ise in price, creating a
loss 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 a merger is announced. Thus, XYZ may
fall close to, or below, the lower "hook." Some arbitrageurs attempt to hedge them
selves by shorting a little 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 a hedge 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 a striking 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: A merger is announced as described in the preceding example: XYZ is to
acquire LMN at a stated 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.
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 a tender 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 a fractional 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, a company 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 a cash 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 a disastrous 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 a waste. 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 a large loss. On the other hand, it may also seem like easy
money to sell naked calls with a striking 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 a higher price for the compa
ny on which the calls were written. If this happens, there might suddenly be a large
upward jump in price, and the written calls could suffer a large loss.
Options can play a more 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 a partial
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
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 of


: 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
a profit of $3 per share if exactly 50% of the shares are accepted at 70 in the tender
offer. In reality, a slightly higher percentage of shares is usually accepted, because a
few people make mistakes and don't tender. Thus, one's average 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 a lot of money.
The use of calls does not provide a more 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 a sub
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"L at 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.
Chapter 27: Arbitrage 453
Before giving an example of that arbitrage, a word about short tendering is in
order. Short tendering is against the law. It comes about when one tenders stock into
a tender offer when he does not really own that stock. There are complex definitions
regarding what constitutes ownership of stock during a tender 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
a strike 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 a tender 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 a locked
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.
454 Part IV: Additional Considerations
If more than 50% of XYZ should be accepted in the tender offer, then a larger
profit will result. Also, if XYZ should subsequently trade at a high enough price so
that the July 50 put has some time value premium, then a larger 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 purc

eBooks\Lawrence G. McMillan - Options as a Strategic Investment_ Fifth Edition-Prentice Hall Press (2012)\Lawrence G. McMillan - Options as a Strategic Investment_ Fifth Edition-Prentice Hall Press (2012).pdf


uld 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 a cash 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 a partial 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 a public 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 a replacement 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
A stock 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 a hedged strategy. The two stocks' price movements are relat
ed historically. The pairs trader would establish the position when one stock was
Chapter 27: Arbitrage 455
expensive with respect to the other one, historically. Then, when the stocks return to
their historical relationship, a profit 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't have any expense except the possible differential
in dividend payout.
The bane of pairs trading is a possible escalation of the stock sold short without
any corresponding rise in price of the stock held long. A takeover attempt might
cause this to happen. Of course, pairs traders will attempt to research the situation
to ensure that they don't often 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 a debit, since both options
are purchased; so there would be a cost 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 a takeover occurs, the put
has limited liability, and the trader's loss 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't return to historical norms.
This would happen, for example, if both stocks dropped a great deal: The call has lim
ited loss, while the put' s profits would continue to accrue. Similarly, to the upside, a
large 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't perform as predicted.
This type of strategy- buying in-the-money options as substitutes for both sides
of a spread 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 a trader 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.
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 a limited 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 a focal point for the work. The investor who does
have a knowledge of mathematics and also has access to a computer will be able to
directly use the techniques


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 a focal point for the work. The investor who does
have a knowledge of mathematics and also has access to a computer will be able to
directly use the techniques in this chapter.
THE BLACK-SCHOLES MODEL
Since an option's price is the function of stock price, striking price, volatility, time to
expiration, and short-term interest rates, it is logical that a formula 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 a result, gained a rather 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
Chapter 28: Mathematical Applications
Theoretical option price= pN(d 1) se-rtN(d2)
p v2
ln(8 )+ (r +2 )t
where d1 = _ r.
V-4 t
d2 = d1 - v--ft
The variables are:
p = stock price
s = striking price
t = time remaining until expiration, expressed as a percent of a year
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 a small
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, a computer can handle the calculations easily and with great
speed. A large number of Black-Scholes computations can be performed in a very
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 a table. Since the normal curve is a smooth curve (it is
the "bell-shaped" curve used most commonly to describe population distributions),
the cumulative distribution can be approximated by a formula:
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) = x if cr > 0 or N(cr) = 1- x if cr < 0
458 Part IV: Additional Considerations
This approximation is quite accurate for option pricing purposes, since one is not
really interested in thousandths of a point 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 a negative
number, the distribution is determined by subtracting x from l.
N(d1) = N(-.67O25) = l -x = l - .74865 = .25134
In a similar manner, which requires computing new values for x, y, and z,
N(d2) = N(-.79179) = 1- .78579 = .21421
Chapter 28: Mathematical Applications 459
Now, returning to the formula for theoretical option price, we can complete the
calculation of the July 50 call's theoretical value, called value here for short:
value = 45 x N(d1) - 50 x e-·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 a point.
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 a small 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 a more 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


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 a negative 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 a weighting factor to call prices. The
weighting factor would be based on the dividend payment, with a heavier 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.
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 a lognormal 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, a normal dis
tribution has a bell-shaped curve. This is the most familiar mathematical distribution.
The problem with using a normal 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 a range only
between zero and infinity. Furthermore, the upward (bullish) bias of the lognormal
distribution appears to be logically correct, since a stock 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 a different 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 a stock has a
greater 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 a large 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 a difficult 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
Chapter 28: Mathematical Applications 461
n = number of days observed
v = volatility
When volatility is computed using past stock prices, it is called a historical
volatility. The volatilities of stocks tend to change over time. Certain predictable fac
tors, such as a large stock split increasing the float of the stock, can reduce the volatil
ity. The entry of a company into a more speculative area of business may increase the
volatility. Other, less well-defined factors can alter the volatility as well. Since the
volatility is a very crucial element of the pricing model, it is important that the mod
eler use a reasonable estimate of the current volatility. It has become apparent that
an annual standard deviation is not accurate, because it encompasses too long a peri
od of time. Recent efforts by many modelers have suggested that one should perhaps
weight the recent stock price action more heavily than older price action in arriving
at a current volatility. This is a possible approach, but the computation of such fac
tors may introduce as much error as using the annual standard deviation does. The
problem of accurately computing the volatility is critical, because the model is so sen
sitive to it.
Computing Lognormal Historical Volatility. The above calculation does
not give the proper input for the Black-Scholes model because the model assumes
that the logarithms of changes in price are normally distributed, not the prices them
selves. That is, the term Pi in the above formula should be changed.
Example: XYZ closed at 51 today and at 50 yesterday. Thus, its percentage change
for the day is 51/50 = 1.02. The natural logarithm of 1.02 is then based on the volatil
ity formula:
ln(51/50) = ln(l.02) = 0.0198
This is similar to saying that arithmetically the stock was up 2% today, but on a
lognormal basis, it was only up 1.98%
If the stock is down, this method will yield a negative number. Suppose that on
the fol


today and at 50 yesterday. Thus, its percentage change
for the day is 51/50 = 1.02. The natural logarithm of 1.02 is then based on the volatil
ity formula:
ln(51/50) = ln(l.02) = 0.0198
This is similar to saying that arithmetically the stock was up 2% today, but on a
lognormal basis, it was only up 1.98%
If the stock is down, this method will yield a negative number. Suppose that on
the following day, XYZ declined from 51 back to 50. The number to use in the volatil
ity formula would then be:
ln(50/51) = ln(0.9804) = -0.0198
462 Part IV: Additional Considerations
A new equation can now be formulated using this concept. It will yield volatili
ties that are consistent with the Black-Scholes model:
V=
n I
where Xi = ln(P /Pi _1); Pi = closing price on day i and X = the average of the X/s over
the desired number of days.
So to compute a IO-day historical volatility, one would need 11 observations. In
the following example, do not be concerned with the complete details if you do not
plan to compute the volatilities yourself; they are provided for the mathematician or
programmer who needs to check his work:
Day XYZ Stock P./P. 1 I I- X-=ln(P./P. 1) I I I- (X.-X)2
I
1 153.875
2 153.625 .9984 -.0016 .000020
3 151 .9829 -.0172 .000405
4 146 .9669 -.0337 .001336
5 144.125 .9872 -.0129 .000250
6 147.25 1.0217 .0215 .000345
7 146.25 .9932 -.0068 .000094
8 149.5 1.0222 .0220 .000365
9 152.5 1.0201 .0199 .000289
10 158.625 1.0402 .0394 .001332
1 1 158.375 0.9984 -.0016 .000020
AVG: 0.0028825 :l;: 0.004455
The average of the Ins (4th column) over the 10 days is 0.00288.
The difference of each In from the mean, squared, is then summed (5th col
umn). For example, for day 1 the term is (- .0016- .00288)2 = .00002. This is the top
number in the far right-hand column. This process can be computed for each num
ber in the "In" column. The sum of all these terms is 0.004455.
Nowv = ✓(.004455/9) = 0.022249
This is a IO-day volatility. To convert it into an annual volatility, we need to mul
tiply by the square root of the number of trading days in a year. Since there are
approximately 260 trading days in a year, the final volatility would be:
V = 0.022249 X ✓(260) = 0.3587
Thus, one could say that the volatility of XYZ is 36% on an annualized basis.
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 a stock or future and its options by seeing how the various volatilities
compare with one another.
There is, in fact, a way 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, a corollary 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 a particular 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, a single 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 a different 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 a volatility 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's daily 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 x is greater than a
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 x is the percentage distance between stock price and strike price and a is the
maximum percentage distance at which the modeler wants to give any weight at all
to the option's implied volatility.
Example: An investor decides that he wants to discard options from the weighting
criterion


.34
.28
.30
.38
Volume Weighting Factor
.25 (50/200)
.45 (90/200)
.275 (55/200)
.025 ( 5/200)
where x is the percentage distance between stock price and strike price and a is the
maximum percentage distance at which the modeler wants to give any weight at all
to the option's implied 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 a formula, 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-
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's implied 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's implied volatilities. Rather, it is a composite 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 a form of standard deviation, and can thus be used whenever a stan
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 a stock. For example, as markets become bullish or
bearish (generating large rallies or declines), most stocks will react in a volatile 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 a day-to-day basis, the
implied volatility for a stock - especially one whose options are not too active may
fluctuate more than the strategist would like. A smoothing effect can be obtained by
466 Part IV: Additional Considerations
taking a moving 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 a
momentum calculation on the implied volatility. For example, today's final volatility
might be computed by adding 5% of today's implied volatility to 95% of yesterday's
final volatility. This method requires saving only one previous piece of data - yester
day's final 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 a discrepancy between the option's actual 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 a transaction, he normally has a belief as to the pos
sibility of making a profit. For example, when he buys stock he may think that there
is a "good chance" that there will be a rally or that earnings will increase. The investor
may consciously or unconsciously evaluate the probabilities, but invariably, an invest
ment is made based on a positive expectation of profit. Since options have fixed
terms, they lend themselves to a more 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 a large number of cases.
A simple 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 a cer
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.
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 Februa


d for simplification. Table 28-5 summarizes these
assumptions.
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. A bull 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 a
return 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 a particular 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 a similar 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 a better chance of making rrwney.
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 a crucial 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 a reasonably 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. A correct expected return computation must take into account all possible out
comes for the stock.
Fortunately, there is a straightforward method of computing the expected per
centage chance of a given stock being at a certain price at a certain point in time. This
computation involves using the distribution of stock prices. As mentioned earlier, the
Black-Scholes model assumes a lognormal 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 a graph of a typical 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 a fairly great chance that the stock will be relatively unchanged;
there is no chance that the stock will be below zero; and there is a bullish bias to the
graph - the stock could rise infinitely, although the chances of it doing so are
extremely small.
Chapter 28: Mathematical Applications
FIGURE 28-1.
Typical lognormal distribution.
60%
0 A Mean (current price)
Stock Price at End of Time Period
469
The chance that XYZ will be below the meah at the end of the time period is
50% in a random walk distribution. This also means that 50% of the area under the
graph lies to the left of the mean and 50% lies to the right of the mean. Note point
A on the graph. Forty percent of the area under the distribution curve lies to the left
of point A and 60% lies to the right of it. This means that there is a 40% chance that
the stock will be below price A at the end of the time period and a 60% chance that
the stock will be above price A. Consequently, the distribution curve can be used to
determine the probabilities necessary for the expected return computation. The
reader should take note of the fact that these probabilities apply to the end of the time
period. They say nothing about the chances that XYZ might dip below price A at
some time during the time period. To compute that percentage, an involved compu
tation is necessary.
The height and width of the distribution graph are determined by the volatility
of the underlying stock, when volatility is expressed as a standard deviation. This is
consistent with the method of computing volatility described earlier in this chapter.
Implied volatility can, of course, be used. Since the option modeler is generally inter
ested in time pe


an involved compu
tation is necessary.
The height and width of the distribution graph are determined by the volatility
of the underlying stock, when volatility is expressed as a standard deviation. This is
consistent with the method of computing volatility described earlier in this chapter.
Implied volatility can, of course, be used. Since the option modeler is generally inter
ested in time periods other than one year, the annual volatility must be converted into
a volatility for the time period in question. This is easily accomplished by the follow
ing formula:
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, t would equal .25 (one fourth of a year), 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 a stock that is currently at price p being below some other price, q, at the
end of the time period. The lognormal distribution is assumed.
Probability of stock being below price q at 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 a computer. One merely has to start at some price - the lower strike in a bull
spread, for example - and work his way up to a higher price - the high strike for a
bull spread. At each price point in between, the outcome of the spread is multiplied
by the probability of being at that price, and a running sum is kept.
Simplistically, the following iterative equation would be used.
P ( of being at price x) = P (below x) - P (below y)
where y is close to but less than x in price. As an example:
P (of being at 32.4) = P (below 32.4) - P (below 32.3)
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.
A partial 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 a point up through
a price of 35. The expected return is compute<l by 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 a large number of them.
In theory, if one had the data and the computer power, he could evaluate a wide
range of strategies every day and come up with the best positions on an expected
return basis. He would probably get a few 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
472 Part IV: Additional Considerations
spreads, some naked writes and ratio calendar spreads, fewer straddles and ratio
writes, and a few 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 a computer 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. A daily 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 a predetermined, 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 a volume 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. A sophisticated programmable calculator
may be able to provide a real-time calculation, but could never be used to evaluate
the entire option universe and come up with a ranking of


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. A sophisticated programmable calculator
may be able to provide a real-time calculation, but could never be used to evaluate
the entire option universe and come up with a ranking 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 a different 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 a minimal 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-
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 a fairly
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, a list 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 a conservative basis, because it has a large 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 a profit potential viewpoint as well. It is a rela
tively easy matter to perform a similar calculation, with the aid of a computer, on all
covered writing candidates.
The ability to measure downside protection in terms of a common 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 a high prof
it potential to choose from. A downside requirement stated in terms of probability
of success removes the necessity of having to impose arbitrary requirements. Typical
I
I I
I i
I ;
I
i ;·
!
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 a certain 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, a uniform
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's individual
preferences, but the general logic is applicable to all cases.
As a first step, one should decide upon a uniform 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


rm
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 a pricing 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.
Chapter 28: Mathematical Applications 475
A final 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 a model, price the option after the stock's decline.
9. Calculate the percentage loss after commissions.
10. Compute a reward/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.
A final 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 a stock will
move more than one standard deviation in one direction in a given time period.
Therefore, in actual practice, one might want to use a smaller 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
476 Part IV: Additional Considerations
vt == volatility for the time period, t
a == a constant (see below).
The constants, a and 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 a year. 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/s if 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/s if XYZ were at 475/s and 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 a real-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 a projected 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, a fair comparison of profitable options
will be obtained.
Step 6 is omitted from this example. It would consist of performing a similar
profit analysis (steps 4 and 5) on all other XYZ options, with the assumption that XYZ
is at 475/s after 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
Chapter 28: Mathematical Applications 477
upward potential was 65/s points, 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/s if 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


he 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. A more moderate assump
tion would be that the stock might be able to move . 7 standard deviation. There is
about a 25% expected chance that a stock could move up at least . 7 standard devia
tion at the end of a fixed 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 a model in their
original paper. However, as has been demonstrated, there is a relationship 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 a method 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 a serious 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.
478 Part IV: Additional Considerations
The listed put' s price 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· 'd d = + - - + IVI en s put 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 a conversion. The arbitrageur's
price is used as the theoretical listed put price estimate.
PUT BUYING
Put option purchases can be ranked in a manner 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 a call. 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:
Chapter 28: Mathematical Applications 479
Theoretical synthetic Theoretical Strike Stock D" .d d = + - + 1vi en s put 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 a synthetic 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 a relative basis, than are listed puts. However, if one is par
ticularly bearish on a stock that has no listed puts, a synthetic put may still prove to
be a worthwhile investment. The recommended analysis can give him a feeling 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 a neutral calendar spread, one is selling a near-term call and
buying a longer-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


ell 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 a second 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's intrinsic 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 a downside break-even point at near-term expi
ration. These break-even points can be estimated with the use of the pricing model.
480 Part IV: Additional Considerations
One method of determination involves estimating the liquidating value of the spread
at successive stock prices. When the liquidating value is found to be equal to the ini
tial value, plus commissions, a break-even point has been located.
Example: If the spread in question is using options with a striking price of 30, one
would begin his break-even point calculations at a price of 30. Estimate the liquidat
ing value of the spread at 30, 297/s, 29¾, 29-5/s, and so forth until the break-even point
is found. Once the downside break-even point has been determined in this manner,
the iterations to locate the upside break-even point should begin again at the striking
price. Thus, one would evaluate the liquidating value at 30, 301/s, 30¼, and so on. This
is somewhat of a brute-force method, but with a computer it is fairly fast. The num
ber of calculations can be reduced by adopting a more complicated iteration process.
A final useful piece of information can be obtained with the aid of the pricing
model - the theoretical value of the spread. Recompute the estimated value of both
the near-term and longer-term calls at the current time and stock price, using the
implied volatility for the underlying stock. The resultant differential between the two
estimated call prices may differ substantially from the actual differential, perhaps
highlighting an attractive calendar spread situation. One would want to establish
spreads in which the theoretical differential is greater than the actual differential
(that is, he would want to buy a "cheap" calendar spread).
Once these pieces of information have been computed, the strategist can rank
the spread possibilities by whatever criterion he finds most workable. The logical
method of ranking the spreads is by their return if unchanged. The spreads with the
highest return if unchanged at near-term expiration are those in which the stock price
and striking price were close together initially, a basic requirement of the neutral cal
endar spread. More complicated ranking systems should tty to include the theoreti
cal value of the spread and possibly even the maximum potential of the spread. A
similar analysis can, of course, be worked out for put calendar spreads, using the
arbitrage pricing model for puts.
RATIO STRATEGIES
Ratio strategies involve selling naked options. Therefore, the strategist has potential
ly large risk, either to the upside or to the downside or both. He should attempt to
get a feeling for how probable this risk is. The formulae for determining the proba
bility of a stock being above or below a certain price at some time in the future can
give him these probabilities. For example, in a straddle writing situation, the strate
gist would want to compute such arithmetic quantities as maximum profit potential,
return if unchanged, collateral required at upside break-even point or at upside
Chapter 28: Mathematical Applications 481
action point (recall that the collateral requirement increases for naked options on an
adverse stock movement), and the break-even points themselves. The probabilities of
being above the upper break-even point at expiration or below the lower break-even
point should be computed as well. Moreover, an expected return analysis could be
performed on the position to determine the general level of profitability of the posi
tion with respect to all other positions of the same type on other stocks. Such an
expected return analysis need not assume that the position is held to expiration. Firm
traders, paying little or no commissions, might be interested in seeing the expected
reswts for a holding period as short as 30 days or less. Public customers might use a
longer holding period, on the assumption that they would not trade the position as
readily because of commission costs. Ratio positions should be ranked either by
return if unchanged or by expected return.
The analyses described for calendar spreads and ratio positions should not be
relied upon as gospel. In the proposed forms of analysis, one is projecting future
option prices and stock prices under the assumption that the volatility of the under
lying stock will remain the same. Although this may be true in some cases, there will
also be many times when the volatility of the underlying stock will change during the
life o


alendar spreads and ratio positions should not be
relied upon as gospel. In the proposed forms of analysis, one is projecting future
option prices and stock prices under the assumption that the volatility of the under
lying stock will remain the same. Although this may be true in some cases, there will
also be many times when the volatility of the underlying stock will change during the
life of the position. If the volatility decreases, the projected break-even points for a
calendar spread will be too far away from the striking price. Thus, a loss would result
at some prices where the spreader expected to make money. If the volatility increas
es, the expected return of a ratio position will drop, because the probabilities of the
stock moving outside the profit range will increase, thereby increasing the probabil
ity ofloss.
The effect of a changing volatility can be counteracted, in theory, by continuing
to monitor the position daily after it has been established. In a straddle write, for
example, if the stock begins to move dramatically, the expected return may become
very low. If this happens, adjustments could be made to the position to improve it.
Such monitoring is difficult to apply in practice for the public customer, because the
commission costs involved in constant position adjustments would mount rapidly.
There is no exact method that would allow for infrequent, periodic adjustments, but
by using a follow-up analysis the public customer may be able to get a better feeling
for the timing of adjusting a position. For example, suppose that one initially wrote a
5-point straddle when the stock was at 30. Sometime later, the stock is at 34. The
expected return of writing a 5-point straddle with a strike of 30 when the stock is at
34 could be computed for the shorter time period remaining until expiration. If the
expected return is negative, an adjustment needs to be made. Adopting this form of
adjusting would keep the number of trades to a minimum, but would still allow the
strategist to determine when his position has become improperly balanced. Of
course, the current volatility would be used in making these determinations. Another
482 Part IV: Additional Considerations
follow-up monitoring technique, using the deltas of the options involved, is present
ed later in this chapter, and has been described several times previously.
FACILITATION OR INSTITUTIONAL BLOCK POSITIONING
In this and the following section, the advantages of using the hedge ratio are outlined.
These strategies are primarily member firm, not public customer, strategies, since
they are best applied in the absence of commission costs. An institutional block trad
er may be able to use options to help him in his positioning, particularly when he is
trying to help a client in a stock transaction.
Suppose that a block trader wants to make a bid for stock to facilitate a cus
tomer's sell order. If he wants some sort of a hedge until he can sell the stock that he
buys, and the stock has listed options, he can sell some options to hedge his stock
position. To determine the quantity of options to sell, he can use the hedge ratio. The
exact formula for the hedge ratio was given earlier in this chapter, in the section on
the Black-Scholes pricing model. It is one of the components of the formula. Simply
stated, the hedge ratio is merely the delta of the option - that is, the amount by which
the option will change in price for small changes in the stock price. By selling the cor
rect number of calls against his stock purchase, the block trader will have a neutral
position. This position would, in theory, neither gain nor lose for small changes in the
stock price. He is therefore buying himself time until he can unwind the position in
the open market.
Example: A trader buys 10,000 shares of XYZ, and a January 30 call is trading with
a hedge ratio of .50. To have a neutral position, the trader should sell options against
20,000 shares of stock (10,000 divided by .50 equals 20,000). Thus, he should sell 200
of the January 30's. If the hedge ratio is correct - largely a function of the volatility
estimate of the underlying stock - the trader will have greatly eliminated risk or
reward on the position for small stock movements. Of course, if the block trader
wants to assume some risk, that is a different matter. However, for the purposes of
this discussion, the assumption is made that the block trader merely wants to facili
tate the trade in the most risk-free manner possible. In this sample position, if the
stock moves up by 1 point, the option should move up by ½ point. The trader would
make $10,000 on his stock position and would lose $10,000 on his 200 short options
- he has no gain or loss. Once the trader has the neutral position established, he can
then begin to concentrate on unwinding the position.
In actual practice, this hedge ratio may not work exactly, because it tends to
change constantly as the stock price changes. If the trader finds the stock moving
Chapter 28: Mathematical Applications 483
more than fractionally, he may have to add more calls or buy some in, to maintain a
neutral hedge ratio. This would expose him to some risk, but the risk is substantially
smaller than if he had not hedged at all. Of course, there would also be certain cases
in which he would profit by the stock price change. For example, implied volatility
could decrease, making the calls cheaper.
A similar hedge can be established by the block trader who sells stock to accom
modate a customer buy order. He could buy calls in accordance with the hedge ratio,
to set up a neutral position.
This process off acilitation is quite widely practiced, especially by brokerage
houses that are trying to attract the business of the large institutional customer. Since
the introduction of listed call options and their applications for facilitating orders,
many quotes for large blocks of stock hav


buy order. He could buy calls in accordance with the hedge ratio,
to set up a neutral position.
This process off acilitation is quite widely practiced, especially by brokerage
houses that are trying to attract the business of the large institutional customer. Since
the introduction of listed call options and their applications for facilitating orders,
many quotes for large blocks of stock have improved considerably. The block trader
(who works for the brokerage house) is willing to make a higher bid or a lower offer
if he can use options to hedge his position. This facilitation with options results in a
better market (higher bid or lower off er) from the point of view of the institutional
customer. Without the availability of such listed options, the block trader would prob
ably make a bid or an offer that was substantially away from the prevailing market
price in order to work out of his stock-only position with a lessened degree of risk
This would obviously present a poorer market for the institutional customer.
THE NEUTRAL SPREAD
The hedge ratios of two or more options may be used to determine a neutral spread.
This strategy is especially useful to market-makers on the options exchanges who may
want to reduce the risk of options bought or sold in the process of providing a pub
lic market. If the hedge ratios of two options are known, the neutral ratio is deter
mined by dividing the two hedge ratios.
Example: An XYZ January 35 has a hedge ratio of .25, and an XYZ January 30 has a
hedge ratio of .50, so a neutral ratio would be 2:1 (.50 divided by .25). That is, one
would sell 2 January 35's against one long January 30, or, conversely, would buy 2
January 35's against one short January 30. Thus, a market-maker who has just bought
50 January 30's in an effort to provide a market for a public seller of that call could
hedge his position by selling 100 January 35's. This should keep his risk small, for
small stock price changes, until he can unwind the position. The ratio for the neutral
spread is not as sensitive to the volatility estimate of the underlying stock as is the
ratio concerning stock and options. This is because the same volatility estimate is
applied to both options, and the resultant ratio for the spread would not tend to
change greatly.
484 Part IV: Additional Considerations
The risk trader can also use the neutral spread ratio to his advantage. This con
cept was illustrated several times in previous chapters describing ratio writing, ratio
spreads, and straddle writes. Ratio spreads are quite popular with member firm
traders and floor traders. Recall that a ratio spread consists of buying options at acer
tain strike, and selling more options further out-of-the-money. The hedge ratios can,
of course, be used by the trader, or by a public customer, to initially establish a neu
tral position. Perhaps more important, the hedge ratio can also be used as a follow
up action to keep the position neutral after the stock changes in price. This strategy
is the "delta spread" described in Chapter 11.
The risk trader is not attempting to establish the spread with the idea of mini
mizing risk for small stock movements. Rather, he is looking to make a profit, but
would prefer to remain as neutral as possible on the underlying stock. He is imple
menting a risk strategy that has a neutral outlook on the underlying stock. He is sell
ing much more time value premium than he is buying.
Example: The purchase of 15 January 30 calls and the sale of 30 January 35 calls - a
ratio call spread - may be a position taken for profit potential. It would be a neutral
position if the deltas were .60 and .30, for example. This spread would do best if the
stock were at exactly 35 at expiration. However, if the stock rose quickly before expi
ration, the spread ratio would decrease from 2:1 to perhaps 3:2. That is, the neutral
ratio between the January 30 call and the January 35 call should be 3 short January
35's to 2 long January 30's. If the trader wants to balance his position, he could buy 5
more January 30's, giving him a total of 20 long versus the 30 short January 35's that
he originally sold. Conversely, if the stock dropped in price, the neutral spread ratio
might increase, indicating that more calls should be sold. For example, if this stock
declines, the neutral ratio might be 3:1. In that case, 15 more January 35's could be
sold, making the position short 45 calls versus 15 long calls, which would produce the
neutral 3:1 ratio.
It would not be proper to adjust the ratio constantly, because the frequent
whipsaw losses on trading movements would wipe out the profit potential of the posi
tion. However, the trader may want to pick out points, in advance, at which he wants
to reevaluate his position before something drastic goes wrong. For example, if the
foregoing spread were established with the stock at a price of 30, the spreader might
want to readjust at 33 or 27, whichever comes first.
By monitoring the spread using the hedge ratio, the trader may also be able to
discern whether he has established too bullish or too bearish a position.
Example: The trader starts with the example described above - long 15 January 30
calls and short 30 January 35 calls - when the hedge ratios were .60 and .30, respec-
Chapter 28: Mathematical Applications 485
tively. Some later time, the stock falls to 27 and the trader needs to reevaluate his
position. The hedge ratios may have become .42 for the January 30 and .14 for the
January 35, indicating that a 3:1 ratio would be neutral (.42/.14 = 3). He now has a
bullish position, because his 2:1 ratio is less than the neutral 3:1 ratio. It is not manda
tory that the trader act on this information. He may actually be bullish on the stock
at this point and decide to remain with his position. The usefulness of the hedge ratio
is that it allows him to see that his position is bullish, so he can make a correct jud


ould be neutral (.42/.14 = 3). He now has a
bullish position, because his 2:1 ratio is less than the neutral 3:1 ratio. It is not manda
tory that the trader act on this information. He may actually be bullish on the stock
at this point and decide to remain with his position. The usefulness of the hedge ratio
is that it allows him to see that his position is bullish, so he can make a correct judg
ment. Without this knowledge, he might still think his position to be neutral, a criti
cal mistake if he indeed wants to be neutral. If the trader's ratio is greater than the
neutral ratio (2:1 vs. 3:2, for example), he is bearishly positioned.
As a final point, it should be noted that the ratio can be adjusted by buying or
selling either option.
Example: If the stock falls and it is desired that the ratio be increased to 3:1, one
might sell more January 35's or might decide to sell out some of his January 30's. A
bullish adjustment could be made by buying on either side of the spread in a similar
manner. In general, one should adjust by selling time premium or buying intrinsic
value. That is, out-of-the-moneys are usually sold and in-the-moneys are usually
bought, when adjusting.
AIDING IN FOLLOW-UP ACTION
The computer can also be an invaluable aid to the strategist in that it can help him
monitor his positions. It is generally necessary for the strategist to have some way of
inputting his positions into an inventory database and also to have some way of iden
tifying different securities that are grouped within the same trading position. Once
this has been done, the computer can simultaneously read pricing data ( either real
time or closing prices) and the inventory database to generate information concern
ing the current status of any position.
A current mark to market (profit and loss) statement is of obvious use in that
the trader can see how he is doing each day. The computer can also easily generate
a set of warning flags that may be of interest to the trader, and could produce a list
summarizing possible positions that need action. In most of the strategies that were
described, it was shown that the strategist should avoid early assignment if at all pos
sible. It is a simple matter for the computer to calculate the remaining time value
premium of any short options, and to warn the trader if there is only a small amount
of time value premium remaining, perhaps ½ point or less. For similar reasons, the
trader may want to have a daily list of positions that are nearing maturity, perhaps
486 Part IV: Additional Considerations
with less than I month oflife remaining in the options. A flag indicating an approach
ing ex-dividend date might also be useful for this purpose.
If the trader inputs another piece of information into the database, the com
puter can help him in another follow-up action. In most strategies that were
described, especially those involving uncovered options, the trader wants to take
some sort of follow-up action based on the price movement of the underlying stock.
If the stock rallies too far, he may want to cover short calls or buy other calls as pro
tection. If the stock declines too far, similar maneuvers would apply to put options or
to rolling down short calls. If the trader inputs the stock prices at which he would like
to take action, the computer can monitor each day's closing price of the stock and
generate a list of positions that have exceeded their upside or downside action points.
The computer can also do more sophisticated types of position monitoring.
Recall that it was pointed out that the deltas of the options involved in a position can
be compared to each other to tell whether the position is bullish or bearish. The
Black-Scholes model can be used to calculate the deltas of the options in one's posi
tions. Then the net position can be determined by the computer, thereby telling the
trader whether his position has become "delta long" (bullish), "delta short" (bearish),
or neutral. If he sees that a position is bearish and he does not want to be structured
in that way, he can make bullish adjustments. The delta spread and neutral spread
strategies very conveniently lend themselves to such types of follow-up action,
although any of the more complicated straddle writing and protected straddle writ
ing positions can be monitored usefully in this way as well.
The computation for determining whether a position is net short or net long
generally involves calculating the "equivalent stock position" (ESP). If one owns 10
calls that have a delta of .45, his equivalent stock position from those calls is IO X 100
shares per call x .45 = 450. That is, owning those IO calls is equivalent to owning 450
shares of the underlying stock, according to the delta. All puts and calls can be
reduced to an ESP and can then, of course, be combined with any actual long or
short stock in the position to produce an ESP for the entire strategy. The resultant
ESP for each of the trader's positions can be printed from the computer along with
the items described above.
Further sophisticated measures can be taken. The computer can generate a
table of results at expiration. If so desired, this could be presented as a graph, but that
is not really necessary. A table suffices quite well, as shown by most of the examples
in this book. Such a picture has meaning only if all options in the position expire at
the same time. If they don't, one may instead want the computer to mmpose a table
of results or a graph at near-term expiration. Thus, in a calendar spread, for example,
one could see what sorts of profitability he would be looking at when it was time to
remove the spread.
Chapter 28: Mathematical Applications 487
Finally, the computer can compute the expected return of a position already in
place. This would give a more dynamic picture of the position, and this expected
return is usually for a relatively short time period. That ti