 300 + 120
Total expected profit $100
As is readily observable, the selection of what percentages to assign to the pos
sible outcomes in the stock price is acrucial choice. In the example above, if one
altered his assumption slightly so that XYZ had a 30% chance of being below 30 and
a 30% chance of being above 35 at expiration, the expected return would drop con
siderably, to 25%. Thus, it is important to have areasonably accurate and consistent
method of assigning these percentages. Furthermore, the example above was too sim
plistic, in that it did not allow for the stock to close at any fractional prices, such as
32½. Acorrect expected return computation must take into account all possible out
comes for the stock.
Fortunately, there is astraightforward method of computing the expected per
centage chance of agiven stock being at acertain price at acertain point in time. This
computation involves using the distribution of stock prices. As mentioned earlier, the
Black-Scholes model assumes alognormal distribution for stock prices, although
many modelers today use nonstandard (empirical or heuristic) distributions. No mat
ter what the distribution, the area under the distribution curve between any two
points gives the probability of being between those two points.
Figure 28-1 is agraph of atypical lognormal distribution. The peak always lies
at the "mean," or average, of the distribution. For stock price distributions, under the
random walk assumption, the "mean" is generally considered to be the current stock
price. The graph allows one to visualize the probability of being at any given price.
Note that there is afairly great chance that the stock will be relatively unchanged;
there is no chance that the stock will be below zero; and there is abullish bias to the
graph - the stock could rise infinitely, although the chances of it doing so are
extremely small.

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470
where
v = annual volatility
t = time, in years
vt = volatility for time, t.
Part IV: Additional Considerations
As an example, a 3-month volatility would be equal to one-half of the annual
volatility. In this case, twould equal .25 (one fourth of ayear), so v_25 = v65 = .5v.
The necessary groundwork has been laid for the computation of the probabili
ty necessary in the expected return calculation. The following formula gives the prob
ability of astock that is currently at price pbeing below some other price, q, at the
end of the time period. The lognormal distribution is assumed.
Probability of stock being below price qat end of time period t:
P (below) = N (In~))
where
N = cumulative normal distribution
p = current price of the stock
q = price in question
In = natural logarithm for the time period in question.
If one is interested in computing the probability of the stock being above the
given price, the formula is
P (above)= 1- P (below)
With this formula, the computation of expected return is quickly accomplished
with acomputer. One merely has to start at some price - the lower strike in abull
spread, for example - and work his way up to ahigher price - the high strike for abull spread. At each price point in between, the outcome of the spread is multiplied
by the probability of being at that price, and arunning sum is kept.
Simplistically, the following iterative equation would be used.
P ( of being at price x) = P (below x) - P (below y)
where yis close to but less than xin price. As an example:
P (of being at 32.4) = P (below 32.4) - P (below 32.3)

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Chapter 28: Mathematical Applications 471
Thus, once the low starting point is chosen and the probability of being below that
price is determined, one can compute the probability of being at prices that are suc
cessively higher merely by iterating with the preceding formula. In reality, one is
using this information to integrate the distribution curve. Any method of approxi
mating the integral that is used in basic calculus, such as the Trapezoidal Rule or
Simpson's Rule, would be applicable here for more accurate results, if they are
desired.
Apartial example of an expected return calculation follows.
Example: XYZ is currently at 33 and has an annual volatility of 25%. The previous
bull spread is being established- buy the February 30 and sell the February 35 for a
2-point debit - and these are 6-month options. Table 28-7 gives the necessary com
ponents for computing the expected return. Column (A), the probability of being
below price q, is computed according to the previously given formula, where p = 33
and vt = .177 (t = .25-V ½). The first stock price that needs to be looked at is 30, since
all results for the bull spread are equal below that price - a 100% loss on the spread.
The calculations would be performed for each eighth (or tenth) of apoint up through
aprice of 35. The expected return is compute<lby multiplying the two right-hand
columns, (B) and (C), and summing the results. Note that column (B) is determined
by subtracting successive numbers in column (A). It would not be particularly
enlightening to carry this example to completion, since the rest of the computations
are similar and there is alarge number of them.
In theory, if one had the data and the computer power, he could evaluate awide
range of strategies every day and come up with the best positions on an expected
return basis. He would probably get afew option buys (puts or calls), some bull
TABLE 28-7.
Calculation of expected returns.
Price at Expiration (A) (B) (()
(q) P (below q) P (of being at q) Profit on Spread
30 .295 .295 -$200
30 1/s .301 .006 - 187.50
30 1/4 .308 .007 - 175
303/s .316 .008 - 162.50

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472 Part IV: Additional Considerations
spreads, some naked writes and ratio calendar spreads, fewer straddles and ratio
writes, and afew covered call writes. This theory would be somewhat difficult to apply
in practice, because of the massive numbers of calculations involved and also because
of the accuracy of closing price data. It was mentioned previously that acomputer will
assume that "bad" closing prices are actually attainable. By a "bad" closing price, it is
meant that the option did not trade simultaneously with the stock later in the day, and
that the actual market for the option is somewhat different in price than is reflected
by the closing price for the option. Adaily contract volume "screen" will help allevi
ate this problem. For example, one may want to discard any option from his calcula
tions if that option did not trade apredetermined, minimum number of contracts
during the previous day. Data that give closing bids and offers for each option are
more expensive but also more reliable, and would alleviate the problem of "bad"
closing prices. In addition to avolume screen, another way of reducing the calcula
tions required is to limit oneself to strategies in which one has interest, or which one
is reasonably certain will fit in well with his investment objectives. Regardless of the
limitations that one places upon the quantity of computations, some computer power
is necessary to compute expected return. Asophisticated programmable calculator
may be able to provide areal-time calculation, but could never be used to evaluate
the entire option universe and come up with aranking of the preferable situations
each day. On-line computer systems are also available that can provide these types of
calculations using up-to-the-minute prices. While real-time prices may occasionally
be useful, it is not an absolute necessity to have them.
One other by-product of the expected return calculation is that it could be used
as another model for predicting the theoretical value of an option. All one would have
to do is compute the probabilities of the stock being at each successive price above
the striking price of the option by expiration, and sum them up. The result would be
the theoretical option value. These data are published by some services and general
ly give adifferent theoretical value than would the Black-Scholes model. The reason
for the difference most readily lies in the inclusion of the risk-free interest rate in the
Black-Scholes model and its omission in the expected return model.
APPLYING THE CALCULATIONS TO STRATEGY DECISIONS
CALL WRITING
One method of ranking covered call writes that was described in Chapter 2 was to
rank all the writes that provided at least aminimal acceptable level of return by their
probability of not losing money. If one were interested in safety, he might decide to
use this approach. Suppose he decided that he would consider any write that provid-

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Chapter 28: Mathematical Applications 413
ed an annualized total return (capital gains, dividends, and commissions) of at least
12%. This would eliminate many potential writes, but would leave him with afairly
large number of writing candidates each day. He knows the downside break-even
point at expiration in each write. Therefore, the probability of the stock being below
that break-even point at expiration can be computed quickly. His final list would rank
those writes with the least chance of being below the break-even point at expiration
as the best writes. Again, this ranking is based on an expected probability and is, of
course, no guarantee that the stock will not, in reality, fall below the break-even
point. However, over time, alist of this sort should provide the rrwst conservative cov
ered writes.
Example: XYZ is selling for 43 and a 6-month July 40 call is selling for 8 points. After
including dividends and commission costs for a 500-share position, the downside
break-even point at expiration is 36. If the annualized volatility of XYZ is 25%, the
probability of making money at expiration can be computed. The 6-month volatility
is 17.7% (25% times the square root of½ year). The probability of being below 36
can be computed by using the formula given earlier in this section:
The expected probability of XYZ being below 36 in 6 months is 15.8%. Therefore,
this would be an attractive write on aconservative basis, because it has alarge prob
ability of making money (nearly 85% chance of not being below the break-even point
at expiration). The return if exercised in this example is approximately 20% annual
ized, so it should be acceptable from aprofit potential viewpoint as well. It is arela
tively easy matter to perform asimilar calculation, with the aid of acomputer, on all
covered writing candidates.
The ability to measure downside protection in terms of acommon denomina
tor - volatility - can be useful in other types of covered call writing analyses. The
writer interested in writing out-of-the-money calls, which generally have higher
profit potential, is still interested in having an idea of what his downside protection
is. He might, for example, decide that he wants to invest in situations in which the
probability of making money is at least 60%. This is not an unusually difficult
requirement to fulfill, and will leave many attractive covered writes with ahigh prof
it potential to choose from. Adownside requirement stated in terms of probability
of success removes the necessity of having to impose arbitrary requirements. Typical

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I
I I
Ii
I ;
Ii ;·
!
474 Part IV: Additional Considerations
arbitra:ry requirements would be including only calls that sell for one point or more,
or stating that the downside protection must be acertain percentage of the stock
price. These obviously cannot suffice for stocks with different volatilities. Rather,
the downside protection criterion should be stated in terms of "probability of down
protection" or, alternatively, in terms of the volatility itself. In this manner, auniform
comparison can be made between volatile and nonvolatile stocks.
CALL BUYING
The option buyer can also constructively use the measurement of volatility to aid him
in his option buying decisions. In Chapter 3, it was shown that evaluating the prof
itability of calls based on the volatility of the underlying stock is the correct way to
analyze an option purchase. One specific method of analysis is described. There are
certain variables in this analysis that may be altered to fit the call buyer'sindividual
preferences, but the general logic is applicable to all cases.
As afirst step, one should decide upon auniform stock rrwvement for ranking
call purchases. One might decide to rank all purchases by how they would perform
if the underlying stock moved up in accordance with its volatility. The phrase "in
accordance with its volatility" must be quantified. For example, one might decide to
assume that eve:ry stock could move up one standard deviation, and then rank all call
purchases on that basis. The prospective call buyer must also fix the time period that
he wants to use. Generally, one looks at purchases to be held for 30 days, 60 days, and
90 days.
The exact steps to be followed in the analysis of profitability and risk can be list
ed as follows:
I. Specify the distance that underlying stock can move, up or down, in terms of its
volatility.
2. Select the holding period over which the analysis is to take place.
PROFITABILITY
3. Calculate the stock price that the stock would move up to, when the foregoing
assumptions are implemented.
4. Using apricing model, such as the Black-Scholes model, estimate what the
option price would become after the upward stock movement.
5. Calculate the percent profit, after deducting commissions.
6. Repeat steps 4 and 5 for each option on the stock.

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Chapter 28: Mathematical Applications 475
Afinal ranking of all potential call buys can be obtained by performing steps 3
through 6 on all stocks, and ranking the purchases by their percentage reward.
RISK
7. Calculate the stock price that the stock could fall to, when the assumptions in
steps 1 and 2 are applied.
8. With amodel, price the option after the stock'sdecline.
9. Calculate the percentage loss after commissions.
10. Compute areward/risk ratio: Divide the percentage profit from step 5 by the
percentage risk from step 9.
11. Repeat steps 8 through 10 for each option on the stock.
Afinal ranking of less aggressive option purchases can be constructed by performing
steps 7 through 11 on all stocks, and ranking the purchases by their reward/risk ratio.
The higher profitability list of option purchases will tend to be at- or slightly
out-of-the-money calls. The less aggressive list, ranked by reward/risk potential, will
tend to be in-the-money options.
Example: Steps 1 and 2: Suppose an investor wants to look at option purchases for a
90-day holding period, under the assumption that each stock could move up by one
standard deviation in that time. (There is only about a 16% chance that astock will
move more than one standard deviation in one direction in agiven time period.
Therefore, in actual practice, one might want to use asmaller stock movement in his
ranking calculations.) Furthermore, assume that the following data are known:
XYZ common, 41;
XYZ volatility, 30% annually;
XYZ January 40 call, 4; and
time to January expiration, 6 months
Step 3: Calculate upward stock potential. This is accomplished by the following
formula:
where
p = current stock price
q = potential stock price

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476 Part IV: Additional Considerations
vt == volatility for the time period, ta == aconstant (see below).
The constants, aand t, are fixed under the assumptions in steps 1 and 2. The first
constant, a, is the number of standard deviations of movement to be allowed. In our
example, a == I. That is, the analysis is being made under the assumption that the
stock could move up by one standard deviation. The second constant, t, is .25, since
the analysis is for a 90-day holding period, which is 25% of ayear. In this example:
Vt == v-ft == .30 {is== .30 X .50 == .15
so
q == 4le· 15 == 41 X 1.16 == 47.64
Thus, this stock would move up to approximately 475/sif it moved one standard devi
ation in exactly 90 days.
Step 4: Using the Black-Scholes model, the XYZ January 40 call can be priced.
It would be worth approximately 81/sif XYZ were at 475/sand there were 90 days' less
life in the call.
Step 5: Calculate the profit potential. For this example, commissions will be
ignored, but they should be included in areal-life situation.
81/s-4 41/s Percent profit == --- == - == 103% 4 4
Thus, if XYZ stock moves up by one standard deviation over the next 90 days, this call
would yield aprojected profit of 103%. Recall again that there is only about a 16%
chance of the stock actually moving at least this far. If all options on all stocks are
ranked under this same assumption, however, afair comparison of profitable options
will be obtained.
Step 6 is omitted from this example. It would consist of performing asimilar
profit analysis (steps 4 and 5) on all other XYZ options, with the assumption that XYZ
is at 475/safter 90 days.
Step 7: Calculate the downside potential of XYZ. The formula for the downside
potential of the stock is nearly the same as that used in step 3 for the upside poten
tial:
q = pe-avt
== 4lc· 15 = 41 x .86 = 35.39
XYZ would fall to approximately 35¼ in 90 days if it fell by one standard deviation.
Note that the actual distances that XYZ could rise and fall are not the same. The

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Chapter 28: Mathematical Applications 477
upward potential was 65/spoints, while the downward potential is about 5¾ points.
This difference is due to the use of the lognormal distribution.
Step 8: Using the Black-Scholes model, one could estimate that the XYZ
January 40 call would be worth about 11/sif XYZ were at 35¼ in 90 days.
Step 9: The risk potential in the January 40 call would be:
. 4- l1/s 27/s Percent nsk = --- = - 72% 4 4
Step 10: The reward/risk ratio is merely the percentage reward divided by the per
centage risk:
Reward/risk ratio = l03% = 1.43 72%
Step 11: This analysis would be repeated for all XYZ options, and then for all other
optionable stocks. The less aggressive call purchases would be ranked by their
reward/risk ratios, with higher ratios representing more attractive purchases. More
aggressive purchases would be ranked by the potential rewards only (step 5).
This completes the call buying example. Before leaving this section, it should be
noted that the assumption of ranking the purchases after one full standard deviation
movement by the underlying stock is probably excessive. Amore moderate assump
tion would be that the stock might be able to move . 7 standard deviation. There is
about a 25% expected chance that astock could move up at least . 7 standard devia
tion at the end of afixed time period.
PRICING A PUT OPTION
Theoretical models for pricing put options have been derived; that is, ones that are
separate from call pricing models. Black and Scholes presented such amodel in their
original paper. However, as has been demonstrated, there is arelationship between
put and call prices in the listed option market due to the conversion and reversal
strategies.
One could use the ba,sic call pricing rrwdel for the purpose of predicting put
prices if he a,ssumes that arbitrageurs will efficiently influence the market via con
versions. Theoreticians will argue that such amethod of pricing puts assumes that the
arbitrage process is always present and works efficiently, and that this is not true. The
"conversion efficiency" assumption could be aserious fault if one were trying to
determine the exact overpriced or underpriced nature of the put option. However, if
one is merely comparing various put strategies under constant assumptions, the arbi
trage model for pricing puts works quite well.

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478 Part IV: Additional Considerations
The listed put' sprice can be estimated by using the call pricing model and the
arbitrage formula. Recall that the arbitrageur must include the cost of carrying the
position as well as the dividends to be received.
Theoretical Theoretical Strike Stock Carrying n· 'dd = + - - + IVI en sput call price price price cost
The "theoretical call price" is obtained from the Black-Scholes model. The carrying
cost is the cost of money (interest rate) times the striking price, multiplied by the
time to expiration. Recall that this is the approximation formula for carrying cost (see
Chapter 27 for comments on present value and compounding). Consequently, ifXYZ
were at 41 and a 6-month January 40 call option were valued at 4 points by the
Black-Scholes model, the theoretical put price could be estimated. Assume that the
cost of money interest rate is 10% annually, and that the stock will pay $.50 in divi
dends in 6 months (t = ½ year).
Theoretical put price = 4 + 40 - 41- (.10 x 40 x ½) + .50
=3-2+½
=l½
This means that if the call could be sold for 4 points, the arbitrageur would be will
ing to pay up to 1 ½ points for the put to establish aconversion. The arbitrageur'sprice is used as the theoretical listed put price estimate.
PUT BUYING
Put option purchases can be ranked in amanner very similar to that described for call
option buying. Reward opportunities occur when the stock falls in accordance with
its volatility. An upward stock movement represents risk for the put buyer. All of the
11 steps in the previous section on call buying are applicable to put buying. The pric
ing of the put necessary for steps 4 and 8 is done in accordance with the arbitrage
model just presented.
If an underlying stock does not have listed puts trading, the synthetic put can
be considered. While all U.S.-listed stocks have both puts and calls at every strike,
there are still situations with warrants, especially in foreign countries, that are appli
cable to the following discussion. Recall that synthetic puts are created for customers
by some brokerage houses. The brokerage sells the stock short and buys acall. The
customer can purchase the synthetic put for the amount of the risk involved, plus any
dividends to be paid by the underlying stock. The synthetic put pricing formula that
would be used in steps 4 and 8 of the option buying analysis is exactly the same as the
arbitrage model for listed puts, except that the carrying costs are omitted:

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Chapter 28: Mathematical Applications 479
Theoretical synthetic Theoretical Strike Stock D" .dd = + - + 1vi en sput price call price price price
When the ranking analysis is performed, very few synthetic puts will appear as
attractive put buys. This is because, when the customer buys asynthetic put, he must
advance the full cost of the dividend, but receives no offsetting cost reduction for the
credit being earned by the short stock position. Consequently, synthetic puts are
always more expensive, on arelative basis, than are listed puts. However, if one is par
ticularly bearish on astock that has no listed puts, asynthetic put may still prove to
be aworthwhile investment. The recommended analysis can give him afeeling for
the reward and risk potential of the investment.
CALENDAR SPREADS
The pricing nwdel can help in determining which neutral calendar spreads are nwst
attractive. Recall that in aneutral calendar spread, one is selling anear-term call and
buying alonger-term call, when the stock is relatively close to the striking price of the
calls. The object of the spread is to capture the time decay differential between the
two options. The neutral calendar spread is normally closed when the near-term
option expires. The pricing model can aid the spreader by estimating what the prof
it potential of the spread is, as well as helping in the determination of the break-even
points of the position at near-term expiration.
To determine the maximum profit potential of the spread, assume that the near
term call expires worthless and use the pricing model to estimate the value of the
longer-term call with the stock exactly at the striking price. Since commission costs
are relatively large in spread transactions, it would be best to have the computations
include commissions. Calculating asecond profit potential is sometimes useful as
well the profit if unchanged. To determine how much profit would be made if the
stock were unchanged at near-term expiration, assume that the spread is closed with
the near-term call equal to its intrinsic value (zero if the stock is currently below the
strike, or the difference between the stock price and the strike if the stock is initial
ly above the strike). Then use the pricing model to estimate the value of the longer
term call, which will then have three or six months of life remaining, with the stock
unchanged. The resulting differential between the near-term call'sintrinsic value and
the estimated value of the longer-term call is an estimate of the price at which the
spread could be liquidated. The profit, of course, is that differential minus the cur
rent (initial) differential, less commissions.
In the earlier discussion of calendar spreads, it was pointed out that there is
both an upside break-even point and adownside break-even point at near-term expi
ration. These break-even points can be estimated with the use of the pricing model.

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