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6 Going Beyond the Basic Definitions: Infrastructure to Understand Puts and Calls

# 6 Going Beyond the Basic Definitions: Infrastructure to Understand Puts and Calls

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338

OPTIONS

In the next chapters, we will give many applications of these techniques.
For example, we can break down European call and put options into their fundamental economic components. This goes a long way towards answering the
following basic question,
Question: What do put and call options provide to investors, in terms of
proﬁt (net of investment cost) or payoff proﬁles, that are not immediately
or easily available to investors otherwise?
The technology we will develop is simple, yet powerful when pushed to
its logical conclusion. It is the systematic use of payoff diagrams and proﬁt
diagrams to analyze option strategies.
1. Payoff diagrams describe the dollar cash ﬂow (ignoring the investment cost)
at the option’s expiration date as a function of the underlying stock price
at expiration.
2. Profit diagrams describe the dollar proﬁts (net of the investment’s cost) at
the option’s expiration as a function of the underlying stock price at
expiration. We will pursue this and give many examples in Chapter 10.
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CONCEPT CHECK 8

a. Draw the payoff diagram for a call’s Intrinsic Value function in deﬁnition
10 in section 9.3.
b. Draw the payoff diagram for a put’s Intrinsic Value function in deﬁnition
11 in section 9.3.
Note that the stock’s price at expiration of the option, ST, is on the horizontal
axis and the payoff is on the vertical axis. Thus the payoff diagram gives the
payoff to the intrinsic value function, in this case, as a function of the terminal
stock price.

MAX[ST–E,0]

ST

INTRODUCTION TO OPTIONS MARKETS

339

9.7 IDENTIFYING LONG AND SHORT POSITIONS IN AN
UNDERLYING
We brieﬂy discussed identifying long and short positions in Chapter 6, Figure
6.1.
Here we will extend the discussion. Derivative securities are all based upon
some underlying asset or scenario. The underlying is the ﬁnancial asset or the
scenario upon which the option is based. There are two kinds of positions
one can take in an underlying, long or short.
To be long something means that we worry about selling it at some point
in the future and therefore the worry is that the price declines. For example,
we could be long 100 shares of IBM Corp. common stock.
Don’t think of long as being equivalent to outright ownership, because long
positions can be more sophisticated. A long position could just involve the
anticipation of selling something in the future. One need not own it today.
For example, suppose that a privately held ﬁrm is planning an IPO, and
therefore it has an anticipation of selling stock at some future date. Therefore,
it is long the IPO and it worries about a price decline, just like an investor
who holds publicly traded common stock.
Investigating the subtleties of this kind of distinction is important for
applications. Developing an extended notion of long (and short) will reveal
that there are many more long (and short) positions lurking out there. Further,
they may very well require the hedge protection that derivatives, ﬁnancial
and real-asset based, can provide.
We have deﬁned a current long position in some underlying as any position
that anticipates selling the underlying at some future point in time, and therefore
is concerned about potential price declines. That is, we work backwards from
the future to discover the current position. If the future anticipated transaction
is to sell, then one is long today (see Figure 9.4).
Short positions are harder to understand, and to identify, but no less
important than long positions. In a classical short sale of common stock, one
borrows the stock from a broker, sells it immediately and then incurs the future
obligation to cover the short sale.
That is, one must buy back the stock at some later time and then return it
to the broker. Here the anticipated future transaction is buying. As an anticipatory
One could deﬁne a short position in an underlying as any position that
anticipates a buy transaction in the future. Thus, short doesn’t mean selling

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OPTIONS

something one owns today. Rather, one can sell something one doesn’t own!
Then one incurs the obligation to buy back the underlying.
An instructive example is the individual who is naked short a call option. The
individual never owned the option, but they got payment from the option
buyer for the option. In return, they are potentially liable for buying the
underlying stock and delivering it to the buyer in exchange for the exercise
price.
Does this example illustrate our deﬁnition of short? First, the option short
is naked short, he owns no underlying stock. Second, the option short worries
thereby forcing the short to go into the market and buy the stock. Yes, it ﬁts.
Next, consider a short real asset example. It is well known that current energy
sources are going to be unable to meet world energy demand 50 years from
today. At that time, we will have to buy energy at possibly prohibitive prices.
That is, we are now in an anticipated buy position, which is the hallmark of
a current short energy position.
Understanding this is the key to hedging the short position. One hedge
vehicle is energy R&D, a real-asset call option. This shows that a government
agency responsible for energy policy must understand the notion of a short
option position.
The point is that it is important to recognize the many implicit short positions
out there and the need to hedge the risk of prices rising. Figure 9.4 illustrates
how to determine whether one is long or short the underlying.
FIGURE 9.4 Long vs. Short Positions

Anticipated Sell,
therefore Long
P0

Price Worry
P1

Time 0, Now

Time 1, The Future

therefore Short
P0

Price Worry
P1

Time 0, Now

Time 1, The Future

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INTRODUCTION TO OPTIONS MARKETS

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341

KEY CONCEPTS

1.
2.
3.
4.
5.
6.

Options and Option Scenarios.
Framework For Learning Options.
Option Deﬁnitions For Plain Vanilla Calls and Puts.
A Basic American Call (Put) Option Pricing Model.
Going Beyond the Basic Deﬁnitions, Infrastructure to Understand Puts
and Calls.
7. Identifying Long and Short Positions in an Underlying with Examples.
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END OF CHAPTER EXERCISES FOR CHAPTER 9

1. (Price quotes)
Go to CBOE.com and update the Merck example in Table 9.2 by
a.
b.
c.
d.

What has happened to the underlier (Merck stock) since 2008?
What are the exercise prices for some of Merck’s liquid options?
Are the increments in Merck’s option prices still \$2.50?
What option cycle does Merck belong to?

2. (Embedded options)
a. List some embedded options that investors are likely to encounter.
b. List some real asset options that ﬁnancial managers are likely to
encounter.
c. Name an option that incorporates futures contracts. Where are these
3. (American option pricing model)
a. State the American option pricing model of section 9.4.
b. What is the methodology used to prove the result of a.?
c. Why doesn’t the methodology in b. immediately apply to European
options?
d. ‘An in-the-money American option is a combination of IV and TP ’. Explain.
e. ‘An out-of-the-money American option is pure TP ’. Explain.
f. What happens to TP as time to maturity, ␶, approaches zero?

342

OPTIONS

g. What happens to an American option’s market premium as ␶
approaches zero?
h. What happens to IVt as ␶ approaches zero?
4. (Long and short positions in real asset options)
Consider the timely topics, health care and health insurance. Most people
consider the status of their own health and its health care.
a. What position do most people have in their own health, long or short?
b. What are some natural hedges for the concern in a.?
c. What is a ﬁnancial hedge for the concern in a.?
d. What position would you take in c. in executing the hedge?
e. How does the issue of rapidly rising health care costs ﬁt into this
discussion?
f. How does the issue of rising health care insurance premiums ﬁt into
this discussion?
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SELECTED CONCEPT CHECK SOLUTIONS

Concept Check 1
a. According to Figure 9.5, under ‘Settlement of Option Exercise’, the actual
shares of the underlying stock would have to be delivered.
b. The option seller always does the delivery.
Concept Check 2
b. See Figure 9.6, under ‘physical settlement’.
c. The option seller always does the delivery.
Concept Check 6
a. The option buyer buys the rights, and only the rights, that the option
confers on the option’s owner.
b. The seller sells those rights, and only those rights, in return for the

INTRODUCTION TO OPTIONS MARKETS

343

FIGURE 9.5 CBOE Equity Option Specifications
Equity Options
Exchange traded equity options are “physical delivery” options.This means that there is a physical delivery of the
underlying stock to or from your brokerage account if the option is exercised.The owner of an equity option can
exercise the contract at any time prior to the exercise deadline set by the investor's brokerage firm. Generally this
deadline occurs on the option's last day of trading.The expiration date for equity options is the Saturday immediately
following the third Friday of the expiration month until February 15, 2015. On and after February 15, 2015, the
expiration date will be the third Friday of the expiration month. If this third Friday happens to be an exchange holiday,
then the last day is the third Thursday of the month. Check with your brokerage firm about its procedures and deadlines
for instruction to exercise any equity options. After the option's expiration date, the equity option will cease to exist.
For additional information on equity options, visit the Equity Option Strategies section of the web site.

Equity Options Product Specifications
Symbol:
The option symbols are the same as for the underlying equity security.Visit the CBOE Symbol Directory for
specific symbols.
Underlying:
Generally, 100 shares of the underlying equity security.
Strike Price Intervals:
Generally, 2 1/2 points when the strike price is between \$5 and \$25, 5 points when the strike price is between
\$25 and \$200, and 10 points when the strike price is over \$200. Strikes are adjusted for splits, re-capitalizations, etc.
Strike (Exercise) Prices:
In-, at- and out-of-the-money strike prices are initially listed. New series are generally added when the
underlying trades through the highest or lowest strike price available.
Stated in decimals. One point equals \$100. Minimum tick for options trading below 3 is .05 and for all other series, .10.
Expiration Date:
Saturday immediately following the third Friday of the expiration month until February 15, 2015. On and after
February 15, 2015, the expiration date will be the third Friday of the expiration month.
Expiration Months:
Two near-term months plus two additional months from the January, February or March quarterly cycles.
Exercise Style:
American-Equity options generally may be exercised on any business day before the expiration date.
Settlement of Option Exercise:
Exercise notices properly tendered on any business day will result in delivery of the underlying stock on the third
Position and Exercise Limits:
Limits vary according to the number of outstanding shares and past six-month trading volume of the underlying
stock. The largest in capitalization and most frequently traded stocks have an option position limit of 250,000
contracts (with adjustments for splits, re-capitalizations, etc.) on the same side of the market; smaller capitalization
stocks have position limits of 200,000, 75,000, 50,000 or 25,000 contracts (with adjustments for splits,
re-capitalizations, etc.) on the same side of the market. The number of contracts on the same side of the market
that may be exercised within any five consecutive business days is equal to the position limit. Equity option
positions must be aggregated with equity LEAPS positions on the same underlying for position and exercise limit
purposes. Exemptions may be available for certain qualified hedging strategies.
Reporting Requirements:
Please refer to Exchange Rule 4.13 for information pertaining to reporting requirements for positions in excess of
200 contracts.
Margin:
Purchases of puts or calls with 9 months or less until expiration must be paid for in full. Writers of uncovered puts
or calls must deposit/maintain 100% of the option proceeds* plus 20% of the aggregate contract value (current
equity price x \$100) minus the amount by which the option is out-of-the-money, if any, subject to a minimum for
calls of option proceeds* plus 10% of the aggregate contract value and a minimum for puts of option proceeds*
plus 10% of the aggregate exercise price amount. (*For calculating maintenance margin, use option current market
value instead of option proceeds.) Additional margin may be required pursuant to Exchange Rule 12.10.
Trading in equity options will ordinarily cease on the business day (usually a Friday) preceding the expiration date.
8:30 a.m. – 3:00 p.m. Central Time (Chicago time).

Reprinted with permission from CBOE.com, 2014.

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OPTIONS

FIGURE 9.6 CBOE Mini Equity Option Specifications

CBOE Mini Options
New CBOE Mini options with physical settlement began trading on March 18, 2013.The new Mini options represent
a deliverable of 10 shares of an underlying security, whereas standard equity options represent a deliverable of 100 shares.
The options symbol for the new Mini options will be the underlying security symbol followed by the number 7 (see
examples in the table below).
In addition, please note that the cash-settled Mini-SPX options (ticker XSP (1/10th the size of SPX, with a multiplier
of 100)) have been listed since 2006 www.cboe.com/XSP.
The Mini options are designed to provide added investment flexibility. For example, for an investor who holds 10 to 90
shares of AAPL or GOOG stock, the new mini options could provide that investor with tools that have the potential to
be more efficient and tailored for strategies such as covered call writing or hedging.

SECURITY

MINI OPTIONS UNDERLYING MINI OPTIONS NOTIONAL VALUE COVERED BY
SYMBOL*
SECURITY
MULTIPLIER
MINI OPTION (IF OPTION WERE
SYMBOL
OFFERED ON FEBRUARY 8, 2013,
APPROXIMATE)

Amazon.com Inc.

AMZN7

AMZN

10

\$2,620 (10 × \$262)

GOGL7

GOOG

10

\$7,850 (10 × \$785)

SPDR Gold Trust

GLD7

GLD

10

\$1,620 (10 × \$162)

SPDR S&P 500

SPY7

SPY

10

\$1,520 (10 × \$152)

Mini options with physical settlement; launched in March 2013–

Mini options with cash settlement; launched in 2005–
Mini-SPX Index
(1/10 the size
of SPX)

XSP

XSP

100

\$15,200 (100 × \$152)

* The options symbol for the new Mini options usually is the underlying security symbol followed by the number 7
(see examples in the table above). However, if there is a corporate action, the symbol could be further modified and
appended with an 8 such as “XYZ8.” Investors can access the options chain for standard and Mini-sized options at the
CBOE Delayed Quotes web page. (Reprinted with permission from the CBOE.com, 2014)

Reprinted with permission from CBOE.com, 2014.

CHAPTER 10

PART 1

10.1 Profit Diagrams

346

10.2 Eight Basic (Naked) Strategies Using the Underlying,
European Puts and Calls, and Riskless, Zero-Coupon
Bonds

347

10.2.1 Strategy 1. Long the Underlying

347

10.2.2 Strategy 2. Short the Underlying

349

10.2.3 Strategy 3. Long a European Call Option on
the Underlying

351

10.2.4 Strategy 4. Short a European Call Option on
the Underlying

355

10.2.5 Strategy 5. Long a European Put Option on
the Underlying

357

10.2.6 Strategy 6. Short a European Put Option on
the Underlying

359

10.2.7 Strategy 7. Long a Zero-Coupon Riskless Bond
and Hold it to Maturity
360
10.2.8 Strategy 8. Short a Zero-Coupon Riskless Bond
and Hold it to Maturity
362

In this chapter, we will ﬁrst deﬁne and then map out the proﬁt diagrams for
the eight basic strategies one can employ using long and short positions in an
underlying, in calls and puts, and in riskless zero-coupon bonds. These are
naked strategies which means that they are fully unhedged and subject to price
volatility.
When considering long and short positions in the underlying, obviously
they need not have expiration dates. However, we are correlating option payoffs

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OPTIONS

with underlying payoffs, and so we use a ﬁnite time horizon and the option’s
expiration date to make the connection.
This chapter is preparation for Chapter 11, which is much more abstract.
It takes us beyond the ﬁrst level of understanding options which was discussed
in Chapter 9.
10.1 PROFIT DIAGRAMS
Before we get into mapping these strategies by proﬁt diagrams, we want to
show where proﬁt diagrams come from. Consider the following historical price
chart of Merck stock over a few months of a historically turbulent period
(source: www.nasdaq.com).
FIGURE 10.1 Merck Stock Price (11/30/2007 through 2/29/2008).

(Actual path in blue, simulated paths in red and black)
03/01/08

3 Month

\$64.00
62.00
60.00
58.00
56.00
54.00
52.00
50.00
48.00
46.00
44.00
42.00
December

2008

February

Volume
71.35 M
59.46
47.56
35.67
23.78
11.89
December

2008

February

Note that the Merck stock price was \$59.36 on 11/30/2007 and it was
\$44.30 on 2/29/2008. Figure 10.1 gives the actual price path from December
2007 through February 2008. The blue path is the actual data. If we were
standing at the end of November 2007, Merck was trading at \$59.36, and a

347

3-month in-the-money Feb. 2008 call option on Merck, Inc. with exercise
price \$50 could be purchased. The horizontal black line indicates the exercise
price of \$50, which is roughly situated at \$50.
Of course, at that date we would not know whether this 50 Feb. call would
end in the money or not. If we could roll forward time, as in the chart, we
would see that the 50 Feb. call remained in-the-money for a signiﬁcant amount
of time (through the 3rd week in January 2008), but then the tide turned.
There was a precipitous drop in the stock price through the entire month of
February 2008. The option ended up out of the money on the expiration
date on February 16, 2008.
Now we move back to November 30, 2007, and we run the ticker tape
forward, imagining other price paths for Merck. There are an infinite number
of potential price paths. Two such paths, one in black and the other in red,
indicate two other possibilities. If Merck followed the red path, the option
would have ended up roughly at the money and if Merck had followed the
black path it would have ended in the money also.
Proﬁt (subtracts current cost) and payoff (does not subtract current cost)
diagrams indicate the proﬁts and payoffs that would have resulted from a given
strategy based on all the possibilities for the stock price at expiration.
10.2 EIGHT BASIC (NAKED) STRATEGIES USING THE
UNDERLYING, EUROPEAN PUTS AND CALLS, AND
RISKLESS, ZERO-COUPON BONDS
Table 10.1 lists the eight basic strategies which we will analyze in sections
10.2.1–10.2.8. All options are European.
10.2.1 Strategy 1. Long the Underlying
Let the current time be t, and let T be the expiration date of the call option.
Only the current stock price, St, is known today. The stock price at expiration,
ST, is unknown and depends on what happens between today and the option’s
expiration date. That is, ST is a random variable, which could be written as
ST=ST(␻).
The proﬁt ﬁgure, ␲T , from long an underlying stock today is also a random
variable equal to the price difference. If the price goes up, you make money
and if the price goes down, you lose money,

␲T(␻)=ST (␻)–St

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OPTIONS

TABLE 10.1

The Eight Basic Naked Strategies

STRATEGY 1

Go long the underlying

STRATEGY 2

Go short the underlying

STRATEGY 3

Go long a call option on the underlying

STRATEGY 4

Go short a call on the underlying

STRATEGY 5

Go long a put option on the underlying

STRATEGY 6

Go short a put option on the underlying

STRATEGY 7

Go long a riskless bond with face value equal to the option’s exercise
price

STRATEGY 8

Go short a riskless bond with face value equal to the option’s exercise
price

For example, if one buys a stock today for St=\$50 and it ends up at time T
(say 3 months’ later) at ST=\$60, then one’s proﬁt is \$10=\$60–\$50. If the stock
price ends up at time T at ST=\$40, then one’s proﬁt is –\$10=\$40–\$50.
The following graphic, Figure 10.2, displays the proﬁts from a long position
in an underlying as a function of the underlying price at expiration, ST(␻).
The underlying price starts at time t at its current value St .
FIGURE 10.2 Strategy 1: Profits from a Long Position in an

Underlying
πT = ST – St

0
St

–St

Underlying Price
at Expiration: ST