Chapter 5
Option Pricing
1
© 2002 South-Western Publishing
Outline
Introduction
 A brief history of options pricing
 Arbitrage and option pricing
 Intuition into Black-Scholes

2
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
4
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)
–
5
Explained the hedging and speculation aspects
of options
The first research that sought to value derivative
assets
The Middle Years

Rebirth of option pricing in the 1950s and
1960s
–
–

6
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)
The Present

The Black-Scholes option pricing model
(BSOPM) was developed in 1973
–
–
7
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
Binomial put pricing
Binomial pricing with asymmetric branches
Arbitrage and Option Pricing
(cont’d)


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The effect of time
The effect of volatility
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
–
–
13
In a well-functioning marketplace, equivalent
assets should sell for the same price
Put/call parity
The Theory of Put/Call Parity






14
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:
–
–
–
–
15
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

16
Covered Call and Long Put

A riskless position results if you combine a
covered call and a long put
Long put
Covered call
+
17
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 )
23
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?
24
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 
32
(1.05) 365
 $77.18
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
27
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
30
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
31
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)
34

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
Binomial Put Pricing

Priced analogously to calls

You can combine puts with stock so that
the future value of the portfolio is known
–
35
Assume a value of $100
Binomial Put Pricing (cont’d)

Possible portfolio values:
$100
$75 + 2($P)
$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

39
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
–
40
Explains why options on Internet stocks have a
higher premium than those for retail firms
Intuition Into Black-Scholes

41
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:
–
42
Requires continuous time calculus (BSOPM)
A riskless investment should earn the riskless
rate of interest
Probabilities and Risk Neutrality

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43
(PupU) + (PdownD) = ert
(PupU) + (1- Pup) D = ert
PupU + (D – PupD) = ert
Pup = (ert – D)/(U – D)
Pdown = 1 - Pup