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Chapter 5
Option Pricing
1
© 2004 South-Western Publishing
Outline
Introduction
 A brief history of options pricing
 Arbitrage and option pricing
 Intuition into Black-Scholes

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Introduction
3

Option pricing developments are among the
most important in the field of finance during
the last 30 years

The backbone of option pricing is the
Black-Scholes model
Introduction (cont’d)

The Black-Scholes model:
C  SN (d1 )  Ke  rt N (d 2 )
where
2
S   
t
ln     r 
2 
K 
d1 
 t
and
d 2  d1   t
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A Brief History of Options
Pricing: The Early Work

Charles Castelli wrote The Theory of
Options in Stocks and Shares (1877)
–

Louis Bachelier wrote Theorie de la
Speculation (1900)
–
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Explained the hedging and speculation aspects
of options
The first research that sought to value derivative
assets
A Brief History of Options Pricing:
The Middle Years

Rebirth of option pricing in the 1950s and
1960s
–
–
–
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Paul Samuelson wrote Brownian Motion in the
Stock Market (1955)
Richard Kruizenga wrote Put and Call Options: A
Theoretical and Market Analysis (1956)
James Boness wrote A Theory and
Measurement of Stock Option Value (1962)
A Brief History of Options Pricing:
The Present

The Black-Scholes option pricing model
(BSOPM) was developed in 1973
–
–
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An improved version of the Boness model
Most other option pricing models are modest
variations of the BSOPM
Arbitrage and Option Pricing

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Introduction
Free lunches
The theory of put/call parity
The binomial option pricing model
Put pricing in the presence of call options
Binomial put pricing
Binomial pricing with asymmetric branches
The effect of time
Arbitrage and Option Pricing
(cont’d)





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The effect of volatility
Multiperiod binomial option pricing
Option pricing with continuous
compounding
Risk neutrality and implied branch
probabilities
Extension to two periods
Arbitrage and Option Pricing
(cont’d)






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Recombining binomial trees
Binomial pricing with lognormal returns
Multiperiod binomial put pricing
Exploiting arbitrage
American versus European option pricing
European put pricing and time value
Introduction

Finance is sometimes called “the study of
arbitrage”
–

Finance theory does not say that arbitrage
will never appear
–
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Arbitrage is the existence of a riskless profit
Arbitrage opportunities will be short-lived
Free Lunches

The apparent mispricing may be so small
that it is not worth the effort
–

Arbitrage opportunities may be out of reach
because of an impediment
–
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E.g., pennies on the sidewalk
E.g., trade restrictions
Free Lunches (cont’d)
A University Example
A few years ago, a bookstore at a university was
having a sale and offered a particular book title for
$10.00. Another bookstore at the same university
had a buy-back offer for the same book for $10.50.
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Free Lunches (cont’d)

Modern option pricing techniques are
based on arbitrage principles
–
–
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In a well-functioning marketplace, equivalent
assets should sell for the same price (law of one
price)
Put/call parity
The Theory of Put/Call Parity

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Introduction
Covered call and short put
Covered call and long put
No arbitrage relationships
Variable definitions
The put/call parity relationship
Introduction

For a given underlying asset, the following
factors form an interrelated complex:
–
–
–
–
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Call price
Put price
Stock price and
Interest rate
Covered Call and Short Put

The profit/loss diagram for a covered call
and for a short put are essentially equal
Covered call
Short put

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Covered Call and Long Put

A riskless position results if you combine a
covered call and a long put
Long put
Covered call
+
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Riskless position
=
Covered Call and Long Put
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
Riskless investments should earn the
riskless rate of interest

If an investor can own a stock, write a call,
and buy a put and make a profit, arbitrage
is present
No Arbitrage Relationships

The covered call and long put position has
the following characteristics:
–
–
–
–
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One cash inflow from writing the call (C)
Two cash outflows from paying for the put (P)
and paying interest on the bank loan (Sr)
The principal of the loan (S) comes in but is
immediately spent to buy the stock
The interest on the bank loan is paid in the
future
No Arbitrage Relationships
(cont’d)

If there is no arbitrage, then:
Sr
S S C P
0
(1  r )
Sr
CP
0
(1  r )
Sr
CP
(1  r )
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No Arbitrage Relationships
(cont’d)

If there is no arbitrage, then:
CP
r

r
S
(1  r )
–
–
The call premium should exceed the put
premium by about the riskless rate of interest
The difference will be greater as:


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The stock price increases
Interest rates increase
The time to expiration increases
Variable Definitions
C
P
S0
S1
K
R
t
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=
=
=
=
=
=
=
call premium
put premium
current stock price
stock price at option expiration
option striking price
riskless interest rate
time until option expiration
The Put/Call Parity Relationship

We now know how the call prices, put
prices, the stock price, and the riskless
interest rate are related:
K
C  P  S0 
t
(1  r )
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The Put/Call Parity Relationship
(cont’d)
Equilibrium Stock Price Example





You have the following information:
Call price = $3.5
Put price = $1
Striking price = $75
Riskless interest rate = 5%
Time until option expiration = 32 days
If there are no arbitrage opportunities, what is the equilibrium
stock price?
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The Put/Call Parity Relationship
(cont’d)
Equilibrium Stock Price Example (cont’d)
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Using the put/call parity relationship to solve for
the stock price:
K
S0  C  P 
(1  r ) t
$75.00
 $3.50  $1.00 
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(1.05) 365
 $77.18
The Put/Call Parity Relationship
(cont’d)

C
P
S0
K
R
t
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To understand why the law of one price must hold,
consider the following information:
=
=
=
=
=
=
$4.75
$3
$50
$50
6.00%
6 months
The Put/Call Parity Relationship
(cont’d)

Based on the provided information, the put
value should be:
P = $4.75 - $50 + $50/(1.06)0.5 = $3.31
–
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The actual call price ($4.75) is too high or the put
price ($3) is too low
The Put/Call Parity Relationship
(cont’d)

To exploit the arbitrage, arbitrageurs would:
–
–
–
–
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Write 1 call @ $4.75
Buy 1 put @ $3
Buy a share of stock at $50
Borrow $48.56 at 6.00% for 6 months
These actions result in a profit of $0.31 at
option expiration irrespective of the stock
price at option expiration
The Put/Call Parity Relationship
(cont’d)
Stock Price at Option Expiration
$0
$50
$100
From call
4.75
4.75
(45.25)
From put
47.00
(3.00)
(3.00)
From loan
(1.44)
(1.44)
(1.44)
From stock
(50.00)
0.00
50.00
$0.31
$0.31
$0.31
Total
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The Binomial Option Pricing
Model

Assume the following:
–
–
–
–
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U.S. government securities yield 10% next year
Stock XYZ currently sells for $75 per share
There are no transaction costs or taxes
There are two possible stock prices in one year
The Binomial Option Pricing
Model (cont’d)

Possible states of the world:
$100
$75
$50
Today
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One Year Later
The Binomial Option Pricing
Model (cont’d)

A call option on XYZ stock is available that
gives its owner the right to purchase XYZ
stock in one year for $75
–
–
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If the stock price is $100, the option will be
worth $25
If the stock price is $50, the option will be worth
$0
What should be the price of this option?
The Binomial Option Pricing
Model (cont’d)

We can construct a portfolio of stock and
options such that the portfolio has the
same value regardless of the stock price
after one year
–
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Buy the stock and write N call options
The Binomial Option Pricing
Model (cont’d)

Possible portfolio values:
$100 - $25N
$75 – (N)($C)
$50
Today
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One Year Later
The Binomial Option Pricing
Model (cont’d)

We can solve for N such that the portfolio
value in one year must be $50:
$100  $25 N  $50
N 2
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The Binomial Option Pricing
Model (cont’d)

If we buy one share of stock today and
write two calls, we know the portfolio will be
worth $50 in one year
–
The future value is known and riskless and must
earn the riskless rate of interest (10%)

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The portfolio must be worth $45.45 today
The Binomial Option Pricing
Model (cont’d)

Assuming no arbitrage exists:
$75  2C  $45.45
C  $14.77

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The option must sell for $14.77!
The Binomial Option Pricing
Model (cont’d)
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
The option value is independent of the
probabilities associated with the future
stock price

The price of an option is independent of the
expected return on the stock
Put Pricing in the Presence of
Call Options

In an arbitrage-free world, the put option
cannot also sell for $14.77; If it did, an
astute arbitrageur would:
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Buy a 75 call
Write a 75 put
Sell the stock short
Invest $68.18 in T-bills
These actions result in a cash flow of $6.82
today and a cash flow of $0 at option
expiration
Put Pricing in the Presence of
Call Options
Activity
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Portfolio Value at Option
Expiration
Cash Flow
Today
Price = $100
Price = $50
Buy 75 call
-$14.77
$25
0
Write 75 put
+14.77
0
-$25
Sell stock short
+75.00
-100
-50
Invest $68.18 in
T-bills
-68.18
75.00
75.00
Total
$6.82
$0.00
$0.00
Binomial Put Pricing

Priced analogously to calls

You can combine puts with stock so that
the future value of the portfolio is known
–
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Assume a value of $100
Binomial Put Pricing (cont’d)

Possible portfolio values:
$100
$75
$50 + N($75 - $50)
Today
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One Year Later
Binomial Put Pricing (cont’d)

A portfolio composed of one share of stock
and two puts will grow risklessly to $100
after one year
$75  2P  $90.91
P  $7.95
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Binomial Pricing With
Asymmetric Branches

The size of the up movement does not have
to be equal to the size of the decline
–
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E.g., the stock will either rise by $25 or fall by
$15
The logic remains the same:
–
First, determine the number of options
–
Second, solve for the option price
The Effect of Time
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More time until expiration means a higher
option value
The Effect of Volatility

Higher volatility means a higher option
price for both call and put options
–
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Explains why options on Internet stocks have a
higher premium than those for retail firms
Multiperiod Binomial Option Pricing
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
In reality, prices change in the marketplace
minute by minute and option values change
accordingly

The logic of binomial pricing can be easily
extended to a multiperiod setting using the
recursive methods of solving for the option
value
Option Pricing With Continuous
Compounding

Continuous compounding is an
assumption of the Black-Scholes model

Using continuous compounding to
revalue the call option from the previous
example:
($75  2C )(e )  $50.00
.10
C  $14.88
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Risk Neutrality and Implied Branch
Probabilities
50

Risk neutrality is an assumption of the
Black-Scholes model

For binomial pricing, this implies that
the option premium contains an implied
probability of the stock rising
Risk Neutrality and Implied Branch
Probabilities (cont’d)

Assume the following:
–
–
–
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An investor is risk-neutral
He can invest funds risk free over one year at a
continuously compounded rate of 10%
The stock either rises by 33.33% or falls 33.33% in
one year
After one year, one dollar will be worth $1.00 x
e.10 = $1.1052 for an effective annual return of
10.52%
Risk Neutrality and Implied Branch
Probabilities (cont’d)
52

A risk-neutral investor would be
indifferent between investing in the
riskless rate and investing in the stock
if it also had an expected return of
10.52%

We can determine the branch
probabilities that make the stock have a
return of 10.52%
Risk Neutrality and Implied Branch
Probabilities (cont’d)

Define the following:
–
–
–
–
–
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U = 1 + percentage increase if the stock
goes up
D = 1 – percentage decrease if the stock
goes down
Pup = probability that the stock goes up
Pdown = probability that the stock goes down
ert = continuously compounded interest rate
factor
Risk Neutrality and Implied Branch
Probabilities (cont’d)

The average stock return is the
weighted average of the two possible
price movements:
P U   ( P
up
rt
D
)

e
down
e rt  D
Pup 
U D
1.1052  0.6667
Pup 
1.3333  0.6667
Pup  65.78%
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Pdown  1  0.6578  34.22%
Risk Neutrality and Implied Branch
Probabilities (cont’d)



If the stock goes up, the call will have an
intrinsic value of $100 - $75 = $25
If the stock goes down, the call will be
worthless
The expected value of the call in one
year is:
(0.6578  $25)  (0.3422  $0)  $16.45
55
Risk Neutrality and Implied Branch
Probabilities (cont’d)

Discounted back to today, the value of
the call today is:
$16.45 / 1.1052  $14.88
56
Extension to Two Periods
57

Assume two periods, each one year
long, with the stock either rising or
falling by 33.33% in each period

What is the equilibrium value of a twoyear European call shown on the next
slide?
Extension to Two Periods (cont’d)
$133.33 (UU)
$100
$66.67
(UD = DU)
$75
$50
$33.33 (DD)
Today
58
One Year Later
Two Years Later
Extension to Two Periods (cont’d)

The option only winds up in the money
when the stock advances twice (UU)
–
There is a 65.78% probability that the call is
worth $58.33 and a 34.22% probability that
the call is worthless
(0.6578  $58.33)  (0.3422  $0)  $38.37
$38.37 / 1.1052  $34.72
59
Extension to Two Periods (cont’d)

There is a 65.78% probability that the
call is worth $34.72 in one year and a
34.22% probability that the call is
worthless in one year
–
The expected value of the call in one year is:
(0.6578  $34.72)  (0.3422  $0)  $22.84
$22.84 / 1.1052  $20.66
60
Extension to Two Periods (cont’d)
$58.33 (UU)
$34.72
$0 (UD = DU)
$20.66
$0
$0 (DD)
Today
61
One Year Later
Two Years Later
Recombining Binomial Trees
62

If trees are recombining, this means that
the up-down path and the down-up path
both lead to the same point, but not
necessarily the starting point

To return to the initial price, the size of
the up jump must be the reciprocal of
the size of the down jump
Binomial Pricing with Lognormal
Returns

Black-Scholes assumes that security
prices follow a lognormal distribution
–
With lognormal returns, the size of the
upward movement U equals:
e
–
 t
The probability of an up movement is:
e t  D
Pup 
U D
63
Multiperiod Binomial Put Pricing
64

To solve for the value of a put using
binomial logic, just change the terminal
intrinsic values and work backward just
as with call pricing

The branch probabilities do not change
Exploiting Arbitrage
Arbitrage Example
Binomial pricing results in a call price of $28.11 and a put
price of $2.23. The interest rate is 10%, the stock price is $75,
and the striking price of the call and the put is $60. The
expiration date is in two years.
What actions could an arbitrageur take to make a riskless
profit if the call is actually selling for $29.00?
65
Exploiting Arbitrage (cont’d)
Arbitrage Example (cont’d)
Since the call is overvalued, and arbitrageur would want to
write the call, buy the put, buy the stock, and borrow the
present value of the striking price, resulting in the following
cash flow today:
Write 1 call
Buy 1 put
Buy 1 share
Borrow $60e-(.10)(2)
66
$29.00
($2.23)
($75.00)
$49.12
$0.89
Exploiting Arbitrage (cont’d)
Arbitrage Example (cont’d)
The value of the portfolio in two years will be worthless,
regardless of the path the stock takes over the two-year
period.
67
American Versus European Option
Pricing



68
With an American option, the intrinsic
value is a sure thing
With a European option, the intrinsic
value is currently unattainable and may
disappear before you can get at it
An American option should be worth
more than a European option
European Put Pricing and Time
Value



69
With a European put, the longer the option’s
life, the longer you must wait to see sales
proceeds
More time means greater potential dispersion
in underlying asset values, and this pushes up
the put value
A European put’s value with respect to time
until expiration is indeterminate
European Put Pricing and Time
Value (cont’d)
70

Often, an out-of-the-money put will
increase in value with more time

Often, an in-the-money put decreases in
value for more distant expirations
Intuition Into Black-Scholes

71
Continuous time and multiple periods
Continuous Time and Multiple
Periods

Future security prices are not limited to
only two values
–
There are theoretically an infinite number of
future states of the world


The pricing logic remains:
–
72
Requires continuous time calculus (BSOPM)
A risk less investment should earn the riskless
rate of interest