Principles of Finance with Excel (Основы финансов c Excel)
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PFE Chapter 23: Facts about option prices page 12
23.4. Fact 4: Bound on an American put option price:
[
]
0
,0PMaxXS>−
Suppose you’re contemplating buying an American put on Microsoft stock. The stock’s
price today is S
0
= 63 and the option exercise price is X=70. Clearly the option should sell for at
least $7. If not, you could easily devise an arbitrage, as illustrated in the spreadsheet below:
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ABC
FACT 4: Lower bound on
American
put price
Microsoft stock price, 15 August 2001, S
0
63
Option exercise price, X 70
Option exercise time, T (in years) 1
Fact 4: Lower bound of American put: Max[X - S
0
, 0]
7 <-- =MAX(B4-B3,0)
Arbitrage
American put option price 3
Buy option -3
Buy stock now -63
Exercise put option immediately: deliver stock and get X
70
Immediate profit 4 <-- =SUM(B11:B13)
If the American put option is mispriced (that is, its price is less than $7), you can make
money by; buying the option, buying the stock, and exercising the option immediately. This
arbitrage profit will not exist if the option’s price is greater than $7.
23.5. Fact 5: Bounds on European put option prices
()
0
,0PMaxPVX S
>−
Fact 5 is the “put parallel” for Fact 1 about calls.
4
4
There’s a crucial difference in the parallel between Facts 1 and 5: Fact 1 applies to all calls, whether European or
American. Fact 5 applies only to European puts. Of course in both cases, the assumption is that the stock pays no
dividends before option maturity.
PFE Chapter 23: Facts about option prices page 13
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ABC
FACT 5: Lower bound on
European
put price
Microsoft stock price, 15 August 2001, S
0
63
Option exercise price, X 70
Option exercise time, T (in years) 1
Interest rate, r 10%
Lower bound on call price
Lower bound of American put: Max[X - S
0
, 0]
7 <-- =MAX(B4-B3,0)
Fact 4: put price > Max[PV(X) - S
0
,0]
0.6364 <-- =MAX(B4/(1+B6)^B5-B3,0)
This fact says that the price of a European put can actually be much lower than the price
of an American put. Look at the example above, in which we look at the price of a put option on
Microsoft stock with T = 1 and X = 70. If our put was an American put, then it couldn’t sell for
less than $7. On the other hand, a European put, which cannot be exercised until date T, can sell
for anything more than $0.6364.
Meaning: An arbitrage with European puts will exist only when the put price goes below
$0.6364. Here’s an example of an arbitrage when the put price is $0.5:
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ABC
Arbitrage proof
Put price at time 0 (today) 0.5
Actions at time 0 (today)
Buy the stock -63 <-- =-B3
Borrow PV(X) 63.64 <-- =B4/(1+B6)^B5
Buy a put -0.5 <-- =-B15
Total cash flow at time 0 0.14 <-- =SUM(B18:B20)
Cash flow at time T
S
T
, stock price at time T
50
Sell the stock 50 <-- =B24
Repay the loan -70 <-- =-B4
Put cash flow 20 <-- =MAX(B4-B24,0)
Total 0 <-- =SUM(B26:B28)
PFE Chapter 23: Facts about option prices page 14
23.6. Fact 6: You might find it optimal to early-exercise an American put on
a non-dividend paying stock
Recall that you’ll never find it optimal to early-exercise an American call on a non-
dividend paying stock. But this is not necessarily true for a put option. Here’s an example:
Suppose that you’re currently holding an option on PFE stock (a fictional company).
You bought the option some time ago, when PFE stock’s price was still healthy. However, at the
current date, the stock has taken a plunge and is selling for $1 per share. Your American put
option has an exercise price of X = 100 and expires in one year. The interest rate is 10%. If you
exercise the option now, you’ll have a net payoff of $99 ($100 minus the current value of the
stock of $1), which—if you invest it in bonds with an interest rate of 10%--will be
$99*1.10=$108.90 in one year. This is more than anyone would have if they waited for a year
until exercise.
Therefore any rational holder of an American put option will choose to early exercise the
option if the current stock price is very low.
23.7. Fact 7: Option prices are convex
To see the meaning of this somewhat opaque statement, we return to the Cisco example
from Chapter 21. Consider the three call options indicated below. Recall that a butterfly strategy
consists of buying one low-priced and one high-priced call and selling two medium-priced calls.
PFE Chapter 23: Facts about option prices page 15
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ABCDEF
August 7, 2001, CSCO closing price 19.26
Stated expiration date
Exercise
price, X
Call price Put price
A
ctual
expiration
date
Days to
maturity
Aug01 7.50 11.90 0.05 17 Aug01 10
Aug01 10.00 9.60 0.20 17 Aug01 10
Aug01 12.50 6.50 0.10 17 Aug01 10
Aug01 15.00 4.20 0.10 17 Aug01 10
Aug01 17.50 2.10 0.40 17 Aug01 10
Aug01 20.00 0.65 1.45 17 Aug01 10
Aug01 22.50 0.15 3.40 17 Aug01 10
Aug01 25.00 0.05 5.00 17 Aug01 10
Aug01 27.50 0.10 7.50 17 Aug01 10
Aug01 30.00 0.10 11.90 17 Aug01 10
Aug01 32.50 0.05 17 Aug01 10
Aug01 35.00 0.05 16.20 17 Sep01 41
Sep01 10.00 9.50 21 Sep01 45
Sep01 12.50 6.30 0.15 21 Sep01 45
Sep01 15.00 4.50 0.40 21 Sep01 45
Sep01 17.50 2.75 0.90 21 Sep01 45
Sep01 20.00 1.35 2.00 21 Sep01 45
Sep01 22.50 0.55 3.80 21 Sep01 45
Sep01 25.00 0.20 5.50 21 Sep01 45
Sep01 27.50 0.10 21 Sep01 45
CISCO OPTIONS, August 7, 2001
CLOSING PRICE ON CHICAGO
BOARD OF OPTIONS EXCHANGE
Suppose that the call option prices for Cisco were different from those actually seen in the
market. In the example below, we show how our butterfly would have looked had the X = 15
call been priced at $2.50 instead of $5.00:
PFE Chapter 23: Facts about option prices page 16
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ABCDEFGHI
WHEN DOES A BUTTERFLY INDICATE AN ARBITRAGE OPPORTUNITY?
Strategy: Buy 1 October 15 Call, Write 2 October 20 Calls, Buy 1 October 25 Call
Call prices
X
Price
15 2.50
20 1.80
25 0.45
Payoff and profits
October Cisco
stock price
Payoff on
October X=15
call
Payoff on
October
X=20
call
Payoff on
October X=25
call
Total
profit
0 -2.5 3.6 -0.45 0.65
5 -2.5 3.6 -0.45 0.65
10 -2.5 3.6 -0.45 0.65
15 -2.5 3.6 -0.45 0.65
16 -1.5 3.6 -0.45 1.65
17 -0.5 3.6 -0.45 2.65
18 0.5 3.6 -0.45 3.65
19 1.5 3.6 -0.45 4.65
20 2.5 3.6 -0.45 5.65
21 3.5 1.6 -0.45 4.65
22 4.5 -0.4 -0.45 3.65
23 5.5 -2.4 -0.45 2.65
24 6.5 -4.4 -0.45 1.65
25 7.5 -6.4 -0.45 0.65
26 8.5 -8.4 0.55 0.65
30 12.5 -16.4 4.55 0.65
35 17.5 -26.4 9.55 0.65
40 22.5 -36.4 14.55 0.65
Butterfly: Profit Pattern
When the total profit line is > x-axis, there's an arbitrage opportunity!
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Cisco stock price, October
Total profit
Notice that the total profit graph is completely above the x-axis. This means that—no
matter what the stock price in October, you will make a profit. This is clearly not logical—
something is wrong with these prices!
You get the same thing if you assume that the X = 20 call option is priced at $3.00
instead of $1.80:
PFE Chapter 23: Facts about option prices page 17
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ABCDEFGHI
WHEN DOES A BUTTERFLY INDICATE AN ARBITRAGE OPPORTUNITY?
Strategy: Buy 1 October 15 Call, Write 2 October 20 Calls, Buy 1 October 25 Call
Call prices
X
Price
15 5.00
20 3.00
25 0.45
Payoff and profits
October Cisco
stock price
Payoff on
October X=15
call
Payoff on
October
X=20
call
Payoff on
October X=25
call
Total
profit
0 -5 6 -0.45 0.55
5 -5 6 -0.45 0.55
10 -5 6 -0.45 0.55
15 -5 6 -0.45 0.55
16 -4 6 -0.45 1.55
17 -3 6 -0.45 2.55
18 -2 6 -0.45 3.55
19 -1 6 -0.45 4.55
20 0 6 -0.45 5.55
21 1 4 -0.45 4.55
22 2 2 -0.45 3.55
23 3 0 -0.45 2.55
24 4 -2 -0.45 1.55
25 5 -4 -0.45 0.55
26 6 -6 0.55 0.55
30 10 -14 4.55 0.55
35 15 -24 9.55 0.55
40 20 -34 14.55 0.55
Butterfly: Profit Pattern
When the total profit line is > x-axis, there's an arbitrage opportunity!
0
1
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5
6
0 5 10 15 20 25 30 35 40
Cisco stock price, October
Total profit
What’s wrong?
Playing around a bit with the numbers will convince you that a condition necessary for
the butterfly graph to straddle the x-axis is:
()
()
(
)
15 25
20
2
Call price X Call price X
Call price X
=+ =
=<
This condition—in the jargon of the options markets referred to as the convexity property
of call prices—says that for 3 “equally spaced” calls, the middle call price must be less than the
average of the two extreme call prices. Another way of saying this is that the line connecting
two call prices always lies above the graph of the call prices:
PFE Chapter 23: Facts about option prices page 18
The Convexity of Call Prices
Call exercise price, X
Call price
Call price: Declines
as exercise price
increases
Straight line connecting 2 call
prices: always above
the call pricing line (this is the
convexity
)
Put prices are also convex. We leave put butterflies as an exercise and let you prove this
on your own. Here’s the way put prices look:
Put Price Convexity
Put exercise price, X
Put price
Put price: Increases
as exercise price
increases
Straight line connecting 2 put
prices: always above
the put pricing line (this is the
convexity)
PFE Chapter 23: Facts about option prices page 19
Summary
In this chapter we have derived restrictions on option prices which stem from their being
related to other securities in the market. These arbitrage restrictions help us bound option prices
(that is, establish minimum prices for put and call options) as well as establish relations between
the prices of various options and the underlying security (as in the case of the put-call parity
theorem).
In this chapter we have dealt with 7 such option pricing restrictions, but there are many
more which deal with cases involving dividends and transactions costs. Understanding the seven
restrictions discussed in this chapter will help you understand not only the pricing of options (we
will have more to say on this topic in the next chapter), but it will also help you understand the
way option traders think—they are constantly busy trying to figure out how to arbitrage option
prices.
PFE Chapter 24: Facts about option prices page 1
CHAPTER 24: OPTION PRICING FACTS
*
This version: November 20, 2004
Chapter contents
Overview..............................................................................................................................2
24.1. Fact 1: Call price of an option >
(
)
0
,0Max S PresentValue X−
⎡
⎤
⎣
⎦
..........................4
24.2. Fact 2: It’s never worthwhile to exercise a call early. ..............................................9
24.3. Fact 3: Put-call parity
(
)
00 0
Put Call PV X S=+ −.................................................11
24.4. Fact 4: Bound on an American put option price:
[
]
00
,0PMaxXS>−.................14
24.5. Fact 5: Bounds on European put option prices
(
)
00
,0PMaxPVX S>−
⎡
⎤
⎣
⎦
...........15
24.6. Fact 6: You might find it optimal to early-exercise an American put on a non-dividend
paying stock.......................................................................................................................16
24.7. Fact 7: Option prices are convex (somewhat advanced) ........................................17
Summary............................................................................................................................23
Exercises............................................................................................................................25
*
This is a preliminary draft of a chapter of Principles of Finance with Excel. © 2001 – 2004 Simon Benninga
(benninga@wharton.upenn.edu
).
PFE Chapter 24: Facts about option prices page 2
Overview
In Chapter 23 we discussed basic option concepts: definitions of a call and a put, the
reasons why you might want to buy or sell an option, and the profits resulting from various
options strategies. In this chapter we discuss some basic facts about option pricing. Our
emphasis is on a set of propositions known as
arbitrage restrictions on option prices. These
restrictions specify relations between the prices of puts and calls and the prices of either the stock
underlying the options or a risk-free asset.
By understanding the option pricing restrictions in this chapter, you can often easily
judge whether an option is mispriced. Here’s an example: Suppose you’re considering buying a
call option on Microsoft stock, which is currently selling for $63 a share. Suppose the option
expires in one year and has exercise price
X = $60. The interest rate is 10%. The option is
priced at $7. Is it a good buy or not? Our first option pricing fact (Section 24.1) will enable you
to say that the option is
underpriced and that it is definitely a good buy.
Notation: Throughout the chapter we use the following notation
0
price of stock at time 0 (today)
price of stock on option exercise date
option exercise price
interest rate
call option price at time 0 (today);
sometimes we also write this as and occasionally
T
0
S
ST
X
r
C
C
=
=
=
=
=
0
00
as
put option price at time 0 (today);
sometimes we write this as and occasionally as
call option price at time
put option price at time
t
t
Call
P
PPut
Ct
Pt
=
=
=