THE USE OF OPTIONS FOR TAX DEFERRAL
by
HARVEY WILLENSKY
B.S., Cornell University
(1971)
M.S., University of Illinois
(1973)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF SCIENCE IN MANAGEMENT
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 1977
Signature of Author.
Depa LRent of Course 15, June 1977
Certified by....
Thesis Supervisor
Accepted by ...........
Chairman, Department Committee
Archives
JUN 27 1977
19ýACL
THE USE OF OPTIONS FOR TAX DEFERRAL
by
HARVEY WILLENSKY
Submitted to the Department of Course 15
on June, 1977 in partial fulfillment of the requirements
for the Degree of S.M.
ABSTRACT
This thesis has presented three strategies for deferring capital
gains income from one year to the next for income tax purposes. All
three strategies involved the establishment of neutral spreads using
options. As the price of the underlying stock changes, losses will
occur on one side of the spread and gains on the other side of the spread.
The losses should be recognised for tax purposes in the current year and
the gains in the following year.
Techniques have been developed for adjusting all of the possible
spreads so that they can be compared at a constant risk level. The metric
used to compare the spreads is the dollar volatility of each side of the
spreads. This is an indicator of the likely tax deduction which an investor can expect.
At the time that the spread is established, it is not possible to know
what the price of the stock will be in the future. Analyses have been
presented to illustrate the potential outcomes of the spreads for a
variety of stock prices.
The precise amount of tax savings which an investor can realize for
a given short term capital loss will depend on the investor's tax bracket
and the type of gains, that is, short term or long term, which are being
offset. The tables and analyses which have been presented in this paper
provide a framework which may be helpful in evaluating spreads for use as
a tax planning tool. However, this information does not specify which
strategy is the best choice for a particular investor.
ADVISOR:
TITLE:
Fischer Black
Professor of Finance
TABLE OF CONTENTS
Page
1.0
INTRODUCTION
2.0
2.1
2.2
BACKGROUND
Definitions
Basic Options Relationships
2.2.1 Call Options
2.2.2 Put Options
2.2.3 Spreads Between Puts and Calls
9
9
12
13
19
28
3.0
TAX TREATMENT OF OPTIONS TRANSACTIONS
30
3.1
3.2
3.3
3.4
3.5
3.6
Simple Purchase and Sale of Options
Purchasing Options and Exercising Them
Purchasing Options and Allowing Them to Expire
Writing Options and Having Them Exercised
Writing Options and Having Them Expire
Straddles
30
31
32
32
34
34
4.0
4.1
4.3
4.4
TAX STRATEGIES
Option Spread Strategies
4.1.1 Call Option Spreads
4.1.2 Put Option Spreads
Spreads Between Stock and Options
4.2.1 Spreads Between Call Options and Stock
4.2.2 Spreads Between Put Options and Stock
Spreads Between Puts and Calls
Comparison of the Strategies
37
42
43
56
65
66
70
74
77
5.0
Appendix
82
6.0
BIBLIOGRAPHY
4.2
1
140
LIST OF FIGURES AND TABLES
Page
Figure Number
1
Call Option Values as a Function of Stock Price for
Different Durations
2
Call Option Values as a Function of Stock Price for
Different Interest Rates
3
Call Option Value as a Function of Stock Price for
Different Stock Volatilities
4
Call Option Deltas as a Function of Stock Price for
Different Durations
5
Call Option Vertical Spread Values as a Function of
Stock Price for Different Durations
6
Put Option Values as a Function of Stock Price for
Different Durations
7
Put Option Values as a Function of Stock Price for
Different Interest Rates
8
Put Option Values as a Function of Stock Price for
Different Volatilities
9
Put Option Deltas as a Function of Stock Prices for
Different Durations
25
Put Option Vertical Spread Values as a Function of
Stock Price for Different Durations
27
Straddle Values as a Function of Stock Price for
Different Durations
29
10
11
Table Number
1
Call Option Prices for IBM and DEC
44-45
2
Call Spreads
47-48
3
Results of Call Spreads If Sold Immediately
51
4
Results of Call Spreads If Held Until December 31
55
5
Put Option Prices For IBM and DEC
57-58
6
Put Spreads
60-61
7
Results of Put Spreads If Sold Immediately
62
8
Results of Put Spreads If Held Until December 31
64
9
Call Option - Stock Spreads
10
11
Results of Call Option - Stock Spreads If Sold
Immediately
Results of Call Option - Stock Spreads if Held
Until December 31
67-68
69
5
LIST OF FIGURES AND TABLES (Cont.)
Page
Table Number
12
Put Option - Stock Spreads
13
Results of Put Option - Stock Spreads If Sold
Immediately
75
Results of Put Option - Stock Spreads If Held Until
December 31
76
14
72-73
78-79
15
Put-Call Spreads
16
Results of Put-Call Spreads If Sold Immediately
80
17
Results of Put-Call Spreads If Held Until December
81
1.0
INTRODUCTION
The Chicago Board Options Exchange (CBOE) began operations on
April 26, 1973.
Since that time options trading has been introduced on
the American Stock Exchange, the Pacific Stock Exchange, and the Midwest
Exchange.
Plans are currently being made to trade options on the New
York Stock Exchange.
Presently, these exchanges deal exclusively with
call options, however, trading in put options will begin in the near
future.
The volume of call options traded has soared since 1973.
Prior to
the opening of the CBOE, all options were individually negotiated and
offered very little liquidity.
The CBOE increased the liquidity of call
options by standardizing the terms of the contract, severing the link
between the buyer and seller by substituting the Options Clearing Corporation as the primary obligor in all CBOE options, introducing a secondary
market for the options, and eliminating adjustments for cash payments of
dividends.
The new options exchanges have followed the example of the
CBOE and provide an active secondary market in options.
Because options trading has become so popular it is important to
know what the tax effects of various transactions are.
This will affect
the use of options as a tax planning tool, and will affect the profitability of various types of transactions.
This thesis analyzes a set of option trading techniques which postpone the recognition of income from one year to the next.
This process
may be repeated in consecutive years so that tax payment is deferred
indefinitely.
The funds, which would have been used to pay taxes, may be
reinvestigated to earn additional income.
The process may be terminated
in any year in which the investor wishes to recognize the income.
For
example, if substantial losses are incurred in a particular year it may
be appropriate to recognize the income which has been deferred.
When
the tax is finally paid, it will be paid in a year of lower income so
the average tax rate will be lower than in prior years for the investor.
There are risks involved in these tax strategies.
First, there are
commission costs which the investor must pay to establish the position.
Before entering a position it must be determined that the tax savings
will exceed the cost of the transaction with reasonable certainty.
Second, there is always the possibility of capital losses when investing
in capital markets.
The objective of this paper is to provide a framework
in which to choose the transaction which provides the maximum tax benefit
for a given level of risk, and to illustrate the potential outcomes of
the transaction.
Once the investor has the alternatives and potential
consequences he is in a better position to make an intelligent choice as
to whether or not to utilize these tax deferral techniques.
The strategies presented in this paper make use of puts, calls, and
the underlying stock.
The strategies assume efficient stock markets.
That is, the stocks are properly priced and the investor has no way of
knowing in which direction the prices will move.
assumed to be less efficient.
The option markets are
This leads to option prices which differ
from the values predicted by the formula.
Although several different
trading techniques are presented, which assume efficient markets, they all
are based on the same concept.
A neutral spread is established.
When the
prices of the securities change, the net effect on the position will be
zero.
Gains on one side of the transaction merely offset losses on the
other side of the transaction.
The key to this set of techniques is that
the losses are recognized in the current year, and the gains are recognized
in the following year.
This provides a capital loss in the current year
and a capital gain of the same amount in the next year.
The first three chapters of the thesis are devoted to background
information. Section 2 presents a series of definitions which are commonly
used in the field of options.
It then elaborates on the definitions,
attempting to present an overview of the major relationships between stocks
and options which are used in formulating the trading strategies.
section discusses some of the tax consequences of options trading.
The next
The
information presented in these two sections establishes the foundation for
the understanding of the strategies which are presented in Section 4.
_
_
2.0
2.1
_
· ___ __
_____
BACKGROUND
Definitions
An option is a contract which gives its owner the right to buy or sell
a stock.
Since the introduction of auction markets for options, such as
the CBOE, the characteristics of options have become standard.
The fol-
lowing list of definitions refers to options which are traded on these new
exchanges.
Call option-A call option is a contract which gives its owner the right to
buy a specified number of shares of a particular stock within a stated
period of time.
Currently the CBOE, American Stock Exchange (AMEX),
Philadelphia Exchange (PHLX), Midwest Exchange (MWE), and Pacific
Exchange (PSE) list options which may be traded.
Put option-A put option is a contract which gives its owner the right to
sell a specified number of shares of a particular stock at a specified
price within a stated period of time.
Although put options are not
currently traded on any of the four option exchanges, they will be
introduced early in 1977.
Contract Size-The standard size of an option contract is 100 shares.
Expiration Date-The expiration date is the date by which an option must be
exercised.
If it is not exercised by the expiration date it becomes
worthless.
There are two sets of expiration dates which are commonly
used.
The first set has expiration dates of the third Saturday of
January, April, July and October.
The second set has expiration dates
of the third Saturday of February, May, August and November.
are originally listed with nine month maturities.
duration of the options decreases.
Options
As time passes, the
After three months another set of
options with nine months duration is listed.
There would now be two
sets of options, one with six months left until expiration and one
with nine months left until expiration.
Exercise Price-The exercise price of an option is the price for which the
option owner has the right to call or put the stock.
This price is
often referred to as the strike price or striking price.
For stocks
with market prices less than $50 at the time the options are listed
the strike prices are in increments of $5. For stocks with market
prices between $50 and $200 per share the strike prices are in increments of $10.
And for stocks with market prices in excess of $200 per
share the strike prices are in increments of $20.
As the market price
of the stock increases above the highest existing option exercise price,
a new option is introduced.
Premium-The premium is the total price of the option.
It is higher for
options with long durations than for options which are about to
expire.
Assume that the current price of the stock is less than the
exercise price of the option.
If there is a long time left before the
stock option expires, the option will have some value because the stock
may go up.
If the stock is volatile, the option premium will be higher.
As the expiration date approaches, it becomes less likely that the option
will be exercised since the stock is lower than the exercise price.
sequently the value of the option declines and approaches zero.
Con-
If the
stock price is higher than the exercise price, the premium is equal to
the difference between the stock price and the exercise plus an additional amount to account for the time remaining before expiration.
As the expiration date approaches, the premium approaches the difference
between the stock price and the exercise price.
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An out of the money
put option exists when the exercise price is lower than the stock
price.
In the money option-An in the money option exists when the exercise price
for a call is lower than the stock price or when the exercise price
for a put is higher than the stock price.
Covered option-A covered option exists when the seller has 100 shares of
the stock for each contract.
Naked option-A seller is said to be writing options naked when he sells
options for which he has no: corresponding stock.
Spreads-Spreads are combination of options of the same basic type but with
different exercise prices or expiration dates.
and the other is sold.
One option is purchased
A vertical spread is a pair of options on the
same stock with different exercise prices.
For example, the investor
might but a Digital Equipment January call with a strike price of $50
and sell a January call with a strike price of $60.
This would be a
bearish vertical spread; if the stock price goes down, the investor
makes money.
spread.
If the positions were reversed, it would be a bullish
A calendar spread is constructed by purchasing an option
with one expiration date and selling another option with an identical
exercise price but a different expiration date.
Straddle-A straddle is a combination of a put and a call, both exercisable
at the same market price and for the same period.
Strip-A strip is a straddle with a second put component.
Strap-A strap is a straddle with a second call component.
2
2
Basic
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The Black-Scholes option pricing formula is a mathematical model
which estimates the value of an option based on the volatility of the
underlying stock, the strike price of the option, the market price of the
underlying stock, the duration of the option, and the interest rate.
The value which the formula predicts will often differ from the market
value of the option.
If we assume that the basic formula is correct,
these differences may be explained by errors in the market or errors in
the inputs to the model.
This paper will not deal with the derivation of the model.
the model will be treated as both given and valid;
vided and option values are predicted.
Rather,
the inputs are pro-
A detailed discussions of the
] [9 ]
model is provided in several papers published by Black and Scholes.[1 '
In this paper it will be assumed that the values predicted by the
model are correct and that the market will eventually adjust prices to
match values.
Under this assumption it is possible to identify options
which are underpriced and options which are overpriced.
One of the criteria
used in choosing the appropriate spread for the tax strategies will be the
difference between the values and prices of the options to be used in the
spread.
The formula may be used to demonstrate some of the basic relationships
in options pricing.
The next section provides an overview of the sensitivity
of the formula to its inputs and of the dependence of option values on the
input parameters.
International Business Machines (IBM) has been chosen
as the underlying stock to illustrate these relationships.
of call options which are traded.
IBM has two sets
One set has an exercise price of $260.
other set has an exercise price of $280.
The
The termination dates of the options
_
__
_
_
are in January, April, August and October.
___
_
At any time there will be three
options traded in each set; the maximum duration is nine months.
It is
assumed, for this set of examples, that there exist put options with the
same strike prices and termination dates as the call options.
2.2.1
Call Options
Figure 1 illustrates the value of the call options with a strike
price of $260.
In this diagram it is assumed that the interest rate is
6% and that the stock volatility (the annual standard deviation of the
return on the stock) is .17.
The interest rate is a reasonable approxima-
tion of the return on short term government notes.
The volatility estimate
was provided by Fischer Black.
The starting point for understanding the pricing of call options is
to look at their value at the time of expiration.
At expiration the options
only have value if the stock has a market price which exceeds the strike
price.
If the stock price is less than $260, an option to buy the stock
at $260 is worthless.
If the stock price is above $260, that option is
worth the difference between the stock price and the strike price.
For
example, if the stock were selling for $280 per share when the option
expired, an option to buy the stock at $260 per share would be worth
exactly $20.
When there remains time before the option expires, the option will
have some value even if the stock price is below the strike price.
There
is the possibility that the stock price will increase before the option
expires.
The value of the option increases with the duration of the option.
The lines labeled "t=3 months" and "t=6 months" indicate the effect of the
duration on the value of the option.
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Figure 2 illustrates that the interest rates does not substantially
affect the price of an option.
This diagram shows the value of a call
option under the assumptions that the interest rate is 2%, 6%, and 10%.
For stock prices in the vicinity of the strike price, a different in
interest rates of 400% (2% to 10%) results in an option value difference
of less than 50%.
This indicates that the value used for the interest
rate in the option pricing formula does not have to be extremely accurate.
Figure 3 illustrates the value of three month call options as a
function of stock price for two assumptions of stock volatility.
assumption is that the volatility is 0.50.
The first
For a stock price equal to the
strike price, this 200% increase in the volatility resulted in approximately
a 200% increase in the value of the option.
Clearly the stock volatility
estimate is more crucial in the option valuation than the interest rate.
This emphasizes the importance of accurately estimating the volatility of
the underlying stock.
Figure 4 illustrates the value of the delta for the IBM 260 call
options with three month duration as a function of the stock price.
[The
delta is the ratio of the change in the option value to the change in the
stock price, for very small changes in the stock price.]
For stock prices
far below the exercise price, a small change in the stock price will not
change the option value significantly.
than one in this range.
Therefore, the delta is much less
For stock prices far above the strike price, the
portion of the premium which is due to the duration of the option has almost
disappeared.
The major factor in the premium is the difference between the
stock price and the exercise price.
price approximately point for point.
in this range.
The option price follows the stock
Therefore, the delta approaches unity
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Figure 4 also li±ustrates tne value or tne six ana nine montn call
option deltas as a function of stock price.
For options which are deep
in-the-money, the longer durations results in higher deltas.
However, for
options which are out-of-the-money, the shorter durations result in higher
deltas.
Figure 5 illustrates the value of a vertical spread between the IBM
options with exercise prices of $260 and $280.
In this case it is assumed
that the options with the $260 exercise price are bought and the options
with a $280 exercise price are sold.
the investor.
Each of the spreads shown consists of two options with the
same expiration date.
to expire.
This results in a positive cost to
The limiting case occurs when the options are about
This is shown by the line labeled "t=0".
$260 both options are worthless.
For stock prices below
For stock prices between $260 and $280, the
option with the $260 strike price increases in value point for point with
the stock.
And for prices above $280 both options increase in value point
for point with the stock; therefore, the value of the spread remains fixed
at $20.
As the duration of the options increases above zero, the situation
becomes more complex.
the stock.
The two options no longer move point for point with
Nor do they move point for point with one another.
The relation-
ships are shown by the lines labeled "t=3 months", "t=6 months" and "t=9
months".
The duration has the same effect on the spreads as it did on the
delta of an individual option.
This is because the difference between the
values of the two options varies directly with their deltas.
2.2.2
Put Options
The value of put options depends on the input parameters in much the
same way as the value of call options.
This section illustrates that the
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effects of duration, interest rate and stock volatility are directly
analogous to their effects on calls.
Figure 6 illustrates the value of an IBM put with a strike price of
$260, for several different durations.
occurs at expiration.
As with calls, the limiting case
If the stock price is greater than $260 at the time
of expiration, the value of an option to sell the stock at $260 per share is
zero.
If the stock price is below $260, the value of this option increases
one point for each one point decrease in the stock.
This is shown by the
line labeled "t=0'.
As the duration of the option is increased, its value becomes greater.
This is because the option assumes a time premium in excess of the amount
by which the strike price exceeds the stock price.
If the stock price is
far less than the strike price, the relative importance of this time premium
diminishes.
The value of the put approaches the difference between the
strike and stock prices.
Figure 7 illustrates that the interest rate does not substantially
effect the value of the put.
This is the same conclusion that was drawn for
the effect of interest rate on the value of call options.
Figure 8 shows the relationship between the stock volatility and the
value of the put.
Just as with calls, the stock volatility has a major
impact on the put value.
Consequently, it is important to have an accurate
estimate of the stock volatility.
Figure 9 illustrates the value of the option delta as a function of the
stock price for put options.
It is important to note that the delta for a
put is negative; the value of the put increases when the stock price declines.
The magnitude of the delta has the same characteristics as the magnitude of
the call delta.
For stock prices far above the exercise price, a change in
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the stock price will have very little effect on the value of the put.
And,
for stock prices well below the strike price, the value of the option is
heavily dependent on changes in the stock price; the delta approaches unity.
Figure 9 also illustrates the values of deltas for options with longer
durations.
The deltas for put options have the same characteristics as the
deltas for call options.
For options which are deep in-the-money, the longer
durations yield higher deltas.
For options which are out of the money, the
shorter durations yield higher deltas.
And the effect of duration disappears
at the extremes; the delta approaches zero for options which are very far
out-of-the-money and unity for options which are very deep in-the-money.
Figure 10 illustrates the value of vertical put spreads.
It is assumed
that the put with a strike price of $280 is long and the put with a strike
price of $260 is short.
The
This leads to a positive value for the spread.
value of the spread is the mirror image of the call spread discussed previously.
The limiting case for the put spread occurs at expiration.
For stock
prices above $280, both options are worth zero so the spread is worthless.
For stock prices between $260 and $280, the options with a strike price of
$280 begin to assume a value, but the other options remain worthless.
The
value of the spread increases point for point with decreases in the price of
the stock.
And, when the stock price is below $260, both options increase
in value with decreases in the stock price; the value of the spread remains
constant at $20.
As the duration of the options increases, the situation becomes more
complex.
The two options no longer move point for point with decreases in
the stock price.
Nor do they move point for point with one another.
The
relationships are shown by the lines labeled "t=3 months", "t=6 months" and
"t=9 months".
Notice that the duration has the same effect on the spreads
as it did on the magnitude of the delta of an individual put.
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2.2.3
Straddles
28
Figure 11 illustrates the value of straddles consisting on an IBM
call and an IBM put, each having a strike price of $260.
This is shown by the line labeled
is when both options are about to expire.
"t=0".
The simplest case
When the stock price equals the strike price, both the put and call
are worthless.
Below the strike price the put increases in value point for
point with decreases in the stock price, and the call remains worthless.
Above the exercise price the call increases in value point for point with
the stock price and the put remains worthless.
As the duration of the options increases, the v-shape of the curve
flattens out.
The value of the straddle increases near the strike price
faster than it increases either above or below the strike price.
This is
because both the put and call are increasing in value in the vicinity of
the strike price.
However, as the stock price differs from the strike price
only one of the options increases in value.
As the stock price moves farther from the strike price, in either
direction, the value of the straddle approaches its value at expiration.
However, the value of a straddle always varies directly with its duration.
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TAX TREATMENT OF OPTIONS TRANSACTIONS
The purchase and sale of options contracts are considered to be
capital transactions for income tax purposes, just as the purchase and
sale of common stock are capital transactions.
However, the tax con-
sequences of options trading are more complex than stock trading because
of the possibilities of exercise and expiration.
Capital gains or losses may be long or short term, depending on the
length of time that the asset was held.
For transactions which are com-
pleted during 1977, the holding period requirement for long term capital
gains or losses is nine months and one day.
is extended to one year and one day.
Beginning in 1978 this period
When computing personal income tax,
long and short term capital gains and losses may be used to offset one
another.
However, long term gains in excess of all losses are taxed at
half the rate of short term gains.
Similarly, long term losses provide
half the deduction of short term losses.
This gives the investor an
incentive to have long term gains rather than short term gains, and it
gives the investor an incentive to have short term losses rather than long
term losses.
Since options, which are traded on the exchanges, have a maximum
duration of only nine months, all simple options transactions will result
in short term gains or losses.
However, if the option is exercised, the
holding period is measured by the length of time that the stock is held.
This opens the possibility of achieving long term holdings.
3.1
Simple Purchase and Sale of Options
Assume that on January first the price of IBM stock is $270 per share.
An investor purchases a call option to buy 100 shares of IBM with a strike
price of $260 and expiration date in April.
The price of the option would
be approximately $1850.
If the stock price increases to $300 in the next
few days, the option would be worth about $4500.
Assuming that the investor
sells the option, he will realize a short term capital of $2650.
If the
stock price had declined to $250 per share, the option would have declined
to about $675.
If the investor sold the option at this price, he would
realize a short term capital loss of $1175.
would have exactly the opposite results.
The writer of these options
When the stock went up he would
experience a short term capital loss and when the stock went down he would
experience a short term capital gain.
If the investor had bought a put option to sell 100 shares of IBM with
a strike price of $260 and expiration in April, he would have paid about
If the stock price rose to $300, the option would decline to about
$500.
$30.
Sale of the option would result in a short term capital loss of $470.
If the stock price declined to $250 per share, the option would rise to $1400.
If the option were sold, a short term gain of $900 would be realized.
As
before, the writer of the option would experience exactly the opposite
results.
3.2
Purchasing Options and Exercising Them
When an option is exercised, the premium which was paid is used to adjust
the tax basis of the stock.
For example, suppose an investor paid $1850 for
a three month call on IBM with a strike price of $260 when the stock was
selling for $270 per share.
If, at the end of the three months, the stock
price was above the strike price, the investor may decide to exercise his
option.
If the stock price was below the strike price, the investor would
not exercise the option because he could purchase the stock on the open
market for less than the strike price.
When the call is exercised, the
investor purchases 100 shares of IBM for $26,000, which is less than the
current market price.
When computing the cost of this stock for tax
The
purposes, the $1850 premium is added, bringing the total to $27850.
holding period for the stock begins when the option is exercised.
No gain
or loss is recognized for the option.
If the investor had purchased a put and exercised it, the tax treatment
In this case the stock would
would be analogous to the treatment of a call.
have to be below the strike price in order for the put to be exercised.
The
holder of the option receives an amount equal to the strike price for each
share he delivers.
The tax basis for the individual delivering of the stock
is equal to the strike price less the premium.
The holding period is deter-
mined by the length of time that the stock was owned.
Again, no gain or loss
is recognized for the option.
3.3
Purchasing Options and Allowing Them to Expire
If an option expires, the holder may write off the entire premium as a
capital loss.
A call option will expire worthless if the stock price is
below the strike price at the termination date.
A put option will expire
worthless if the stock price is greater than the strike price at the
termination date.
3.4
Writing Options and Having Them Exercised
The writer of an option receives a premium.
This premium is held in a
"deferred suspense account" for tax purposes until the entire transaction
is completed.
No tax effects are recognized at the time the option is
written.
The writer of a call optionmust deliver the stock at the strike price if
the option is exercised.
The tax basis for the sake of the stock is adjusted
by the amount which the investor receives for the option.
For example,
the writer of the three month call on IBM received an $1850 premium in a
previous example.
When the option is exercised, this investor must deliver
the stock at a total price of $26,000 for the 100 shares, even though the
stock may be selling for a much higher price.
The option writer records
the sale price of the stock as the sum of the $26,000 received for the
stock plus the $1850 received for the option, a total of $27,850.
If the call writer had owned the underlying stock, he could deliver
from his inventory or he could purchase new shares on the open market and
deliver them.
If the calls had been written naked, the writer would be
forced to purchase shares on the market for delivery.
The holding period
for capital gain or loss on the entire transaction is determined by the
amount of time that the stock was held prior to exercise of the option.
The writer of a put option must purchase the stock at the strike price
if the option is exercised.
The cost basis for the stock is adjusted by the
premium received for the option.
For example, the writer of the three month
put on IBM in a previous example received a $500 premium.
When the put is
exercised, the writer must purchase 100 shares of IBM for $26,000, even if
the stock price is substantilly lower.
The option writer records his cost
basis as the $26,000 paid for the stock less the $500 received for the
option, a total of $25,500.
The put writer may have sold the stock short.
of the put uwuld serve to cover
his short position.
In this case the exercise
If the option writer did
not have a short position in the stock, he would simply be forced to buy the
stock from the option holder.
In the first case, a short term capital gain
or loss would be recognized at the time of exercise.
In the latter case, the
holding period of the transaction would depend on the time the stock was
owned before it was sold.
_
3.5
__
Writing Options and Having Them Expire
If an option expires, the writer records the entire premium as a
short term capital gain.
A call option will expire worthless if the stock
price is below the strike price at the termination date.
A put option will
expire worthless if the stock price is higher than the strike price at the
termination date.
The writer's position in the underlying stock has no
effect on this transaction.
3.6
Straddles
The tax code defines a straddle as a simultaneous combination of an
option to buy and an option to sell the same quantity of a security at the
same price during the same period of time.
The tax treatment for the
purchaser of a straddle is the same as if he had purchased a put and call
separately.
The writer, however, has two alternatives for allocating the
premium received between the put and the call for tax purposes.
The premium
may be allocated based on the respective market values of the two parts of
the straddle.
(Rev. Rul. 65-31, 1965-1 CB 365)
Or, the writer may allocate
55% of the total premium to the call and 45% of the premium to the put.
The latter alternative is only permitted if the writer uses it for all
straddles that are written.
(Rev. Rul. 65-29, 1965-2 CB 1023)
Usually the writer of a straddle will only have to fulfill one side of
the straddle contract.
If either or both sides of the contract are exercised,
the writer must pay taxes on the premium allocated to the exercised part as
if it were a put or call not sold as part of a straddle.
The unexercised
part is treated as a short term capital gain at expiration.
If the straddle
expires wholly unexercised, the entire premium is treated as a short term
capital gain.
To illustrate the taxation of straddles, consider the investor who
writes the put and call used in previous examples.
the call and $500 for the put.
joint premium.
He receives $1850 for
He has two alternatives for allocating the
He can allocate the premiums based on their market values,
or he can allocate $1292.50 to the call and $1057.50 to the put.
writer faces four possible outcomes from this transaction.
sibility is that the call is exercised and the put expires.
The
The first posThis would
occur if the price of the underlying stock was about $260 at expiration.
If the percentage allocation approach is chosen, the $1292.50 would be
added to the exercise price received for the stock and is taxed as if the
call had been written independently of the put.
The $1057.50 allocated to
the put is taxed as a short term capital gain.
The second alternative is that the price of the underlying stock is
below $260 at expiration.
In this case the premium allocated call is
realized as a short term capital gain, and the premium allocated to the
put is subtracted from the price paid for the stock.
The put side of the
contract is taxed as if it were written independently of the call.
If the put and call are both exercised, each option is treated as if
it were written separately.
This could happen if one of the options was
prematurely exercised and then the stock price changed allowing profitable
exercise of the other option.
The final possibility is that neither side of the straddle is exercised.
This would occur if the stock price exactly equalled the strike price
at the time of expiration and in this case, the entire premium, $2350, would
be taxed as a short term capital gain.
36
Previously, it was necessary for the writer to identify which parts
of the transaction comprised the straddle and which are independent options.
This is no longer true for straddle strips or straps.
This was necessary
to determine which parts of the combined option were qualified to receive
capital gains treatment upon expiration, and which would be taxed as
ordinary income.
This is no longer necessary because all unexercised
options are now treated as short term capital gains.
4.U
CI·~~ ~Cllllr~~h-rCI~
TAX STbKA'TEGILES
The intent of the following tax strageties is to defer the recognition of capital gains from one year to the next for income tax reporting
purposes.
It is assumed that the capital markets are efficient, and that
the transactions which comprise the strategy are short term capital events.
The objective is to establish a loss in the current year and recognize the
offsetting gain in the following year.
The losses may be used to offset
capital gains which were recognized during the current year for tax
reporting.
The following sections discuss three basic strategies.
The first
strategy involves setting up neutral spreads between options of the same
kind.
That is, the investor would buy one set of calls and sell another
set.
Or, the investor would buy one set of puts and sell another set of
puts.
The second strategy involves spreads between options and the underlying stock.
Here, the investor would establish a position in the stock
and take a position in the option which would neutralize the net holdings.
For example, the investor might purchase stock.
To neutralize this posi-
tion he could either write calls or purchase puts.
Alternatively, if the
investor sold the stock, the situation could be neutralized by buying
calls or writing puts.
It is important to note that the ratio of stock
to options will not necessarily be one in order to have a neutral spread.
This would only be true if the delta of the option were unity.
Since this
is not usually the case, a neutral spread will be composed of more options
than shares of stock.
The exact ratio is determined by the option's delta.
The third strategy consists of spreads between puts and calls.
A
neutral spread is established by either buying puts and calls or selling
puts and calls.
The ratio of puts to calls is determined by the deltas
of the options which comprise the spread.
The neutrality of a spread is guaranteed by setting the delta of
the spread equal to zero.
However, the delta of an option changes with
changes in the price of the underlying stock.
As the stock price varies,
the ratio of the number of options on each side of the neutral spread
changes.
This introduces a risk into the transaction.
measured by the gamma of the spread.
The risk may be
This is the change in the spread's
delta divided by the change in the underlying stock price for small moves
in the stock in the short run.
A gamma of zero would mean that the
spread's delta was independent of the stock price.
Therefore, the spread
would remain neutral (delta equal to zero) no matter what happened to the
stock price.
Gamma is an adequate measure of risk if the comparisons are being
made among spreads on a single stock.
However, if it is necessary to
compare a spread on IBM with a spread on Digital Equipment Corporation,
it is necessary to take into account the variability of the stock because
it affects the variability in the delta of the spread.
This is accom-
plished by constructing the dollar curvature of the spread.
The dollar
curvature is equal to the gamma times the stock price squared times the
stock volatility squared.
It is a measure of the frequency with which
the spread must be adjusted in order to keep it neutral.
curvature may be positive or negative.
The dollar
In either case, lower magnitudes
of the dollar curvature imply lesser adjustments to maintain neutrality
and, therefore, less risk in the spread.
38
The actual number of option contracts or shares of stock which are
used in a spread is determined by the delta of the spread being confined
to zero and the magnitude of the dollar curvature being confined to unity.
The former constraint is required in order to have a neutral spread.
latter constraint is arbitrary.
The
As long as all spreads have the same
dollar curvature, they will have equal risk.
The value of unity was
chosen because of its computational convenience.
The algebra used to
derive the actual numbers is quite simple, and will be provided in the
discussions of the individual spread strategies.
The decision of which options to buy and sell is determined by the
excess value of the spread.
The excess value is the difference between
the market price of the spread and the estimated value of the spread.
All spreads are defined so that they have a positive excess value.
The
sign of the excess value may be reversed by reversing the buy and sell
decisions on the options which comprise the spread.
For example, if
option A has a price which is less than its value and option B has a
price which is greater than its value, it would be reasonable to purchase
option A and sell option B. This spread would have a positive excess
value.
If the buy and sell decisions are reversed, the excess value
becomes negative.
The commissions charged by a brokerage house are part of the costs
of establishing the spreads.
As an approximation, these are calculated
using costs of $0.375 per share of stock and $8.50 per option contract.
These are fairly representative of actual costs.
The cash outlay for a spread depends on the margin requirements as
well as the particular options involved.
fairly complex.
The margin requirements are
The purchase or short sale of stock requires margin of
50% of the market price of the stock.
the full option price as margin.
consider on the sale of an option.
of 30% of the stock price.
The purchase of an option requires
There are several different cases to
Naked options have a margin requirement
This is reduced by the funds received on sale
of the option and by the difference between the stock price and exercise
price of the option.
For in-the-money options the absolute value of the
difference between stock price and strike price is added to the margin
required.
For out-of-the money options, this amount is subtracted from
the margin requirement.
Covered options do not have the same margin requirements as naked
options.
If the short option is covered by stock, the margin is equal to
the difference between the stock and exercise prices.
investor has purchased stock.
In this case, the
Part of the outlay for the stock is offset
by the income from the option premium.
If the option is in the money, the
difference between the stock price and the strike price is added to the
funds required.
be negative.
If the option is out of the money, this difference will
It then serves to reduce the margin requirement.
If the short option is covered by another option with a shorter
duration, the short option is considered to be naked.
If, however, the
long option has a longer duration than the short option, the margin requirements are similar to the normal covered option case.
The investor has an
outlay for the long option and receives income from the premium of the
short option.
This is adjusted by the difference between the strike prices
of the two options.
For calls, the long option's strike price less the
short option's strike price is added to the cash required.
Therefore, if
the short option has a higher strike price than the long option, the difference will be negative, and will reduce the cash required.
The case is
reversed for put spreads.
The difference between the short option's
strike price and the long option's strike price is added to the cash
required.
In this case, if the short option has a higher strike price
than the long option, the cash required will be increased.
Spreads between puts and calls are treated as if they were independent
transactions.
If the options were purchased, the cash required would equal
the sum of the premiums.
If the options were written, they would be
treated as if they were naked.
The option pricing formula includes interest income on the value of
the option.
This means that if two options, or two spreads, have the same
risk, the difference in their values ought not to be a consideration in
choosing between them.
The investor will often have to leave margin money with the broker.
Frequently, the broker declines to pay interest on this money.
This lost
interest is an opportunity cost of leaving the money with the broker.
For example, if an investor sells stock short for $1000 and has to put up
$500 margin, interest is being lost on $1500.
Interest is being lost on
the proceeds of the sale and on the cash being left with the broker.
It
does appear, however, that the importance of this lost income is decreasing,
because brokers are beginning to pay interest on account balances in excess
of $2000.
The dollar volatility of an option is a measure of how much the option
position is expected to change.
It is equal to the number of options times
the delta times the stock price times the stock volatility.
For a neutral
spread, the dollar volatilities of the two sides will be equal.
they predict price movements in different directions.
However,
In order to achieve
the greatest impact of the spreads for the purposes of the following
strategies, one ought to choose the spreads which consist of option positions with the greatest dollar volatility.
4.1
Option Spread Strategies
The establishment of the spreads is based on two conditions.
the spread's delta is zero.
is unity.
First,
And second, the dollar curvature of the spread
These two conditions defined the size of the position on each
side of the spreads.
Let:
Dl= delta of one option 1
D2= delta of option 2
Gl= gamma of option 1
G2= gamma of option 2
P= stock price
V= stock volatility
N1= number of contracts of option 1
N2= number of contracts of option 2
DC- dollar curvature
The variable P is determined from market data.
Dl, D2, Gl, and G2 are
determined by the Black-Scholes pricing formula.
The volatility, V, is
determined by historical price movements of the stock.
The dollar curva-
ture is a function of the other variables and will be set equal to unity.
Thus, Nl and N2 are to be determined.
Option 1 has a delta equal to Dl.
If the position consists of Nl
options, the delta of the entire position is N1 times Dl.
Similarly, a
position consisting of N2 options, having a delta of D2, would have an
aggregate delta of N2 times D2.
If a spread were to be formed between
these two positions, the spread's delta would be:
Delta=Nl*Dl-N2*D2
In order for the spread to be neutral, the spread's delta must be zero.
This leads to the following relationship between N1 and N2:
D2
N1= D
D1N2
Using the same line of reasoning, the gamma of the spread, GS, is
equal to Nl*Gl-N2*G2.
Recall that the dollar curvature is defined to be
the spread's gamma times the stock price squared times the volatility
squared:
DC=GS*P2*V2=1
After a little algebraic manipulation, this information leads to
concise expressions for N1 and N2:
D2
N1=
(Gl*D2+G2*D1)*P V 2
D1
N2=
(Gl*D2+G2*Dl)*p2 2
4.1.1
Call Option Spreads
IBM stock has six call options associated with it.
Table I lists
the values and deltas of these options for several different strike prices.
There are several assumptions underlying this presentation.
volatility is assumed to be 0.17.
The stock
The interest rate is assumed to be 6%.
And the date is assumed to be December 5, 1977.
On that date there will
be options outstanding with termination dates in January, April and July
1978.
Table I also lists the values and deltas of options on Digital Equipment Corporation (DEC).
Although DEC has more than fifteen option
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This is
a sufficient number of options to illustrate the use of the strategies.
Table II illustrates the potential spreads which may be formed among
the IBM call options and the spreads which may be formed among the DEC
call options.
Since each stock is assumed to have six options, there will
be a total of fifteen possible spreads for,.each stock.
The spreads are
formed on December 5, 1977.
All of the spreads shown in Table II have dollar curvature equal to
unity.
All of the spreads are also neutral.
The sign of the dollar cur-
vature depends on which option is bought and which is sold.
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price of the spread and the model price of the spread.
which option is bought and which is sold.
excess value always being positive.
It depends on
The spreads are defined by the
The excess value of the spread is
merely the excess value of the long position minus the excess value of
the short position.
$58.71.
The first spread in Table II has an excess value of
This means that the cost of the spread is less than the value of
the spread.
If the buy and sell decisions were reversed, the spread would
have a negative excess value.
The rightmost column is the dollar volatility of each side of the
spread.
The dollar volatility is equal to the number of options in the
position times the option's delta times the stock price times the stock
volatility.
It provides a measure of how much price variation can be
expected in the position.
same dollar volatility.
Both sides of a neutral spread must have the
The price changes on each side offset one another.
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The objective of this strategy is to establish the largest possible
loss in the current year and offset it with a gain in the following year.
This is accomplished by choosing an options position with the greatest
dollar volatility.
tuation.
This position has the greatest expected price fluc-
This does not imply additional risk because all of the spreads
are constructed to have the same dollar curvature, and, therefore, the
same risk.
High dollar volatility simply means that both sides of the
spread have large price variations.
However, the price changes always
neutralize one another.
The column labeled "commission" lists the transaction costs required
to establish the spread.
The investor must have a reasonable probability
of obtaining a tax advantage which exceeds this cost.
The column labeled "cash" is the sum of the margin requirement and
the commission cost.
to enter the position.
This is the amount of money which would be required
The investor could either pay this amount in cash
or leave stock or treasury bill with the broker as security.
The column labeled "lost int" is the amount of interest income which
is foregone if the spread is held for one year.
This figure is determined
by applying the 6% interest rate to the difference between the margin held
by the broker and the price of the spread.
For example, if the spread
costs $800 and the broker a total outlay of $1000, the lost interest is
based on a pricipal amount of $200.
However, if the investor sells the
spread, in the sense that he ought to receive $800, and the broker requires
$200 margin, the lost interest would be based on a priciple amount of $1000.
Table III illustrates the possible outcomes of an investment in one
of the spreads shown in Table II.
This Table is included to show the
results predicted by the model if the investment could be sold immediately
for the value specified by the model.
Since the pricing formula predicts
option values which do not equal the market prices, the model predicts
profits or losses even if the stock price does not move.
This example examines the behavior of a spread formed between two
sets of IBM options.
The spread is established on December 5, 1977.
The stock price is assumed to be $280 per share.
It consists of the
purchase of 2.059 contracts of the April 260 option and the sale of 1.888
contracts of the January 260 option.
The value of the April 260 option
is $28.19, according to the model, however, the market price is assumed
to be $29.00.
This assumption is arbitrary.
It has been made for the
purpose of demonstrating the use of the models.
260 option is $22.73.
Its price is $24.00.
The value of the January
Consequently, both options
are overpriced by the market.
Table III contains a variety of information related to the profitability of the spread.
The rightmost column lists the assumed price of
the stock at the time the spread is ended.
The remaining columns list
the results of the spread for each of the assumed prices.
This gives the
investor the opportunity to see how much of a tax gain he will have, and
how much his expenses can be.
The second and third columns state the configuration of the spread
being examined.
The number of option contracts being bought and their
expiration date are listed in the second column.
The number of option
contracts being sold and their expiration date are listed in the third
column.
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The fourth column lists the commission costs for the entire transThis includes buying and selling costs for the options shown in
action.
the previous column.
Since the commissions are assumed to depend only
on the number of options in the position, the costs are constant.
The next column lists the changes in the predicted value of the
long position.
This is determined by multiplying the excess value of
the option times the number of options in the position.
If the options
were correctly priced, the change ought to be zero when the stock price
is assumed to be $280, in this example.
However, the option was overpriced.
If it is assumed that the option were to suddenly become correctly priced,
the price of the long position would decline $189.77.
The sixth column lists the changes in the predicted value of the
short position.
case.
This is determined in the same manner as the previous
In this example, the price of the short option would decline
$248.50.
The column labeled "profit" lists the net change in the spread's
price minus the commission costs.
This is an indication of how much the
investor can expect to show as a capital gain or loss on the position if
the entire position were liquidated.
However, the strategy is not to
liquidate the entire position in the current year.
spread is to be liquidated.
Only one side of the
The fifth and sixth columns indicate which
side should be liquidated and what potential deductions can be achieved.
If the stock price remained at $280, the model predicts that the
long position would decline in value by $189.77.
loss on this side of the spread.
This would result in a
The model also predicts that the short
side of the spread would decline $248.50.
This would result in a gain
equal to the excess value listed in Table II.
The strategy, in this case,
would be to sell the long position and recognize the loss in the current
year.
At this point the investor would have two alternatives.
He could
either re-establish a neutral hedge of allow the short position to remain
uncovered.
In either case, the margin requirements would change.
The
last two columns in Table III indicate the deltas of each side of the
original spread.
This figure is arrived at by multiplying the number of
options times its delta.
For the example being considered, the investor
would have 1.888 January 260 contracts naked.
For each one point move-
ment in the stock, the value of the option position could be expected
to vary $172.75.
This information should be helpful in deciding whether
or not to accept the market risk until the second leg of the spread is
lifted.
Table III provides insight into the differences between market conditions and predicted conditions. It demonstrates that as the price of
the underlying stock varies, the potential of the strategy increases
significantly because both sides of the spread change in price.
For stock
prices below the initial price, losses will occur on the long side of the
spread.
For stock prices above the initial price, the losses will occur
on the short side of the spread.
In either case, the losses are offset
by gains on the other side.
The spread remains fairly neutral over a wide range of stock prices.
This can be seen by observing that the position deltas offset one another
over the range of stock prices listed in the table.
$280, the deltas are essentially equal.
struction of the spread.
At a stock price of
This was specified in the con-
As the stock price is varied, both the long and
short deltas move.
However, they move more or less in tandem.
If the
stock price were to increase to $290, the deltas would not be equal.
The long delta would be $187.16 and the short delta would be $183.94.
This means that for small stock price movements around $290, each one
point change in the stock would have only a $3.22 effect on the value
of the entire spread.
The preceding discussion was intended to illustrate the differences
between the market conditions and forecast conditions at the time the
spread is established.
It is unlikely that the stock price would vary
significantly enough during the initial day to warrant selling one side
of the spread.
In addition to the lack of price movement, there is a
second reason for delaying the closing of one side of the spread.
It is
preferable to leave the remaining side naked to avoid paying commissions
for the options required to form the new neutral spread.
If the action
can be delayed to the last day of the year, the risk can be reduced.
The
loss can be established on the last day of the year and gain the recognized
on the first trading day of the new year.
Table IV presents the same analysis as Table III.
However, in this
case it is assumed that the position was held from December 5, 1977 until
December 31, 1977.
Because both of the options have come closer to expira-
tion, their values and deltas have changed.
The spread is no longer neutral.
If the stock price was $280 on December 31, the same price as on December 5,
the spread would no longer be neutral.
This can be seen by observing that
the deltas of the positions are no longer equal.
The long position has a
delta of 175.55 and the short position has a delta of 183.65.
stock price rises, the spread becomes more neutral.
As the
The spread is
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essentially neutral when the stock price is $288.
Above that price, the
The spread is still fairly neutral over
spread is no longer neutral.
the entire range of stock prices shown in this table.
The use of this strategy may be illustrated using Table IV.
Assume
that on December 5, 1977 an investor is considering entering into the
spread shown in the table.
At that time he has no idea what the stock
price will be at the end of the year.
The investor does believe that
the option prices will adjust to the values predicted by the model, or
at least come closer to these values, as the expiration dates approach.
Table IV provides all of the information this investor requires to make
an intelligent decision whether or not to enter the spread.
If the stock
price remains unchanged, the investor will have a deduction of $520.32.
His remaining option will move $183.65 for every one point change in the
price of the underlying stock.
It the price declines to $270, the inves-
tor will have a $2148.92 deduction and the remaining option position will
have a price change of about $160.22 for every one point change in the
underlying stock.
In this case the investor would have invested $1473.64
to achieve a $2148.72 deduction in the current year, and he would have
achieved a $471.56 capital gain after commissions.
4.1.2
Put Option Spreads
Puts are not currently traded on any of the organized option ex-
changes, however, they will be actively traded in the near future.
When
these options are traded, they will offer the same tax opportunities as
the calls which were just discussed.
The tax strategy for using spreads between puts is exactly analogous
to the strategy involving call options.
deltas of hypothetical put options.
Table V shows the values and
It has been assumed that the puts
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will have the same strike prices as the calls which are currently traded.
This table includes values for options on IBM and DEC.
IBM and DEC.
Table VI is directly analogous to Table II.
It lists all possible
neutral spreads on the two stocks and provides information concerning
the excess value, commission costs, margin requirements, lost interest
and dollar volatility.
Market prices for the puts have been arbitrarily
chosen to illustrate the approach used to apply this strategy.
These
prices are shown at the top of Table VI.
The spreads are ranked according to the dollar volatility of each
side.
Recall that the spreads are neutral.
tility of the entire spread is zero.
Therefore, the dollar vola-
The dollar volatility shown in this
table applies to each side of the spread.
It is an indication of how much
one can expect the price of one side of the spread to fluctuate during a
one year period.
Naturally, the spread will not be held for more than a
few weeks, so this is merely a convenient method of ranking the spreads.
The examples listed in Table VI are based on the assumptions that the
price of IBM is $280 a share and the price of DEC is $50 per share.
The
spreads are initiated on December 5, 1977.
Table VII shows the predicted results of a spread if it could be sold
immediately for the value predicted by the model.
The spread consists of
a long position in 2.097 contracts of IBM January 280 puts and a short
position in 5.697 IBM 260 puts.
Referring to the line corresponding to a
stock price of $280, it can be seen that the long position was overpriced
by $49.60 and the short position was overpriced by $262.50.
If the option
prices are adjusted to the values, the result would be a potential tax
deduction of $49.60 in the current year.
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As in Table III, it can be seen that the position remains fairly
neutral over a wide range of stock prices; the delta of the long position is approximately equal to the delta of the short position.
It can
also be seen that the potential deductions grow as the stock price varies.
However, the side of the spread which will be used for the deduction is
reversed from the previous case.
Here, for increases in stock price, the
deduction is obtained by selling the long position rather than repurchasing
the short position.
For decreases in the stock price, the deduction is
obtained by repurchasing the short position.
Table VIII shows the results of the put spread on December 31, 1977.
Because both options have come closer to expiration, their values and deltas
have changed.
The spread is no longer neutral.
It is assumed that the
prices have adjusted to equal the values.
The
The investor has used $36077.45 as margin to enter this spread.
cash outlay is high because of the large number of naked April 260 options.
Each naked option contract requires approximately $8400 in margin.
If the
stock price declines to $270, the investor will have the opportunity to
deduct $579.33 for tax purposes.
ically have a $203.26 profit.
In addition, the investor would theoret-
If the stock price increased to $290, a
$1033.43 deduction would result.
The decision of whether or not to establish the neutral spread is
supported by the information provided about the delta of the remaining side
of the spread.
For the latter case, when the stock price increases to $290,
the long option would be sold to recognize a loss.
position would have a delta of $43.29.
The remaining short
Each one point change in the stock
price would cause a $43.29 change in the value of the short position.
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4.2 Spreads Between Stock and Options.
The same line of reasoning which was applied to spreads betwen options
can be applied to spreads between stock and options.
A neutral spread is
constructed by setting the delta of the stock position equal to the delta
of the option position.
The size of each side is determined by setting the
dollar curvature equal to unity.
The delta of a stock is identically equal to one.
stock is equal to zero.
The gamma of a
The other variables required to define the spreads
are:
D2= delta of the option
G2= gamma of the option
P= stock price
V= stock volatility
N1= number of shares of stock in the spread
N2= number of options in the spread
DC= dollar curvature=l
The delta of the spread becomes:
DS=N1-N2*D2
In order for the spread to be neutral, the spread's delta must be equal to
zero.
This leads to the following relationship between N1 and N2:
N1-D2*N2
Since the gamma of the stock is equal to zero, the spread's gamma
simply equals the gamma of the option position, N2*G2.
The dollar curvature
is equal to the gamma times the stock price squared times the volatility
squared, and is defined to be unity.
N1 and N2:
This leads to concise expressions for
N1=D2/G2*P2*V2
N2=1 /G2*P2*V 2
4.2.1
Spreads Between Call Options and Stock
Table IX lists the spreads which can be formed between IBM and its
call options, and the spreads which can be formed between DEC and its call
options.
The spreads are formed on December 5, 1977.
assumed to be $280 per share.
The price of IBM is
The price of DEC is assumed to be $50 per
share.
When forming these spreads, the investor can either buy the stock and
sell the calls, or sell the stock and buy the calls.
overpriced, they ought to be sold.
be bought.
If the options are
If they are underpriced, they ought to
The column labeled "excess" indicates how far the option prices
differ from their model values.
The column labeled "lost int" lists the amount of interest that would
be lost on money held by the broker.
Although this figure is not directly
relevant to the establishment of the spreads, it is of interest because it
specifies an opportunity cost for this strategy.
The rightmost column lists the dollar volatility of the spreads.
As
previously stated, this is a measure of the expected price fluctuation of
each side of the spread and is therefore an appropriate quantity by which
to rank the spreads.
Table X illustrates the possible outcomes of one of the spreads if
it were liquidated on December 5, 1977, the same day it was initiated.
has been assumed that the option prices adjusted to equal their values.
This table provides information about the differences between the market
conditions and the predicted conditions.
The results of this table are
straightforward and directly analogous to those of Tables III and VII.
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Table XI lists the results of the same spread if it is held until
December 31, 1977.
Assume that an investor bought 5.03 shares of IBM
stock on December 5, 1977 and simultaneously sold .062 July 260 call
contracts.
The cash outlay was $519.47.
The possible outcomes which
the investor would face on December 31, 1977 are listed in this table.
If the stock price declined to $270 per share, the investor would have a
deduction of $50.30, and a net capital gain of $6.99.
Alternatively, if
the stock price increased to $290, the investor would have a deduction of
$39.65 and a net gain of $6.35.
The investor could realize his losses in the current year in exactly
the same way he did in the previous strategy.
That is, the investor could
simply liquidate one side of the spread in the current year and delay
liquidation of the other side until the following year.
However, if the
spread shows a loss in the option side of the spread and a gain in the
stock side of the spread, the investor has another alternative for ending
the spread.
Payment dates for stock transactions are five business dates
following the purchase or sale.
actions are the following day.
However, payment dates for option transConsequently, if the investor closes both
the option and stock positions on the second to last trading day of the
year, the losses on the options would be recognized in the current year and
the gains on the stock would be recognized in the following year.
This
eliminates the market risk associated with holding an uncovered position
into the new year.
4.2.2
Spreads Between Put Options and Stock
When constructing spreads between stock and put options, the inves-
tor will either buy both the stock and options or sell both.
Table XII
lists the put option-stock spreads which can be formed for IBM and DEC.
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The spreads are formed on December 5, 1977.
The assumptions used in this
example are identical to the assumptions in the previous examples.
Tables XIII and XIV list the possible outcomes for a particular
spread.
Table XIII is based on closing the position on December 5. It
assumes that the option prices adjust to their values immediately.
XIV is based on holding the spread until December 31.
Table
If the price of
the stock declines, the value of the puts will increase.
Since this
spread has a short position in the puts, losses will occur on the put
side and gains will occur on the stock side.
If the stock price rises,
the situation is reversed.
The decision to sell the puts was made because they were overpriced
when the spread was established.
However, by setting up the spread in
this manner, the result will be a capital loss on the spread for all of
the stock prices shown.
If the stock and puts had been purchased, the
excess value of the spread would have been negative and the profits on
the spread would have been even more negative than shown in the table.
4.3
Spreads Between Puts and Calls
Spreads can be formed between puts and calls on a stock.
The spread
is constructed by either purchasing both sets of options or writing both
sets.
A straddle is a spread consisting of one put and one call, each of
which has the same strike price and expiration date.
spreads can be formed between the options.
It will soon have six put options.
However, many other
IBM has six call options.
It is therefore possible to form a
total of thirty-six unique spreads.
The spreads formed in this strategy are neutral and have a dollar curvature equal to unity.
The number of options on each side of the spread
is determined by the same equations that were used in the first strategy.
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All of the possible spreads are listed in Table XV.
This table contains
the same information that was provided in Table II, VI, IX, and XII.
Tables XVI and XVII list the possible outcomes of a spread if the
positions are closed on December 5 and December 31, respectively.
The
analysis of these tables is exactly the same as their counterparts in
previous strategies.
4.4
Comparison of the Strategies
The examples provided for each of the three strategies have all
used the same stocks, the same time interval, and the same risk (dollar
curvature).
defer income.
Each of the strategies has the same objective, that is, to
Consequently, the examples provide a convenient medium for
comparing the effectiveness of the strategies.
On December 5, 1977, the investor is faced with the problem of
establishing a spread to defer income.
He realizes that there are a
total of 78 possible spreads that he can construct between the calls,
puts and stock if he restricted himself to IBM and DEC.
The best spread
for any situation will depend on potential tax savings, the investor's
risk aversion, and potential capital gains.
In the series of examples
which have been presented in this thesis, it is clear that the best
spread is between the IBM April 260 and January 260 call options.
This
spread provides the highest potential tax deduction for the risk.
How-
ever, a different spread might be chosen if the investor has a cash
constraint.
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5.0
APPENDIX
A variety of computer programs have been written to support the
analyses performed in this study.
interactive.
These programs were designed to be
The programs prompt the user for any data which are required.
The programs have been loaded in two modules due to the core limitations of the computer being used.
The data file is named DASTA.
These modules are named *MAl and *MA2.
All of the subprogram codes are presented
at the end of this paper.
The program *MAl provides analyses of the call spreads, and the call
option-stock spreads.
After date and interest rate information is input,
the program asks for a subroutine request.
analysis is desired.
This informs the program which
The following is a list of available subroutines for
use in *MAl:
Subroutine
1
terminates program execution
2
supplies the call values and deltas for a single stock
price
3
supplies the call values and delta for a range of stock
prices
4
presents an analysis of all possible spreads between call
options on a particular stock
5
presents an analysis of all spreads between the stock and
the call options.
The program *MA2 provides analyses of the put spreads and the put
option-stock spreads.
It also has subroutines which present an analysis of
spreads between put options and call options.
In addition, *MA2 has
subroutines which present the possible outcomes of any spread for a
range of possible prices of the underlying stock.
The subroutines are:
1
terminates program execution
2
supplies put values and deltas for a single stock price
3
supplies the put values and deltas for a range of stock prices
4 presents an analysis of all possible spreads between put
options on a particular stock
5 presents an analysis of all possible spreads between put
options and stock
6
presents an analysis of all possible spreads between put
options and call options on a particular stock
7 presents the possible outcomes of spreads between put options
8 presents the possible outcomes of spreads between put options
9 presents the possible outcomes of spreads between call options
and stock
10
presents the possible outcomes of spreads between put options
and stock
11
presents the possible outcomes of spreads between put options
and call options.
PROGRAM:
*MA1
INTEGER
0F NGOP: SR
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M :( :' OC..
:FORMAT:
1.0
.):'.
ST'I OC
Cii SI N NAME i:,:I. : CD2i.• C 3•2•I':
)
READ ( 1 20) STOCK
3)
t T .)T
F 0 H"(S
11O 30 : I :::: . ENT
20
0 40
Efi . S 'T0 C1 ) GO T O
)
ME
IF (NAME(
CONTc:iNU.E
WRI Ti'E: ( :i., 5):f)
FORMAT'T( 'ST OCK NOT
250
13'5
4
GO
H0= T
G0
WRITE(1 :1.-50)
FORMAT("LOWER
50
5,)1
55
IN DATA SET' )
TO": 10
PRI(1E
")
F."ORMAT "'t..0PPER P Ii:E: 'C)
READ( 1
) :12
FORMAT(:1 :3)
N
0=NOP
Fi
V=VOL. H )
Hb ) )
L. (fo 1 .1
CD :. -CALD
TD (I 3) )
3=
2CAi...D (I
(1C
WR::I TE( 1 200)
200
FORMAT(4w.2X-,
1
CALl... F1PR I1CES
60
WRITEE (1:i. 6.0
FORM1: T ( / :!5
r CD2::
ICDi:t.
:E
SPR
TX
.•K
1.1.0
)DO :100 X:::: :I.:1.1:2Y: 10i
WR ITE ( 1.:,1.:1.0) X
I CI ' .1XY :3
70RMAT('PRICE::
)
CI)3
'
tRI:CE
'
A4 :1.5 X A4 :1.5X A4
K6X
0 !1
DO) 1N,3)
JD)/360
M
T::: 'il( ly I )..
...( V YY OP ( H I ) Ty F" V~ . Dy1 G ):1.
C A.....L.
) -J--- ))/360
1yI ::I1.
T=: ( D 11(
G2.'.2)
V2,:2,'1
CALL. VAL.(V.y Y OPN( HYi+1f ) T yRF,
T (
T::: (TD
70
f80
100
. ) --JD )/360
. 1-..2
3)
H +2) 1vT'Y : V3, D3'
Y ,PO' iy1
CAl...i... VAL.. (
y,V2 VD2, V3i:'3
::1.
,):1.
:1. ')70)OP:.i' I:I)
' 1E(
WRIT
3 (4X:, F. o 3Y:4X, F4. 2))
FORMAT(:1.5XF7 .
NT INUE
C(Of
C;O NT IN UtE
RP.T uj;
DI
1EN4
1(:.L.9 8)
SUBROUTINE 4
SU
Q3
1:OUT :NE
YNUP
P
PSPH( ( NiAME:: Lt
. 8 ) NotP (10)
" (1
1
Y
.'... P
D
:
v :F
R*ENT)
JD
i
.8) p N1.( 3
:i 00:': :~
pO(
S 3IM ENSI N A3:
18)
1.
v
I
t
N2(3
):.
l
1
iS.
i: N 1VC1...(10) NAi E(10):'EXS(3
I)I: MENS O)
[K(3 18 18 ).
3 EX(1 8) EL.. S(3 18 18) vRANK
3:DIMENSION P (18)
8)
:
:1.
i
•
v
3
I
(
L
...
AS
18)
8 :.
H
i)I:MNSI 0ON C SiAS4
:1:MENSION CO ( 3 , :1.8 :'1.1.
ii:ENT
I NTiEGER
i. :
18o
i: . P::
REAL
REA:l .. N:1. rN2
1
E1. C'2, NAM E,: STOCK
SOUU ...E PRECISIO N Cu CDI:C
NI S:= 3
1.01:.
E T
DOi 600 K:i 1: EN
S 5 :: 1 1.8
D0 6 J5". 1.18
Ni. (K 1 P J)=0()
N2 (K I J)y 0
) =0
:K
J
v 32
1:
DEXS (
' I:J 0
1:FEL '-'.3:.K
RANK ( 1K : :, j ) )=0
CONTINUE
CONTINUE
CONTI NUE
( 500 )
WRf :: TEE J.
F ST OCKS•
F":*'ORMAT('NUMBER OFL
NS
.0
51.0
:1.
REl0JAI(
FO:RM AT ( :2 )
DO :1.000 IN:I.:X::,::::1 ,NS
: Y)
: 1.
WRITE(
1
FORMAT ( STOfCK'
6
5
600
500
5:1.0
R EA ( 1 ,2)
2
1.5
8
16
9
1.0
1.
I.
30
40
20
)
STOCK
FORMAT(A3 )
IF
)
(STOCK.
EQ
NAE(I))
GO TO
FORMAT (FA3 )
CONT:I:NUE
:i. . 6 )
WRITE :1.
SET')
)
NOT IN DATA
ft("STOCK
ORMA
F
GO TO 101.
H.:: I
WRI ITE (1 . 10)
:
FORMAT ('ENTERF CURRE NT PRICES')
:1.:.)
WiFRI: TE 1,.:
REA' D( 1,11) X
F MAT( KFS. 4
F:
:O 20 I: :1.,N
( I ))
C I=CAL . TDFly
KI30) CDhOP (K-i I)
WRI:TE (1
')
A'4 :1.X YF' '7 +3
F 0,R AT (A4
iREAD ( :1. 40) P( )
F:ORM AT (F7. 4)
: UE
CONT IN
81
D: o 70 i::: .N
T== (TD(H I,)--JD)/365
CAL.L VAL (V X 0 P (H :.i ) T, R v Vi.7 ) 1 YG1 )
EX(1)= VI-P()
70
CONTINUE
NN=: N--:
DO 50 I: I.PNN
Tr= ( TD ((H Ii)-JD )/3,:Z6.5
CALL.. VAL ( V X OP (Hv I
I
TYFR pV IYE,y1G1)
SII=T
:1
DO
C) 60 ,.:::=I
i:N
'::-=
(7'11 (1D.,
J)'.
-.JD )/ 3• t5
CALL V •....(iL
V X%.,
OP ( H YiJ)v T R Y V2 y 1)D2
,
BVG2)
*:A:2 )fV
( 2 /1 1:GA I:2)
2"'G2*At :1):A (X:
N:1 (I
H,i ,J) :::
)
IF (iN: (::: : J)) . L.T. 0 ) Ni (H-yl .J)=-N1.(H: I ,J)
(H Iy J..l)::: (ui: (,G
(N2
i :.:1)
2- G2*
i;
:1.
) *:(X*V
* )*2)
:V
IF: (iN2(1-1i:i:y J), I...T'i0
N2(IHtII yJ)'=--N2(H, lJ)
GO TO 280
3000
A1..I(IHv.IvJ)::::(CfLASH(H.i y..j.l)-100*(N1 (HJ I ,J)
P ( )) R
(H:J
i,,
.
C P(T I.:).N2
90
50
C
0N
tI::
N I.J.
F..
DO
80
3I=
.Y:.:
.NN
IF)
F
90
I."
1
:
:
G
ICON
: :::: 1:1
+
TINUE
0 ET E11
CONTINUEI
)0 90 J =II pN
: (HY I
J) : (N
EI:
XS:(H. I,:
90
80
C 0..ONTINUEI.1
10
CONTINUE
00
.) EX i( ) N2 I(1- I J) EX (J).)*100
:H=
:iENT
'DO150
:
0
GO TO 1500
IF* (NP1(H :i.2),. EE. 0
NT
400 M=HE::::
DO :1.
.I2). EQ
0) GO TO :1.400
IF (N1 (M
N::NOP
(H j:i=:1.Y.PNN
)
) K ::::
.
DO
C :1.
:L
50
DO
60
J=
... 1v N
NT==(
:
NOP(M)
I
NNT=
`NT"
:1.
M) :1:2.:--:!::
IF (H1.
EG
M) 1 C2=I.
IF (H. I...T.
DO :1.
70 K::'12vNNT
IF (K, EQ. :1:2) KK= J
T
12) KK=K+:
IF (KI. GT.
I.F (!"H..T. M) KK=4K'+:1.
K I ,NT
DO 1 8 0 L;..:"
, ))
t..S
1( MK 1.L.
E.! !C L.
14.1 :I.
YEJ
,))
IF (1) L
.J)=RANK ( y l,:,J)+,:1.
CRANK (H y
.I.F
130
16
33.
U
(. LE 1.........
CONTIN:f114 UE
C
01 i1T IN
i:
1-':
i[NIi
C
D.E .S
H
lI
.J
i..i
il
0 NT I PNUE
1) (1' ) :
1.
103
200
12)I=N
AIU = I2.1,...
.1 (
1500.
.!
T
UE
M. ( 1 Y6 .2 )'
Will
TE
L,2
I290 ).
T' ::. :1. *
i
!-
.':'. J., I <0)b5: 'Q:"
fl0} ": 1:
e.2
DO
.():::
J=1..! .NN
j.,
2
'ir:
t1
1'9·'
f·:::i
....
,
• I.
,
"):-:(A B S (N'f
.'.•'1:;, (Hy
.1) ) +A B S(
H, -.
))
1
(H3...1J
-
B S.i
j..
i
:1
i.NI
.J,J: +,J 1.
'J
N
D'O} 27 Oi':l
ED
RA N(K
3F ( 1.
G0 T 0 12 ;7"<
B
0 ML.--..
t...
C..
SN
lii
' HiIPJ
i'Hi .
:T'.. t1'3 H
B i:::
::::
F:',.J
i
3
S TS
0
H JI)
S... i
0
P
Fn]P.
.13
:2
3
H
J ,
...
.
.l..
,.H
C......1
.,+----"A
• C,
":"(BN. C0
G0 T 0 5 T
S.
P4 KS"F
LT.
"... E 1C"-f
N)
B1J).1
111 0B
S::
N+
N
BN
C,*( . 34X--SS) ) *:.00
575
CASySH (
i v J.) ::::I H
M1(H v J) + (BN BP
95
N
( :1.
+ 3 X-SS-SP ) 10.
GO TO 3000
630)
CD ::::=
CAL..
(HJv ,) )
C:2=CAL.) (TI (Hlv ) )
GT. 0)
IF (N1(4i ,J. K)
300
RI TE(i.v300)
i
P H J
N2
2(H y,
,JJ,")
Y
CNAME (H y)
vN1 (H y,
J yK ) yC 114J
Jy ),
J
) :"A..I !(H:'
(,
F J.J 'K ) CO1M (H •i J K) CAS3H ( [y-vl
CC 2p OP (H y K) PEXS'.H
K)
(HI
Jv
CDEL.S
0) WRITE ( 1 :-300)
IF (N2(Hi J K). GT
(NH J.
iK) v
:N2
( H ,J, i
K ) )CD2 OP (HK) vN i.
CNAME (H)
CCE1,0P(vF(,J),vEXS(H.,JvK) vCOM(H, J.K), CASH(H JvK)vAdi. (HC.'JNK) -,
C.1)I.L.S ( H .J v,
K)
7. 3 ',5
4Xv F1S ,2.
FORMAT (A3;., 1 X y2 (F9 ,,
3,yX y )A4y1X 1,
F7.3,y 4X) ,FT t y2,4X y F7
C 4XE F,,.2 FX .ý,
F 8 .'
3)
270
2 60
CO' TIN
IN EUE
C O NT INUE
255
CONTI NUE
250
CONTINUE
RE TURN
El N1:1
SUBROUTINE 5
L
I ' 1 NF?0U
"S
?4E
1:3F" " '. A F
PiE11
b4 ). V0( L
I Ef )
0 M
E 1 S 1rI 0O1N"
1.
Ei
i'- i
I..NT
FTEG.
ER
. ...
f' .. I.
'.i.
i ii
::I
D.
REiel L F."
R E- L.
to2
:)•
: U1:L.E. RF::
E..i:::
::
I.1 ::.
.
0
c.
1..I
"'j
i)i0
i :::
COM
. ! () 0
.N
..
i )
1
I 0?1MK
AJ
. C"
H I.., ,
*1
=
0t3II
TI
I.Nf ( -Iy2) 71CC.
isi ,t r
NP)
READ(1?2)
F'
STOCK
j•'F
)
WRITEI
::::
t:)
NN ...N
:N
1....
. .
0i-(
pq
?4lP4" 1
- 1
U :: i..( H)
",
')
.1.:
R e)
1.
, :1. x
iT:
3:
:E
1 : 1. ,
C1
,.:"1:CF'
to. ):
{:
f 1.1 .A '
F2
CC)
.'
T1:.:.
NI
II70
I=1
)
Ti 0i::
P-:
(H)
N
T
T)
yI;-J
C(1 fL.i";':."
I::if
..fV FAH
'..
':.:,
F
":
V ,X
:-:
,0"P:• (i::
I f'
):0
I
, "'
i1
R1AW
::
E
.
: !E- fN..2
,::1.
A:i.
:y:' f
.
.1.
F 0 F?:f.TV 1
J.
P
...
yF B
)P
) E1 b
:: ::.
f "2
. F.
.... ...
I
0
I .0F.
:..
,
:.....
':
ý-,J'
( F:
*.*'
I . .
0
, E
!,O
i71."•::
J> - :"
: ::: ..
::.
,,
'.:::.
i
)
7
3
II N
DO IS T 1 104!
SA L.L. V•A, L.(. • X
(N 24 ( 4E111
I
if
1F : ( X
T ).A
to.
41
4
2 •.....
:: :. 0 0
:' .
1.
!;
0
1.1i..
•
H
T
J. ::
1.
'
4800
i
:
fP
1i'
:1.
f:
( -
~
4/2
)
2(H:
H
-5.N= H T
:X
2
:
H:
./2+A2.I....
+A2+0'
CA
HH4 I"I V. (X::
+0"(fl 1
yI
-•' X./.2 ..:P.. I
AL" (H y IH:.
•"•
) ( C SH•( H y I
50
,R
2
J,=:--::
f O
25 1
1'
0. y
(01
:-f
. I , 0 'N 2. o(H
L.". :)N )
0
HNH:
t%"1
TF- (A
t*,)
:;.
)
MOM
F
(
1 1E
OT
1 000 N'
COT
i
M
P
US.1
It)
"',
.,
IF (Ni14
1
it.
1EU
0 7.0
IS0 .i. P-N
'F () EE 13
to. I
1F
MI
"0"
L:(
.~F 1"
..
(IE
,:i"..
..
!y•1:1"IKvJ)
2
.....
i,
L1::'
T
(ly":)
vl.
7.106•1.
i :i 14iN
fU
0 ) G.0 "Y
0' 7 0
E
H y
0
:
.L
S 1..
p
(.
,.
•
C ::
.N 1 IN
" ..
,NTIN1
....1.
- 1
U
DELS(Eb I)
j . 1 ..
.....
i
"
C ' ,.,,
,. 7X., 'L . I ,T
NT'
't".
",
,
C2X I 'lD£1...
VOL. ' )
DO. 250 I=::::: NUM
DO 2 55 JJ:1.
IU ENIT
IF (N1(H
(.i )
.
N =NOP ( 11 )
NN :-- 1.
DO 260 J::::
1 ,N
:IF:(i. E
GO TO 260
975
T0) GO TO 255
R ANK;( HJ)
GO TO 975
CDI=CAL (...TD (H .J))
IN
:1 ( :hii"
J ) :=1.
~ 0 N 1 (--0 J)
IF (NI(H.J. GT. 0) WRITE(,300)
C:NAIME (1H)- N (IHv .j) Y N2 (H, .)
:,
0J)
(CCD1
'.i
:1.Y 0 PF'
( 01 y J vEXH:
X E3 1j
) (CASH( J.;
.
H!
: JJ)
C :ELS
...(H! v.J )
IF (N2( HlyJ)) GT . 0. WRITE (1.•.' 350j )
CNAiME (H) vN2 ( -I i ) CD1
i: Of' ( J) N:.J( J
CEXS (t1"
H JJ ) yCOM (H v.,
J ) ,CASH1(H p ) yJ
ALI ( H, J )
Al...:I:
(H J)
350
C PDELS(H J)
X A4 :LX
.
1F7. 3 4X,>F7? 3
F"ORMATj'A,,
C v.:I(
. X F :1.0. 2 :. X
3X1SHAR
" . X .v
`ES
F95. 3 :.
C4XF: 27.
2,4X vF7' 2 y4XP:'
F7 2, 4X
O pF7.
. :2)
FO3RMflAT(A3 ,1X.YF9 .3 :A.
X4
X
y:1.X v FS.024X'F11.27:Xv ' SHARfES 'X7Xv
26 0
255
250
F7.
CF7 34X,
S
F72,4X
7.
CO NT I NUE
CONTINUE
CONTINUE
300
12 5
o P *r
: U1' *:~
RlEE
N UR:'
,4X,4?
7.2
X
, 2 4X, F7..2)
100
PROGRAM:
*MA2
101
I:NTEGER:NOPt: SR, H ENT EXPCYC
DIM:ENSION NAME(0), VOL ( O. NOP(1O)
DIMENSION EXP(2,~3) .OP((10-9). TDI(10..9)
DOUBL.E PFRECISION NAME
EXP::
( 1 3 ).:::3561
EX P(2:2
:1):::414
:::596
EXP (2'3)
READ
J::"NOP (50END25)
(I )
NAME(I)
DO 10 K :i
1::-. 3
READ
OP(fj::
L
K":.=
1 (5s.40)
.
L
NOP(
K)
(I K)
:
1K
0FI::::
L .0P
AOP
(:1:
LIt
L+ :1.
C)F
. y
)2 :
TD ( :: :K+1 ) EXP ( CYCC.'32)
.EXP
( CYC
:
TD ( I Kt) .='
.4mý
1.0
CO NT INUE'
20
25
30
40
45
CONTINUE
EN"T"= I
F:ORMAT ( A3. :.X, F3. 2 1X, 12 :X1 :12)
FORMAT (F6 3)
WR:I:TE (:1.
50 )
READ(:i1 ,.:.60)0)
DAY.
0i4ONTH
R
WRITE (. ,,70 )
:
F :1READ)( .,80) SR
FORMAT("ENTER DATE:,
INTEREST RATEN)
1X
,, F3 , 2)
X,
FORMAT (12 :1.
SUBROUTI::NE DO 'YOU WA NT')
FORMAT(i/ 2):
p
:
S:IF
( MONTH
EQ,
,.D::=DA Y+ 1
IF
N T'l1 1- E Q 2)
( (:30
1:i:F (MONT.I
IF:' (
( MONT'. E
E Q,. 3)
IF
N'.'ii'TH.
4)
:NT H. E U 5)
IF (MOi
JD=4DAY+.151
:i: :: (MONT I1 EQ.,:
EQ.
IF (MONTH.I
IF::(MO(NTH·1 EQ,, 7)
46
50
60
70
80
8)
::F (MONTi-1.
IF (MON'TH
IF
(MONTH
(I3F ( MONTH0
206
207
4F1
E .:9)
EQH.
JD=:.:.Y..+..j.273
JD=
)A Y4.
5 04
J.
fD:::=:O AY.133)4
EQl
E
2
12)
GO TO (2:201)Y.206, 207,208,209:y210.y
C211A :1.
2:1, 213 ,214 v 215.
5 . SR1Z
C AL...
I..(NAME Or NOP, VOL. TD ,
JD PR ENT
(GOTO':46
CALL VAIP ( NAME
: ONOP OLV TD
'JD
NT
'
CY.
G::
0 T"D 4 6
G0.. ..
011
N C
G.i
0 T 0 4.
S
AL....
C
i:-is
T1 R 4::L
" "
G'0 T 0 4A6
(1
*TO 46
EN1:1
N
E
0
N
1...
E T
103
SUBROUTINE 2
SUBROUTINE PR'
iP (NAP ,E OPF NOP vV0., TD
v I .r :R ENT )
104
20
DIMENSION NAME(aO) .NOP( O):F:'('
:i.8)•, VOL(10),vTD(10.18)
DG
1 P31 M
iEE FREC
) f IS
1i F.:
114 0 P::
t,: i 1
(). )- C"2,
c:y Ch C113
: y STOCK
S yV OL
DOUBLE
iONI ()NAME
CD
INTEGER H. .'ENT
WRITE ( 1, 10)
FORMAT(STOCK: )
RiEAD (120) STOCK
FOR MAT (A3)
30
IF (NAM
i:(1
E
CONTI P
NUE
35
1WRITE` ( :! 35)
FOR.M'AT (' "S"T"OCK
10
E:. S' 1TOCK)
)
NOT
(1 T]
" 40
IN4ATA) SET')
WRITE
50)
FORMAT( "'PRICE:
i
)
: 55 ) X
R:;'EA 1(::!
F'ORMAT (F7 :.3)
N:::=
NOP' ( )
W&=COL ( H )
to
(" ( , ) )
CD1 ::::CALD
50
55
CD:2=CAL. (T
( H 2))
WR ITE (1 .v 00)
100
6
FORMAT(32XF'PUT PRICES')
W I:TTE (. 1 0) CD1:!. C 22 , C )13
FORM AT (/
STR KE
t
PRICE ' 6X A4 :i. XA:
F
4 5:i. X A4)
Di(.*
8 0 I"
E
1 :.<y N :y
=:
3
'Ta (
11h :ip, ) -0J
X: )// 365
T :: ( "' : ( 1! v:i: +42) H) /3:iJ:
'65
y
V3 y:13, G3 )
2) ,.4 :RF
CALL. VALP V. X y DPF H y, 2
70
FORATF7.3.,3(4XHF,:. .A4Xp.F4.2))
80
C?4ONTINUE*';i
T*UFIl
105
SUBROUTINE 3
y,' E1NT>
VOL. p U) .. ;V
E:P A ..YP
.U..•ARP"('N....
TI!I fE
IE A, jN: Al E ( 1
N" P)0
0.18 y V0L. 10
... .. ..
if0 U L.,E F" it C
20
30
1T
GER
W
F!
T E:(F1
XI11
0 A! 14 A
: ,CT 2
Z ST "
FNT
:
!
::
I:GIIA TA : )
i 30 : : :.i
IF
tEi
Ap
I ). END
CO p NU
TP4 EIN
Of3f CK ) (O
f 0. 40
''i TOI :'
41
to50
HiO:::
I
W '"r
i :( is!0
F:0 R. ST
L 0 Ili ET:. F,IT:CE : )
P41
T.C
F; (.
H"]..
C...
: .:
.
D::I,i<
i
b
:;1.
5l::
C1i13 CAL. )i7 (1,H 5
WRIT:11:
Ei...yi2 t'.) )
000 4.:
F"• .TE:
T. 4t'2X
Y ""O
1 6G)
Ci. PRIT1
C C
i EE
y
Y
60
FO,
0.R!S:T
::1
t':
JT ..
.
/ 1.X- ::C >3 Ej "(i'
R
E
C).....
L.L.
. 0"0 C Q
END
....3
:
A L.F` V Y 1E.
,.,
1(HPl
0 U . ..•.,UE.:
..
.,+
6XX
+!);
: TY
R
YMA l
A4
1.15X
y
) 10
106
B
107
SUBROUTINE 4
J IT .OfERF"0 F:: N A HiEE 0 P N."0
L.
II 13,
II N'
SI
I.,N P0( 8 - E X (!8,
)i) L
i) l: EN S
N C SH ( , 8, 8) , ...
RF?
E(IIf):L P
', .L N 1 .N.2-
108
EA
NT)
T .!::
, L.,U
, )
. ) , 1 3iy13i.
.
E1) .TD 1)
(
i v 18u 1.,4
13(3A
2'5v. v1..
I
1U
LEK
PF?
EE
C1:
I:i
S 0Np CDv C1)1i )2'to:, :"
0
NU H::a:
iD.O
600K:--;
:ENT
N
....
:' i::: ::...l
)
...f I
E.."
[E
J
I ::
I
.f
,:=0
i:XAN (K v::-:y
Ci0O -" 1
f
0J
i
REE A-1
''
11S1
M)
1. )()
i
)
100
toD FEX
. :•4 S
MRHT
E1K)
1,:2
16.
?
A.*
1.
)0lAT
F..C. fi T( : A,, E*:
A! , I' .0.:" .. 10
F
( !::;
i" E (C2- ' }.T
F,1i:MAT..'
I5''1
Af.'..I'
.
1A' 1
't
'
.
,i'
.
F • 01 AT ( 1 Of'T11
J 1 j:C)F
i i - ):i
SI:: CA ( L. 15 1) 'T .C
RE
A
H
',
N D.,
i
•: "
CU ?IRE
•H
3
F0'0
M)
' iF. A
•4 4,-.
:.F;..'
I Fo
i: TE
i.);- X F,
J ý:;
20
2·
,.::
CON•TN
UE
U"
{2 {:
i2NUi
. 1:
SE
HE: S' 0 C
70
109
DO 70 PI:=N
T== (TD(H, I )-..i
D)/365
CAL.L VAL..P(
P FlXy) .M
T IR:- 1 iI G1
EX ( I )=V1- P ( I)
CO NTINUE
NN::N- 1
iDO110
50 I :1.
. NN
T= ('TD) (:1. )3D
6/365
CALL. VAL6P( X:::OP(:ON ,:I):T
vRPV1.ivW G!)
II=
P+
DO 60 J"=:I:v -N
T= ('TD(H v J)-,JD) /365
X OF (H'Y J) Tf !: V2 ':1D2
CALL VALP V(V,
' G2)
N 1 (H I ,.J
):=12/ ( (G .
(XA'V )* *2
2- G2D:
1 )
IF (NAi(H:tCl:.,I J).
T..
. 0)
N2 (H y,
I: J) = (DI/( (G:1. 1).2"G2*D1)*(X*V)**2P:t:
It!:
il: :,.)f L..J 0) N2("1 YI2 .
:1.F(N2(I
N2 (H1
Eil
l.*
ASi. 0 0*NKX* V
)ELS (HF1I .,J)-=N:1.
(H :i:J)
EXJI:EX (J)N2(H,
(ly
3000
60
50
'4
J)
IF (EXI:.
GE. EX: ) N2( II
1:i J "..I)
N2 (1 l I.
:F J)
IF (EXJ . GT. EX )
NI (Hl' 10
J)0 =-N:1 ( Hl I:J)
GO TO 280
A... (H P
i: .))= (CASH (H el:i:
.J)-:1 O0:, (N:1. (H : i, J) *
CP( :(:
)+A 2 (H Y
I J) 3 (J
CO NT"INUE
:
i',O NT
80 N:i
I
,NN
4S1 1.YN1.
0 J):0
DO 9:0 ,.J= IrN
90
80
1000
EXS ( HYs 1i
CONTINUE
i f
) =: N41. (H r:
CONT I N.jE
C.0 N T I NUE
D:!O :1.500 I:1::%i
1 ENT
IF ('N:1(HY:02). E Q...
1) :1.l400 P!:.HENT
IF (N.1 (MP:!.
1 2). EQE.
N=NOP (H)
) G
'T
0 1.400
NN .N
.:1.
DO 150 I3=1:::':.vNN
1 i:.:I+1
I1O 16
.6 0N .0I
NT= NO
CP)
NN T::':i'-'
T--1
IF (i.
., ED
M) :1:2:::::
I:F (C·
UT .. ) :I2:::::1.
i...1
.IO :170 K :::::2 NNNT
:i:F::
(K. E
2) KK::':
2.
IF (K G
12))T KRK=< +j.
IF (H.,(KK=0
LT
M:)
K+
130 1.
8.0 L...
KK,NT
1
If' ( TO SDE
S H,. y•I
J). 1iC...E
i ELS.
5 M
iiK
CARAN
H IJ
MANK (H•ip
y. J )+:t1
..))
IF
180
:1.70
:1.
60
150
(DEL S( Mii i.y.
K L)
1..
CRANK1 (M K
CO NTINUE
COfNT INUE
CO)NT :INUE
DEL.S (H
.,
)=RANK ( M,
r
I J))
L.) +1
CONT1INUE.
1)
A( I )"=:i*A J(I200
CONTINUE
1400
CONTINUE
:1.
500
NUM= A (N) / (2 A ( N 2 ) N
NU
CON
NTINU
650
F:' OR i ATo 50X
(
, "PUT
S REASz; '
/ )
WR I TE i.
(
290)
290
1FOR AT (/ ::.3X ' L..0ONG '21 X 'S3ifiORT'
T 1.3X v ' EXCESS'` 3X :.
C' COMMI SSION(
X ' CASH
6X' ' ... )OST I:N' ' v2Xv 1'
6X
DOL.. VOL.
).
DO1'C
25f0 T
:1:( N
NUM
1DO 255
!-H='I.'1 ENT
:1: ( Nv:1.(H
l:i. 2) . EQ
0)
GaO "TO 255
NN=::::N--:i.
DO 26e0 ,.::=
:. NN
1DO 270 K=:= JJ!:,N
IF (. E
RANK (ilJ:K))
280
GO TO
'
630
GO TO 270
BULL.w=0
COMiI(HI :.I
J)=(ABS(N:i. (Hyf I J) )+ABS(N2(H-• :1J ) ) )V8 5
BEAR::.:O(
IF' (NI. (PI
H
525
B N=1 N2 ( i"i ..
J)
525. 550 55(0
1)
B S:::':
F' ( H ,J)
SS=OP (HF I:)
BP=P ( . )
SP
F9:P ( I)
TB='TD( H.' J)
550
GO TO 625
BN::::N .H i :i:
J,'
SN=.-"N2 (H1p I yJ
BS=OIP (H y 3:)
SS=TOP ( H, J)
BP:P
(I)
S P: P (
J)
TB=TID (H I )
T S= T]D (H ,J)
625
IF (T'B•
LT. TS) GO TO 575
BC :CO:
M (HPC 1 J)'+( BN* BP"-SN SP) :"::1.
00
IF
(BN, LNT. SN) GO TO 2000
) : B C + (SS- "BS) *SN* : f00
IF (BN . LT SN.) CASH(H I ,J)=:::BC+'( (SS-3'S) BN+(SN"BN)
CASH (H :y.
2000
110
111
:*( SS-,7*X ))*100
GO TO 3000
5•75
630
) )* 100
4 (BN E P+SN (SS-- ,, .7X-SP
:COM (H . j Jj ) +.
CASH (H,
H pjvij:
GO TO 3000
J =CAL .D:(TY
D (I : J( ))
C:
:2=
CAL. ( TL ( Hv K )
(N 1(H J K).K GT. 0)
) ::Ci
CNA fME
(H) Y
,N1 I(Hy.J YK)
C
SIF
30(0
CC1)12 s,0FP (H
K
CDE1:
...S (H f
K
WRI TE(1i. 300)
OP (H j) , N2 (H JsK)
I v CASHi(H , J ,K)
X S (H
H
J , YK) ) P:COM (H J.K)
00)
0 ) WRITEE (1.,
•JK)
GT
:IF (N2. (H,
1 it
J
H y!,K)' v1.N:l.
P2(1 HY JYiK) CD2, OFP H:
tCNAME
itE (Hl)
·
"
) PCOM (Hy J :. t) v tSH
CC::1.
P(1.H J) YEXS CH J K.l
CD:ELS ( H v J K)
Y : X v F7.:3 Y4X)
9.3 • rY
2 ( 0I
F RM AT(A3 1
C4Xy 1F'8 2 3XF 83).
270(
CONTI NUE
260
CONTINUE(
250
CONTINUE
Si"ENJ
N
H J
'Li
L (HF J. 1K)
'!I
AL
(1H . J v K) ,
F S o.2 Y.
4X Y1F7~. •Y4X •vF
.2:
112
SUBROUTINE 5
SU.BROUTINE PSSP (NAME 3,Ofp NOC -. VOL
C
T El JD R y S
ENT )
DIMENSION AT9) 1NOP(::'(10 ) . OFP(10 9) TD(i10:. 9)N: (N:1.0
9)
LDI: ME NS ION VOL..( 1 ) ,,NAME ( 10) I:EXS(1.P0 9) N2
0 •9)
DIMENSION P (9) EX (9) ' ELS ( 10Y 9) RANK (09)
I)!IMENSION Al ...
I (10 9) COM(10,9) ,CAS HA(I'9)
INTEGER
F:)
. ENT
REA L.. N :1 N2
'C)2
C
)DOU 1iLE PFRECISIO N CI1:D ! i. Y
N iS:= 3
) 1
NA1 E STOC
iDO) 600 K=:iENT
DO 5 I=:-1-9
ItXS(K
1)=0
5
600
500
510
01
.
2
1.15
8
:1.6
9
1-. N2(K I,
:I) u
)
)
K.'
(
S
DEI
I 0'"
:1::s.
.
(
ICIEL..S
RANK (K.: ):'0
CONT INUEI
CC)NT 1: NUE
TE (1 500 )
WR IF
:. STOCKS-1,.0KI
FORMAT('NUMBERi
" )
READ( , 5:!. 0) NS
FORMA (1
1' 2)
DO 1000()IN'EX"-"1,NS
W I:
TE (i i)
FORMAT("STOCK-')
READ(:12) STOCK
F ORMA
)
T(A3)
., T ChE!N
D : 8) IE:.[
( .l
, NAM E (I) ) GOCTCO) 9
IF ( STOCK .KEC
FORMMAT ( A3 )
CONTINUE
6)
WR ITEE(.
K NOT
STOC
'
1A $SET')
IN DAT
FORMAT(
0
GO TO 101
!H=::
I,
N=INOP(
)
NN 1:N 1 :a I
10
1. 1.
30
40
20
..
,10)
WRITE(I
*
FORMAT ( "ENTER CURRENT PR:ICES' )
WR
:I:TE
I ( :1..
:1 )
READ ( l:11) X
F7ORMAT (F8 .4)
:iO 2(0 I=::1.
:I::')
CD= CA L.D (TE (H
WRITE (:1. :30) C1, 0P (H . )
)
1 p'
F OR1MAT (A4 : X pF7. 3
READ (1..(Il40 ) P
)
FORMT F7 .4)
C O NT INUEl
::DO7 0 I: 1..
I
I :1-JD:/365
T= (TF ((H,,T
.l ) YT"R,. :1.D:IAG )
OFP ( 1H
CAUl... VALP (V .X ::
113
114
7 COff114 E:
i
lb1~5 ':~'
'1'
T T)HY
''
)
Q
16U.Fi
)
C' 1.. L.F
11Y1
.QVY0
yTYFY'
.v)1yG1
~.4
1 F,
IllI
1-4y1
1
Nt2
11 *( :
)
F
1
..
I
1-
J'F' (Ný*2
G
it
..
L
()
X
:E
T)..
F,
0
2
liii
1
"
Fi
:1
G
0'F5
F
F
0
.)0
1
15
ST H
AMF
,F
. A.
EAR
...
1 )M
:!N~ hi1
H
1
1-.q.0
N:. !--r·I
T' Nfl. h
1:
-1
X ( :1
F,
1
5
P
id3
5.
FE to T1
F:**F'
K )
FFMI
.I
I.
1NUIT
H LF ý 11 ) L.Q.yit
1)
0I 3Y
6I K:i .i L.Yff-
F51
H1
(ih~is~
41
FF..C.H.:C
I
0:)
1
FF·
800 f)r)1.6)13
S(i
A1.)
T
1 -1
H
2
0)
11
:7 11.'ý.'
:::j iS
...
.IY
~'f~i
x .1::
2 1j ! :F
1
T*I..*
E: XS::ý
F HF
1. '2
41
4)*
WF
Fj1
ofX/
4 M !Y1 +J
F?
Af:
NIt i y F..,f.i
I!.`, j :1
10 T1toUE
C0Wr 114 UE.
;1 to
71 0CIff 1:N
IUE
sý
~"719
t4 F.:1.
11:Y2
C1X 10 . 4)0 .,"
2i
F"
1~j~
J
I)O 250 I:1.".'l.1NUM
11DO
255
=::
1EN T
IF (NI(1H,.1i),
N'".NOP
( H)
E ,, 0) GO TO 255
N N C..-F 10
:00 260 ,..=:
.N
IF (1,, ElQ. R:ANl'.(HY,J).)
.0
975
GO TO 2.0260
C CAL . TD ( ,..))
GO
115
97.5
N1 ( H, J.:::: :100,:,N1 (NA
H U)
CNACUIE (i.i)
i N1j (! J) v' N2(Hi ,.i
)
CCc i.
Pcr3
(,jq
) EXS ( 1... ) Cl"( i 1,".•J) , CASH(
300
•R•2•
F
"i A3•
260
2C P4X
rE7l
E .'
CON "I N
I E
255
CONT'INUE
250
CONTINUE
RETU..RN
END
4XF'72
.:IXF•
..
4X
a2: :1'
'S
pl.27 2y
.I<
4Xj
RE '
:p
y XI.l
v ,..i) , AL..:I ( H, ,J)
9)X F9 3
)
i.,• X
.
7.J,
116
SUBROUTINE 6
V,0..v T
J.13 y ? ENT)
117
SUBROUTINE STRAD'( NAMEE, 0R
F' yNOP .j'
) IMENSION PEX(9) .''
EX (9)
Op ( 10).
( 10,9) T (10 9) N 1(3,9 9)
]IMENSION
:
VOL... (10)
r AE (10) vEXS3
(3 ' :.
9P)V M
N2 (3 9 9)
9 v"CiIELS (3 ,9 9)
9
RAN (3 . V'9)
1)IMEE"N.S 1ON PC(9
F
,)
pP(9) EX(9)
I)i: MEKNS ION CASl
(3 9; 9)iA ..
I( 3 ' 9 9 V COM( 3 VP 9
TS VIE NT
1- TB
B TSV"
I NTEGE I::'
REAl...
.y
SU3 N N12
E
REAR
S
N
y2 :
N
v
NUiM:=0O
1DOC
600 K: 1. ENT
DO 5 1 =1 99
D)O 6 J.:::
19
,,)'0
J
(V
NI(K.
N2.(K
IlV P J)":'
EXS (K
J ) ::
6
600
500
5:1.0
101
DE LS ( K -IV3. ) 0::::
RANK (, V J , J ) .':i0
(DCI)
NI FN( Tl
CONTINUE
CCO NTI'NUE
WR I TE (1 ::500)
FCORMfT ('NUM.iER OF ST OCKS1:
I IREA('I 1.V•5:1. 0) NS
FORMAT(I2)
I )O1000 iNI)
: EX=
:::: i.
W RITE(I1. )
)
VNS
is )
FOCt
c pi A T ( STOCK
:i ADt(1:V.2 ) STI (OC1
2
825
S
:1.6
9
FORM!AT (A5 )
IF (STOCK.. EE . NAIE( I)) GO T OI9
P
FOR
M
'OAT
(
A3)
CONTINUE
C 0 N T1IF
N UEEI
WRITE
E ( :,16)
FORM
PI
AT ('STOCK) NC
COT I N DOAw
"TA SE T'')
GO TO 10..1
11::::IF
N:= NO:0I( )INI
.')
v11
( )I
:1: !
'i
0 2 C.)
REA('
1 :1.A(FS..
1.).X
X
O
FORMAT
7
D0 20 I= 1 .4)
N
v
C DJ=CA.T'
( H l:.))
30
'CA
FO0RMAT CA"4 .1
X, F 7 .3 1.
READ(1 ,40) PP ( )
Hy 1 . ..
Tl..
)
40
20
FORMAT'm "(1F.7 .4)
CONTINUE.
)i•:.i
7 0 :.1::
. N
i" (T'f
I )- ) ) / 365
CALL VA ...
(V Xs
FH yl:i.
I
CEX(' :1-: :. C
70
118
' yV 1)
:i..
G:)
)
CA(1
.L VALP.( V X, CO (H y ) T Ri V~i.•.Y .v G
PEX( :1:
)=
-PP:
F ,I(V1
C3::iNTI4
:i
NUE
1:10 5Q0
D)
)
I:::::.
N
6 0 J=.n
.f1N
i I)
t -JD:
TI: = ( TD(r
)/365
CALL VAL.( V X y OP (1H,.I) v:
T I1 R !;V :11.1:1
: ,v1.
C :.)
T J= (T
1i-y
..J)
-J:1
. )/365
TJ (Fl.
RF V ' Dl G2
N )
(2
CAL;.L. V1L.P V X, OP(H J) N
(H y !: ..
I)n::
( 2
t:G :.
All) 12-...G
t
2 AI.)* X V )* AR
2 )
J ) -J)
:=.N. (•W1.
I .
p
:i:::'F
(N .&1
H
J )
W!".'0) N .( ,:: •,ly
N2( iil:i:
.• '(.
/4( G I 2- 2 01 ) (X
) 2 ))
J
Tl(
. 0y N2(
) .*:.00*X*
H2
) yFl
(+
I1
((i\l-*l
2 -N1
1-1
J( E H:1.
Iy)::::
J)I.
.T..
H).lpI
J)=
IF
(N2(H
lv
=
Ht
:.I:.:,)*5
lvJ)
:DELS(H
59
,2 5
5
3)pJ
:)
4
N2(H"IlI
J)
(EXS(H•,:I.,J)
H.
GE.J 0) GO TO
C .159
N1(1. I,y1:iJ) - N 1(H I" J)
N2
:I:
. ) -N2 (H)
yI,. J)
J ):-EXSHT
EX( H,
COM (H1
Al
J ): (ABS (PNt1(H .y J
+ABS( N2 () ly-•.)) )V•8 5
525.550,550
IF (NC(14
I"I,:.J) .) 5
CAS ((Hl,1
3 ,J ==(N 1.
(H y :,J )* C ( 1 )+N 2 ( 1I
H I .J):kPP (.J)
(( As3
X .P4(N
(II:: J) +N2 (H,:.
: y J. )
I:\1
, J):( X"-fIOP (Hy ::))
EC-N2PH
l I
) (OP (i:-'i )-X ) ) :•10
0+C)OM (H' vIl
: J)
(3GO1
TO 6:
:
I:)+÷'N2 H
CASH(l:..
H v J ) =COiM (H Ii:J.. I(N H: :I:
00
CPP (A ) )*i.
61
AL.I (Hy Iy J )= (CASH(H(0
.I
3 J)-:.00 (NI.(1H.l yJ)*
CPC (I:
)0+N2 (H Iv5 i)*:5:'(F' J) ) ) MR
60
CONTINUE
50
CONTIN U E
(0
1.000 CONTINUE
'550
TiO
:I:F
:1.
50(0 H :
E'NT
(N:i.(Hi:.v2) . Ef
0) GO TO :1.500
D0
= P4 :5.
1=
o0( :i:::Ii.
1 -N
N
):.I
5:
DO 16:i.
0 J= 1:i.
N
.0 1400 M ::: H E NT
N' :N C)F:'
( M)
IF1 (NN:Il.(
y,:t.2).
L 1:::: 1
IF
(M.
Ell!.
H)
E1.:l(
L.I= I
IF ( . E . H) ...2::::=J
1:1
: :1t.
70 K= L...:1.
I , NT
IF (K. GT. I) L...2:::::1.
G80 L..=L.2.,NT
DO
0)
G(0 TO :1.400
J -X
:i.
80
:i.70
IF ( DELS (iM:K1• ))* L..
.T IDE!...S (HI
CRANKI(M
N K,1 J ) =RANK (M, K L..) +1
CONTI :i:NU f:
C1O:
CN'TINUE
:1.
400
i 60
C:ON TI1 NUE
CONTINUE
L50
1.500
CO NTINUE
NUM=NUMN+'A
CONTINUE
.DO I 10
50 1 H--:
H
:1.503
1.502
1.50:1.
2000
290
280
630
.,J))
ENT
IF (N:. 1(Hl,:I.2). E:: 0 ) GOC) TO 1.50 1.
CF1( 1
N NOP
:::: 1 N
)O :1.502 :=
iDO:1.503 J: 1.,N
C)ONTINUE.
CONTINUE
CC)NT INUE
WR I T
TE 1:
i2000)
FORMAT (50X v " PUT" -CAT.L.L SP RE ADS')
WR:i':TE ( :1.
v 290)
FORMAT( / y 3X v ' CAl..I...'
21 X• ' PUT ' :I.
3X 'EXCESS'
3X
C COMM IS SIION ' ,5X ' CASlH ' v 6X, ' i...COST I NT'
C2X ' I' 0
1L.. o VOL.. '*)
DO 250 :I:: 1. NUM
DOC) 255 H :Ii.
,,ENT
IF (N::.(H :l1 2GO
2)
EQ
O0)
TCO 255
N\::::
NOP (1.)
DOiC260 ..J::::f N
DO 270 K::::I.vN
IF (I . EQ.o RfANIK (H J K))
'
GO TO 280
CGO TO) 270
CD :.=CAL..C'( T (H J) )
CD2:=CALD (TD(HNK))
WRITE( :1.
300)
CNAME 1(HI)N:1. (H!i J :.K) CD .POP (H :.
J) N2 (1-Y J ,K)
CC D)I2 fOP ( 1., '*),)EXS (H v J '•
) :.,
COC"i (H JJ, K ) ,
(H J Ri) ACAiSH
AL... (H, .,J
K)
C IDELS (H yJ K)
F)RPMAT (A3: X2(3 F 9
:.Xl'
X A4
X:F7.
A.
a
3; 4X) PF8
0 2 4X F7]3.4X>FP2:
C4X :S,
F *.2 v 4X v'
F 3)
CO NT INUE
CONTINUEi:
CONTIN UE
CO TINUE
N
'
300
270
260
255
250
END
120
SUBROUTINE 7
SUBROUTINE RES ( NAME OP FNO
VO L. T, J RfENT )
121
DIMENSION NAMIE(:10) NOP1F
) P'(0
(10 .18)
VOL(1 0) v i(10,18)
UBl
I...: PRECISION NAME CD:1 I:C: c2D3 s STOCK'
iý
I:NTE:GE:R X %- ENT
10
20
REAL N1. N2
WRITE (1. 10)
FORM7MAT ( "'STOCK:)
RE:1AD( :. 20) STOCK
F: ORMATf(A3)
iO
0 30 :I= 1.
iE NT
IF::'
(NAME (:1). EQ:
Q
30
35
40
S50
GO T O) 40:
C=I
WRITE( 1i50)
F ORMAT ( 'L... WER PRI
F CE ':I )
1REA I(f1.Y55)
5'.'
55
STO CK)
CONTINUE
WRI TE ( 1.v 35)
FORMAT(7
lM T ' ST OCK) NOC I:N I ATA SET )
TI:1
FORM,',AT
(:t:
N::
::fNOP 1i
)3 )
V= ::::.V
L A...
((11 )
I
C.1 =CALD(
L TD
H)
i ))
C
2':"CALt.
== CAL ...(T.
(TN
))
CD
I:3::::
(.1( H
' f2.))
S60
WR I:TEE (:1 6.0)
F ORMAT( 'LONG OPT
I :ON )
WR ITE(1 :II.y870)
(80
REA: "( :i.
P80)
FPORMAT
(F
7 .4 N:1.
WRITE (1 90)
90
F:ORMG
('
I. I )) .
READ:( 1,100. J1
100
115
F ORMAT
f.(1 :2)
WR::TE ( 1. :1.
15)
FOR AT'PRCE
1
A I::
READ
( 1.,
80 ) P :.
F
SWR( TE (1, 120 )
!20
01
F'ORiM AT ( "NUMBER SOL[
READ: (1.E80) N2
WRITE (:.v 90)
READ:
(v. 100)
")
J2
WRITE ( 1,115)
READ (i.
) P2
WR I TEE (1. 2000)
CI::i. =CAL.A)
(TD(JQ
H
l J 1i) )
C::12=CAL. (T.DT
Il( y.J2) )
2000
F:'ORMA T(55X
RESULT
OF" CALL S FREA DS')
1
50
WRITE( 1 . 50)
X
1 X, ' STOCK' I 2X.r
FORMAT(/
i...DNG'
3XC 3X
C'COMMISS IONS'
LONG
C.,9X
SHORT'
F. 'i :V
"'SHORT'
?X' PROF IT'
C'LONG'I.6X'SHORT")
WRI TE ( v160)
160
9X 'C AL.L...
4X :'F'
" A
F ORMAT(X:.
DR
i
I
:F'RICE' .12X
,
'CA . L.
5X 'DEL...TAr'/)
i...TA'3
CHANfGE ':' 9X v 'DE
4X
Cf'5X V 'CHfNGEi'
DO 200 X: ..I3:.12
)-.JD)/365
T='::
(T (
J1,.
(1)
T R y•W . 1 ,v
CAL. VAL.(V Y ,OP (H,yJ ) :,
T:: ( TD ( H y J2 :1-JD) /i 6K5
CALL.. VAL.. (V :Y OP: ( Hvyj2V ) . VT Ry v2 y D2 ' G2 )
-')4
1 7,
A'"" ( ABS (NIL) +ABS (N2
P )* ( V :.-P1:.)
B= ( :1. 0*N:.
BB"I( 100 2 ) ( 2--P2 )
=D 1 Ni. 100
210
200
D2=:o 2 AB S (2 )42I.00
( H , J2):.'
.I (H: ,.i) N2.
2. CD2 v'!C::
CD:. OP
WRITE(1 2. 0 ) X :N:i..I
P
CAy B .i By.C y Di1 D2
FORMA'T('7.a.3.
3 X 2 ( ::F7 :I.
X A .'3
LX F7.3 .4X) 6(FE 2 .3 X))
CO TINUEI
RETURN
E,'<ND
122
123
SUBROUTINE 8
NESSUBROU
RESP(NAE, 0 " NO
:V0...:TD J
R ENT)
DIMENSION
S:
NAME(:1
)
)0N!P(1 K)F:QP(10 ,18)
3
.)
"L(
OUBL.E PRECISIOFN NAMEi C1::I)
CD2 C' 3:STOCK
INTEGER:(3FE., X 1-.1
H ,ENT
REA L...N:F1. N2
WRITE((1:.
y 10))
FORMAT( 'ST'OCK
i' )
20f: STOC
D(REA
T'
FORMAT(A3)
DaO: 3 0 I= :...
EN'T
IF (NAENI). EQM.. S TOCK
10
20
GO 'TO 40
30
CONTINUE
WRI:TE( 1i.
35)
35
FORMAT(
i 1
S TOCK NOT IN DATA SET'.
40
H3=I
1. 50 )
WRIIEE(::
50
FORimAT("I "OWER P
:1
WRITE (1 5)
3GO TO :10
:51
ICE :)
FOi:0RMAT ' Ui::'
PER P:
RICE
.:')
R:IEAI) ( :1.55) 12
55
FORMAPT (I3)
N %::NOF:( 1;)
WRITE (1
0)
FORMAT("LONG OPTION')
60
FORMAT' . NUMBER
W70
'
0
90
FO RM AT (F7 . 4)
WRITE ( 1790)
FORMAT(!U 1[E')
:1.00
) J1.
SREAD ( . 1003
F:ORMAfT (I 2 .:
90.75
F1'ORMAT"(
1.:1.0
REA1(
,
M
JP
::!0
)
(1:.::
WRI:TE
FORMAT("SHORT OPTpAION" )
WRMI TE ( 1 120)
FORMAT('NUMBER SF11:.. :')
W R :I:
TET ( :.
120
)9
PRICE
1AID
: ")
READ (:1.
,80) N2
WR(ITE(:1 r 190)
REAMIE( 1,1.00) J2
WRITE ( 192000)
.,J ) )
CK 2=)CA ...
. (TD ( p2 J .2
))
CD :.
-CAL.
1 :. (TD (
K
2000
FORMAT(50X, "RESUL..TS OF PSLT SPREADS")
0)
:. •I
124
K
(
1F)
. ...
17
i5
. ... .....
W125
. .
•AT (
F
125
S.T0
. . CI
iX
..12%v L.C. ':!
. X
-
'.T'
X.
H10 RT ,
C3I6X y.,•'1S
is il`,(iiy I..fSj1
A660.,,
..F 0•y",
1C
XE
C
v, :'i 3
. .FS
14U
'
C;
'C
,:C:"
HA 4GE
....:: X y is
GiE. j..so X:i)LE L
DD20
XI!
1i..i...
i... i
210
200
Y
.
P H .11
) :: .
-I *.."• .,
TIB....
H1 J1210
IN
:,T
1. T!.
42
f )V
CO. ::B.Bi
Y 1 D2
3 6V.
-F1
F"D RMP!f) T(F 3.l3 X 2(
C!NTIN0
P4U E-:3f
RETU RN
ENpq
D
"T
... (1
1
Y :;
:. ...
14
1:P ( " y .:1.)y
:,F 2 a:
C; 1 :.!
13
0P
i (:
Hy .
...
..
,*
:"•
X ):
6..
t
,2
: -..
-2,,
X
126
SUBROUTINE 9
a p :1.:: ' NO: vV y.. TEl J , R ENT)
S U 0 1UT
:I:IN Ei F:E CS (NAMEE
VOL(. 10):
:1 0 ) y:P (1 0•18)
1I1:
MESN IO N NAME1 (10 ) yNO P(
S OCK
CX:2 (.C: 3 STI
C.tC
N AMi: .i4
:C1.
EE
E :PRE
i:
: OUBL.3.
127)
. 8:3
T~'(1 0M1':
I::
NTEGER X.Hy
F ENT:
N
SREAL. N1., NN2
10
WRITE(I1
!0)
FOR
C) ATI ( 'STOCP
I
''
:=.
1i EN
f:i30
IF (NAME () . EQ~.
35
40
150
51
40
ST OCK) GO ""0TO
(3;:ONTINUE
c0
WR I TE ( :1 3 5 )
FIORAT('STOCK NOT I N DATA SET)
0
GO TO :1
H =:I
WR ITIE (1i 50)
: ')
0RICER
FORMAT (' L.OWE,
,:
5I5
•
READ (i .'1 ) I
(1. 51.)
::
WRITE
::
F::RMAT ( 'UPPER PRICE
T2
:.".*"55)
RE AD( 1,
N=NOP"(H
Co
=0CAL.) (Tl (H 1.)
ElC
oCi:'
::
2= CAL(.
2:)
)
60
CD3:::CAL.:D(TDi(Hp3))
WRITnE(1'C 60) 1
S
)
FOUiRMAT ( " STO'
70
FOR"AT('.UMBER
i70(
80 :.
1 00
1. : t:, '
REA
(1 k
F
Cf:'uI Miti'
C80)
(F7:1.
4)0
FORMAT
'NU
;, N
11i',,i
:) . Fl:,1,
FI
FORMA'(
F90
F ORMAT (12)
1.1.5
WR1 TE ( 1. 115')
FORkMAT ( 'NRICE PAIW
11 0
120
: "HARES::')
:.ft
hJS,
i
)L
MREA
F T 80) P10
FCORMAT 'OP 'ION') (
y 1205)
WR ITE ( i.
O F:TIONS
C P1
FORMAT :il NUMBER F0N
'
.1
[N2'80"
SZREA-(:
)
:100) J.2
REA():1.,
y. 115 )
WR I TIE ( vI
v 80) P"2
READ (:1.
2000)
WRITE (:1.,
,5'X y"1 ESU LIPTS OF CALL OPTION-STOCK SPREAT)S")
FORMAT(1: -5
2000
0 ..
WR I TE (, , 1f'....0
,' 8X,
.7X'
"
:.'
7 X OPTI'ON"
Fl
ORMAT (/ iX 'STOCK ' .2X, STOCK
51.
5X
,1
OtTI
0O" :5X ' :OFIT'
X
CtCOMM I $0 1ONS ',p5.X Y"X'STOCK":: 6X
y6
6X 'SHORT " )
L 0 NG:.3'
C.,"
128
I ey
F CY:1*1\i:
'
1f.)*'
1HXI
CPI,*I
· X
C
I..;;
i
I
F-I
'IS
to ".A I. 13 I :
13 ) 1
...
..!.''
;
BS it
11
'5,
)
3
0
C;IF.3yIXý-f.'-i
, A,
200 CCj.11
",:1:
N,F..
R
PEER
14T
Df
%
2
129
SUBROUTINE 10
)
T'D: JD1 R, EN T'E
1NOF'
0P V0L...
EPS ( NiAME p0
t
SUBROUT RINE
SIME'ENS
S'1
tiNAc (,)vE(10.
) NOP(10))0 OP(:.01
iS)0 VO L..(10)
DOUB1L.E
1
PPF'RECISION
R,
S 1Ci3 NAE CD:1 C::.2
.STOCK
'
iI, N 2
'
: NREAL i'..
WRI TE (1: 10)
10
20
40
50
w I TE :i.50)
F::
ORMAT
("
. .OWERPRICE"f!.
35
TD(10"v 21 )
FORMA' ('STOC : )
!:READ,(' 1 20) STOCK
)iA35
F OR MAf"T' (
.1O 30 I'. v :.'ENT
E1 STOCK)
FCGO
!
IF' (NAME(I4).
C:ONTINUE: N
WRITE:T F (1. v 35)
FORMAT('STOCK
' SSET') NOT IN IA'A
GO TO) :1.0
HN= I
30
130
TO( 40
)
RE1AF:(
( :t1.
55) 1:1.
W 1RITE (151 )
. PER
51
FOI:RMAT
55
FORMAT (W13)
FPRIE:2
V:: V .. (H )
2) )
I ( ( ,D
(T)D
C.2:= CA :I..
C: 3=CAI..D (Ti(1H1,3 ) )
.R 1 TE
I
' . 60 )
60
70
FORMAT( STOCK" )
WRFI2TE (1. 70 )
FORAT(
( ' SNUMBER OF S- ARES
REAID( :1., 80) N 1.
80
90
FORMiAT (F 7
)
F OR AT ( T•)
1.00
FORMAT (12)
WRI TE ( 1 , 115)
F ORMATM'T'(
1.15
PAI
' )
)
READ (1
:. 80
P:1.
;)
:1210:)
W RI TE (
:110
F:ORMAT '
::TION'
120
FORi"AT('NUMBER
1REA(i (:.
1 80) N2
90)
WRITE (1 .'
OF
)1PT1ONS
)
READ( 1 :1.
t00) J2
Wi IT E ( , :1.
115
:1
. )
SREA
1(
(.,
i 80) P2
2000 )
WRITE (1. 2v
2000
150
F
S )
1 N-- ST'OCK SPREA3
'.
F) PUTf G PTIO
F ORMAT( 55 X, 'RESULTS
WRi TE(1 :1.
1 50)
10ON'
hSX S
.17X 'OPT
0
:12X.
X 'STOCK':
F ORMAT( I/:1 X 'STOCK:'
IT''
X,
PROFf
"
C' COMMISSt
i
ONS0' , 5X. ' ST OCK' 7X "' OTION ' :4 X '.
'
)
'
7X(
'
PUT
C " ST'OCK
131
WRITE (I 160)
160.
FORMAf (1X 'PRICE
1
v53X
AI.D',
C(,6X / Ci--IA NG E' 6X 'C:HAN GE', 6X ' DELTA'
DO 200 X=
,:iI.
12
T:= ( TD(H .J2)--JD )13615
CA1..1.. VA...P(V:: Y OP(H
y ' J2)'T'R 'Vy2,D2:.G2)
CD22=CAL.. D (TD ( 1'....2 ) )
Aeo 1=ABS
(N1:) .7 5 +"ABS (N2 ) •:8
Ef 5
B= (N4I tA
(X--P1)
BB= (101N2)2
C=:=13 +13 81-- A
(V2-)P2)
D 1 :::
ABS
'
(Nl
i )
Us? p It p 11I3
y 111. E12
D2=ir-ABS C(N2)*BD2*iO00
210
200
FORMAT(
C):'
(F7 3 6
6X
7'
.X
HAFS/,E: ' 6
6rX
CF73 :XA3
,r
XF7 3: 4X6(F .23X))
CONTINUE
.
'
ELTA' ,)
132
SUBROUTINE 11
SUBROUT
I NE RESS ( NAME 0P ! NOF v VO
T i.' , J r' R ENT)
DIMENSIO
:
N NA ME ( 10 ) yNOP(10):. OP(0 1, :18) VOL( 10)
SOUBLE
13
PR ECISION
11
NAMEE CD i1: CD2 C:v :ST OCK
13
IREA l.
N1
F ,'N2Y
R j. n I. N :1.v N2
WRI TE ( 1. 10 )
1.0
FORMAT (, STOCK'1:
20
READ ( 1, 20 ) S"I"OCK
F3
ORMA
i
( 3
.S0
30
35
40
50
30
55
I::
.'ENT
IF" ( NAP!E ( '1
CONTINUE
I). ElQ
STOCK
C
) C(GO"1 TO 4,)
WR 3:IT"E ( :35)
FTORMAT( STOCK NOT I1N DATA S ET')
3GO TO 10
:I::
TH=
FORMAT( LOWER PRICE :'.")
READ (1 .,,55) 11
R
I::'CE )
WRITTE (1
51
)
151 )
" C"
21:
EE
FORMAT ( 'UPPER F1::p11,
FORMAT:1.
(5 )
N::NOi'(H)
CDI::::
jCALD
k)1::X:r:V0
L1...
H( (TD
T' (H,5) I2F)) ))
CD2=CALD
(TD(HT
CD:3= CALS (T..D (H 3 ))
60
WRI:: TE (1.v6
:..
v,0 ) OFIv ON'
FORMAT('CAL..
WR ITE (1 6
70)
70
80
90
1.00
115
FOR7ATC
T
'N IUiBER
R
REA: ) ( :1.80()
N1
FORMAT (F7O .4-)
WR:i:TE (1 90)
FORt:
If 'Y' i :If t)
00) J1:l
READ (i
FO
C)RM AT (I2)
WRITE (11 115)
FORPAT('PRICE PAI.O:'1
REA )( :1. 80) PR.
WR :1 TE ( :1 11 0 )
110
FORMA
G I, A' (T'"
WRITE (1 :1.20)
120
F:pORMAT( 'NUMBERNf
SREA (1 S 0 ) N2
WR :I
TE( :1( 90)
OPTI OT'
")
READ(i,100) ,.2
WFPITE (. : :115)
2000
1.50
READ(!180) P2
WRITE (:1.-, 2 000 )
F ORMAT (5i5X , RES1LTS OF PUT--DCAL.L SPREADS')
WRITE (1 1.50)
FORv,
0 mAT (/ :1X., 'S. OC',,
1 '. :i2X.2 'CAL.l...'
: i9X
X ' P19
UX
.':
133
'TD(10
:1.8)
134
C C 0MM
160
SSI
S 1• NS
.i.
13 X ,' C; ..
7X.
PUT
SX
P PR
O' F"1 T
X:;..I
WR
TE(1,160)
1
'Ai f '
'A
X
:1.X , -'PRICE',5 X
FORMAT (I
17X"
.
' EL TA'i X 'D:iLEL..TA''
A
6XX 'CHANGE'
C6X 'CHUANGE' BX
DO 200 X= 1112
Y=X
T:::: ( Tm ( y-,Ji) - Ji
.. /365
G:1)
.
O:: ( H J ) T R V:i.1
CAL.... AL (V Y :
T=*(TDD H J2 )-iJD ) 365
, '(H, J2)
A...( ( Y'YOy0
CALL...
TY ,'.7V2, 2D
G2 )
A= AABS (N1.1-N2)
2 Ai.7,
(V -F:1.
B=:::
( 100 N1)
BB:(100N2) (P2-v2)
i:
00
. ":::=
ABS (N1) *0 8:I.*
2:=1ABS (N2)I CD2i100
1J ) )
CD12::=CAL...D (TD ( .'
Ci:2=CAL.:(T.D (H, J2) .)
WRIiE (1,210) X, N i:CL :.1OP:: H . :
. N2
J2)
C 2, OP H ..
210
FORMAT(F7.3,.3X 2(F7..3. :X A3,1X,.F7.339 4X) V6(F:" 2 3X))9
200
CON TIN UE
I:f
E T U RI
PN
P4. D'
EE
SHRJT
135
SUPPORT FUNCTION PROGRAMS
.JSUBROUTi:NE
Ai...
V X C
W•y
ON ,N 1ND
REAL.
13 2:: k)
t
2(S
OA= AL..OG (X/C,)
DI=(AL.G(XiC)+((R'"+.2/2i2)*T))/SfRT(3.2,T)
i2 =X 1 -- .T
JS.Q
A( 2 T 2:
.T)
:XII N.t:::
,2I=3 f
V 8
T
W= X*N(D'.I)- CI:EXF:P(- RT),'fNCD3i.2)
N :TO
N ( :e 1 )
IR
L0 PN.
EEpDT
136
137
FUNCTION N(X)
lit)
(RJ ?1v1
v.
j
P==.47047
Ai= .4480242
A2:)=-
05
to 2
o8798
Y=:::
X iS.QRT (2,)
IF (Y) 100200,200
10 0
200
Y= -Y
:1. :1. :1.+'F,
Z:::: (
AI:M )+(A2
N= ( Z+) /2
M=
:: 1
..(P. Y) )
Z= 1 - C t(AIM.o
( )+ )(A2,k
M:toK2) f A3,A M
N= (1+fiZ )/2
ND
.4.
Mi2)+(A3 i'
' i*M )
3)
EXP(-Y
2))-
,EXP(-Y**2)
FUNCTION C'ALJ (X)
.OOUBLE PRECISION CALDO
i
C!ON(1.2)
iOUJBLE F'RECISION
REAL MON
'AUG " , JULY' ,
0C
SEPT :..
EC - NOV
E
V , OCT
fATA MON/
JAN'/
. MAR' , 'FEB,
C "JUNE
MAY , 'APR'
Y:='X--36-i.:
I1 (Y. LT
IF (Y. LT.
IF (Y. iT.
IF (Y. LT.
IF (Y. LT.
lF (Y
LU .
IF (Y
LT.
IF ,Y. L.T.
IF (Y1. T.
I:F (.Yo L.T.
1 F ( YA LT...
IF (YA. LT.,
RETURN
j:;E
Ni:'D
365) CAL.D ::::HON(1)
334) CALi.i=N0(2)
304) CAL. P!1ON( 3)
273.- CALD=O)
:1
(.I)
243) CALD.":
MON(.5)
212) CA..)t.=
j
ON(6)
:1.8:)
CAL.=.ION(7)
15:i..
) CALD:: MON(3 )
10) CAL.-:MON(9)
i0) CALD.=)HON(10.)
59) CAI...Ti.MON ( :1.
1
:)
3:1.)CAL):= ON( !2)
138
SUBROUTINE
REA1...
A
N1y NDI
VALF" V y XC
Y
T Ry W.y,
G)
B=SQRT
(S2T)
S= A...
(G
X/C )
1 .= I(AL. OG (X/.) ( (RC+S2i/2) T) )/S3RTf (S24T)
)2 :1)D 1 -- SQFRT ( S2T )
P'I'=3 . 142
G= EXP -..
(I(1 **2)i2)/(X*VISQRT(2*PI'Af*T))
W= - Xi* ( DI) .
N,
=N(DD1
.)
RETURN
EN4D
.EXP
-: AT )
(-
2.
139
6.0
BIBLIOGRAPHY
1. Black, Fischer, "Fact and Fantasy In the Use of Options," reprint
from Financial Analysts Journal, July/August, 1975
2.
, On Options,
"What to do When The Puts Come," February 9, 1976
3.
"All That's Wrong With The Black-Scholes Model," February 23,
1976
4.
"The Long and Short of Options Trading," March 8, 1976
5.
"The Long and Short of Options Trading (Conclusion),"
March 22, 1976
6.
"What Happens to Stocks When Options Start Trading," April
19, 1976
7.
"How We Came Up With the Option Formula," June 21, 1976
8.
"How We Came Up With the Option Formula (Conclusion),"
July 5, 1976
9.
, Scholes, Myron, "The Pricing of Options and Corporate
Liabilities," Modern Developments in Financial Management,
Stewart C. Myers ed., Praeger Publishers, NY, 1976, pp 229-246
10.
Gallager, Michael C., "Options Trading Markets Offer Unique Tax
Savings Opportunities for Investors," The Journal of Taxation
(JTAX), May 1975, p. 258
11.
Gross, LeRoy, The Stockbroker's Guide to Put and Call Option
Strategies, New York Institute of Finance, NY, 1974
12.
Herskowitz, Elliot, "How Tax Considerations Enhance Opportunities
In Listed Options," JTAX, October 1975, p. 212
13.
Malkiel, Burton G., Quandt, Richard E., Strategies and Rational
Decisions in the Securities Options Market, MIT Press,
Cambridge, MA, 1969
14.
Miller, Jarrott T., The Long and Short of Hedging, Henry Regnery
Co., Chicago, 1973
15.
The Options Clearing House, Prospectus in Traded Put and Call
Options, October 29, 1976
16.
Rosen, Lawrence R., How to Trade Put and Call Options, Dow JonesIrwin, Inc., Homewood, IL, 1974
140
141
17.
Shepherd, William G., "The New Game in Options," Business Week,
February 3, 1975, p. 53
18.
Soll, Lauren S., "The Income Tax Ramifications of Securities Options,"
The Tax Adviser, August 1974, p. 484
19.
Tax Treatment of Options to Buy and Sell Stock, Securities or
Commodities, Hearing Before the House Committee on Ways
and Means on HR 12224, April 5, 1976
20.
Tax Treatment of Transactions in Options, Report prepared for the
use of the House Committee on Ways and Means, April 13, 1976