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4 A Basic American Call (Put) Option Pricing Model

# 4 A Basic American Call (Put) Option Pricing Model

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INTRODUCTION TO OPTIONS MARKETS

333

Some might think that there is another right which is to sell the option to
someone else. This could be called the liquidity option. From a valuation
perspective, liquidity is not a separate right distinct from 1. and 2.
On the other hand, liquid options could command a liquidity premium
and the bid–asked spread could be lower for liquid vs. illiquid options. This
is more of a transactions costs phenomenon, as opposed to an embedded rights
feature.
The ﬁrst right is worth the option’s intrinsic value, and the second right is
currently worth the option’s time premium.
Denote the current (time t) value of the American call option by CtA and
let PtA denote the American put option price at time t. The basic American
option pricing model says:
CtA=(Call Intrinsic Value)t+(Call Time Premium at time)t
=IV(CtA)+TP(CtA)
PtA=(Put Intrinsic Value)t+(Put Time Premium at time)t
=IV(PtA)+TP(PtA).
The idea behind this American option pricing model is simply that, if the
ﬁnancial instrument we are attempting to value—an American call (put) option
in this case—has an implicit or explicit payoff associated to it, then its price
must incorporate the PV of those payoffs. Otherwise, there would be
mispricing and an arbitrage opportunity.
That’s the idea, but how do you prove our American option pricing model?
The proof is ‘by deﬁnition’. One simply defines the time premium, TP(CtA)
in the case of calls or, TP(PtA) in the case of puts, as the difference between
the option’s premium and intrinsic value. That is, define the time premia by,
TP(CtA)≡CtA–IV(CtA),
TP(PtA)≡PtA–IV(PtA).
This is a neat trick, but only a trick. This American option pricing model
is interesting because it uses only the basic deﬁnitions. However, it is not wellsuited to European options because, for a European option, ordinary intrinsic
value is not immediately realizable, since you can’t early exercise a European
option.
We will discuss an analogous model for European put and call options in
Chapter 11, in the context of rational option pricing.

334

OPTIONS

At this point, we can immediately apply this American option pricing model
to examine market option prices. Note that CBOE individual equity options
are all American (see concept check 1). Therefore, it is legitimate to apply our
American option pricing model.
The most important stock option exchange in the USA is the Chicago Board
Options Exchange (www.cboe.com). One can obtain the full speciﬁcations
of equity call options traded on the CBOE by going to the site and hitting
‘products’, then ‘equity options’ (see concept check 1).
A standard stock option is the right, but not the obligation, to purchase
100 shares of a publicly traded common stock at a ﬁxed price, the strike price,
at or before a ﬁxed date, the expiration date.
Let’s consider the nearby call options for Merck Inc. To obtain the
following list of price quotes go to www.cboe.com, select ‘products’, then ‘options
on single stocks and Exchange Traded Products’, hit ‘quotes and data’, then hit ‘delayed
quotes classic’ and insert the symbol Mrk. Then choose ‘list near term at-the-money
options & Weeklys if avail’.
The quote date and time for this particular quote was Feb 22, 2008 @ 15:54
ET as indicated. Just around ﬁnancial crisis time. You are asked to do an update
in End of Chapter Exercise 1.
One sees, in Table 9.2, that numerous Merck options were traded. Here
we have taken the nearby options only. These are the options that mature in
the closest several months to the observation date. In this case, these are the
call and put options on Merck maturing in March and April of 2008.
Note that Merck has an option series based upon exercise prices changing in \$2.50 intervals around \$45. This procedure captures options that are
in-the-money and out-of-the-money. As the stock price changes, new
options will be issued with new exercise prices reﬂecting the current stock
price level.
Note that the underlying (Merck stock) is traded on the NYSE, not on the
CBOE. So, we have no guarantee that the stock price observation was made
at the same time as the observation for Merck options. This leads to the nonsimultaneous price quote problem.
Prices in alternative markets need to be compared at the same time. If you
think you have discovered an arbitrage opportunity by reading the Wall Street
Journal, or other summary data, the opportunity probably is not real. However,

1.65

0.60

0.20

5.60

2.95

1.35

0.70

08 Mar 45.00
(MRK CI-E)

08 Mar 47.50
(MRK CW-E)

08 Mar 50.00
(MRK CJ-E)

08 Apr 42.50
(MRK DV-E)

08 Apr 45.00
(MRK DI-E)

08 Apr 47.50
(MRK DW-E)

08 Apr 50.00
(MRK DJ-E)

Net

–0.20

–0.30

0.0

0.0

–0.10

–0.20

–0.40

0.0

0.70

1.40

2.55

4.10

0.15

0.55

1.60

3.40

Bid

Reprinted with permission from CBOE.com.

4.10

08 Mar 42.50
(MRK CV-E)

Calls

Last Sale

0.80

1.50

2.65

4.30

0.20

0.60

1.70

3.60

Vol
30

24

0

0

702

26

267

0

Open Int
15176

5905

2595

646

15476

7651

2355

786

08 Apr 50.00
(MRK PJ-E)

08 Apr 47.50
(MRK PW-E)

08 Apr 45.00
(MRK PI-E)

08 Apr 42.50
MRK PV-E)

08 Mar 50.00
(MRK OJ-E)

08 Mar 47.50
(MRK OW-E)

08 Mar 45.00
(MRK OI-E)

08 Mar 42.50
(MRK OV-E)

Puts

Last Sale
5.40

2.85

2.15

0.90

4.20

2.70

1.25

0.42

45.63 –0.50
Bid N/E Ask N/E Size N/ExN/E Vol 9241648

+1.19

0.0

+0.45

0.0

0.0

+0.50

+0.25

+0.12

Net

Feb 22, 2008 @ 15:54 ET

5.10

3.30

1.95

1.00

4.60

2.55

1.05

0.35

Bid

MRK (MERCK & CO INC)

5.20

3.40

2.05

1.10

4.80

2.65

1.15

0.45

Merck Options Price Quotes

3

0

52

0

0

45

604

260

Vol

TABLE 9.2

6078

6755

8611

3182

19148

17273

14110

2804

Open Int

336

TABLE 9.3

OPTIONS

A Particular Merck Option Price Quote

Calls
08 Mar 45.00 (MRK CI-E)

Last Sale

Net

Bid

Vol Open Int

1.65

–0.40

1.60

1.70

267

2355

Reprinted with permission from CBOE.com.

the stock price of \$45.63 on the observation date gives us a rough, but not
exact, idea of the stock price at the quote time.
Consider the price quote for the second call option in Table 9.3 which
was excerpted from Table 9.2. We will discuss each component of the basic
call option price quote in Table 9.3.
a. 08 Mar refers to the expiration date in March 2008.
b. The exercise price of the call is \$45.00.
c. C is the CBOE’s expiration month code for March calls (O is the code for
puts).
d. I is the strike price code for 45.00 (http://en.wikipedia.org/wiki/Option_
naming_conventionStrike_Price_Codes; accessed May 27, 2015).
e. E indicates that the option was traded on the CBOE Chicago Board
Options Exchange (www.cboe.com/delayedquote/quotehelp.aspx; accessed
May 27, 2015).
f. The Last Sale gives the asked price at which this option was last traded
prior to the observation time.
g. Net represents the change in the asked option price relative to the previous
price and sell options at the bid price.
h. The Bid and Asked prices were \$1.60 and \$1.70 for the call.
i. A standard equity option always refers to 100 shares of underlying Merck
stock, unless it is a mini-option (see concept check 2). In that case, it refers
to 10 shares of the underlying stock,
j. Volume represents the number of such options traded at the observation
date which is 267 (each based on 100 shares of Merck stock).
k. The Open Interest, 2355, represents the number of outstanding options
(counting one side only) in which investors have taken a buy position as
observed on the observation date and time.
This particular call option matures on the Saturday immediately following
the third Friday of the expiration month, March 22, 2008. This was the CBOE
rule until February 15, 2015. After that date, the expiration date became the
third such Friday.

INTRODUCTION TO OPTIONS MARKETS

337

This option entitles the option buyer (the long) to purchase 100 shares of
Merck common stock at \$45 per share (100*\$45=\$4500), on any business day
before the expiration date.
This call was in-the-money because the stock price was \$45.63 which is above
the exercise price by \$45.63–\$45.00=\$0.63. So, based on the asked option
price of \$1.70, the option’s market premium per share is \$1.70.
Since intrinsic value is \$0.63 per share, using our basic American option
pricing model, the rest of the option premium \$1.70–\$0.63=\$1.07 must be
That is, options investors are paying \$1.07 per share for the right to exercise
this option on any business day prior to the expiration date, except the current
date. That is, \$1.07 is the delayed exercise premium per share. The price they
are paying for the ability to exercise this option immediately is \$0.63 per share,
=\$0.63+\$1.07.
To get these corresponding values for the entire option, multiply the per share
amounts by 100 (for 100 shares).
The basic American option pricing model also explains why out-of-themoney options command a positive price prior to expiration. While out-ofthe-money options have zero intrinsic value, they will have time premia at
all times except at expiration. From this we see that the value of a call option
must equal its intrinsic value when it matures.
n

CONCEPT CHECK 7

a. Explain in detail the 08 Mar 42.50 (MRK OV-E) price quote in Table
9.2.

9.6 GOING BEYOND THE BASIC DEFINITIONS:
INFRASTRUCTURE TO UNDERSTAND PUTS
AND CALLS
In the ﬁeld of options and derivatives in general, the deﬁnitions are more or
less straightforward. Learning why these objects are important, recognizing
options in far-ﬂung scenarios, and knowing how to use derivative securities,
all require a knowledge deeper than the mere deﬁnitions.
Our goal will be to provide some techniques of ﬁnancial engineering
(synthesizing complex derivatives) for obtaining this deeper knowledge.

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.
n

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