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
Basic Option Pricing Models






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The theory of put/call parity
The binomial option pricing model
Binomial put pricing
Binomial pricing with asymmetric branches
The effect of time
The effect of volatility
Arbitrage and Option Pricing




9
Finance is sometimes called “the study of
arbitrage”
– Arbitrage is the existence of a riskless profit
Finance theory does not say that arbitrage will
never appear
– Arbitrage opportunities will be short-lived
Option pricing techniques are based on basic
arbitrage principles
Efficient market - equivalent assets should sell for
the same price – ‘the law of one price’
The Theory of Put/Call Parity





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Covered call and short put
Covered call and long put
No arbitrage relationships
Variable definitions
The put/call parity relationship
Put-Call Parity Theory

For a given underlying asset, the following
factors form an interrelated complex:
Call price
– Put price
– Stock price and
– Interest rate - risk free rate
– time to expiration
.....the price of European style puts and calls on
the same stock with the identical strike price
and expiration dates have a special relationship
–
11
Put – Call Parity Theory
‘When European options are at the money
and the stock pays no dividends, relative
call prices should exceed relative put prices
by an amount approximately equal to the
riskless rate of interest for the option term
times the stock price.’
….a theory about how to price options
relative to other options
12
Put-Call Parity theory - 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

13
Put Call Parity theory - Covered
Call and Long Put

A riskless position results if you combine a
covered call and a long put
Long put
Covered call
+
14
Riskless position
=
....stock price at option expiration has no impact on
the profit/loss position - ‘riskless’
Covered Call and Long Put
15

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....and will be quickly taken
advantage of.
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
No Arbitrage Relationships State of Equilibrium

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 (S)(r)
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

If there is no arbitrage and assuming
European at the money options, then:
Sr
S S C  P
0
(1  r )t
Sr
CP
0
(1  r )t
Sr
CP
(1  r )t
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No Arbitrage Relationships
(cont’d)

If there is no arbitrage, then:
CP
r

r
S
(1  r )t
–
–
The call premium should exceed the put
premium by about the riskless rate of interest
for the option term times the strike price
The difference will be greater as:


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
The stock price increases
Interest rates increase
The time to expiration increases
The Put/Call Parity Relationship


We now know how the call prices, put
prices, the stock price, and the riskless
interest rate are related
What about when the strike price is
different from the stock price ie. In/out of
the money options ?
K
C  P  S0 
t
(1  r )
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The Put-Call Parity Relationship
K
C  P  S0 
t
(1  r )
…where the strike price is less than the stock price S0
(in the money call) the call price will reflect this intrinsic
Value
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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?
22
The Put/Call Parity Relationship
(cont’d)
Equilibrium Stock Price Example (cont’d)
23
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
Put-Call Parity – another look


Another way to illustrate the theory
Instead of the covered call and long put with
borrowing – an equivalent situation is:
–
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A share of stock plus a put is equivalent to a call
plus an investment in risk free bonds
The Binomial Option Pricing
Model

A theory about establishing the option price from
the factors that influence it:
–
Another simplified pricing model - used to illustrate
various influences on option prices:




–
–
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Stock price
Strike price
Interest rates
Stock price volatility
The simplifying assumption is: a single period binomial
model – there are only two possible outcomes and one
time period
The stock can go up or down in price over the period
Binomial Pricing Model

Assume the following:
–
–
–
–
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
….we want to construct a riskless portfolio of stock and
options – one where the future value of the portfolio is
independent of the future value of the stock.
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The Binomial Option Pricing
Model (cont’d)

Possible states of the world:
$100
$75
$50
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Stock Price
Today
Stock Price
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 (no risk)
–
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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
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Total Investment
Today
Total Investment
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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Discounting at 10% for one year 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

The option must sell for $14.77 otherwise there
would exist an arbitrage situation:
–
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If the option is selling for more than $14.77 sellers would
step in or if the price was less than $14.77 buyers of the
option would find it attractive
The Binomial Option Pricing
Model (cont’d)

The option value is independent of the
probabilities associated with the future
stock price and hence:
The price of an option is independent of the expected
return on the stock!
The price of an option tells you nothing about the
future path of the stock price!
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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 Option Pricing Model

36
Establish a riskless or hedge portfolio where
no arbitrage opportunity exists - state of
equilibrium
Binomial Put Pricing (cont’d)

Possible portfolio values:
$100
$75 + 2($P)
$50 + N($75 - $50)
Today
37
One Year Later
Binomial Put Pricing

Solve for N – how many put options to
purchase
–
–
–
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$50 + N(25) = $100
25 N = $50
N=2
Binomial Put Pricing (cont’d)

A portfolio composed of one share of stock
and two puts will grow risklessly to $100
after one year (discounted to $90.91 in
today’s dollars)
$75  2P  $90.91
P  $7.95
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Back to Put-Call Parity
Let’s reconcile:
 Using binomial theory we arrived at a call
price of $14.77 and a put price of $7.95
 Does this reconcile with the put-call parity
theory?
40
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

More time until expiration means a higher
option value
–
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….remember an option can have time value and
or intrinsic value
The Effect of Volatility

Higher volatility means a higher option
price for both call and put options
–
E.g. high tech vs utility type stocks
….work through the same call option example with
smaller range of possible outcomes (lower
volatility)
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Black-Scholes Model - the next
evolution


Pricing logic remains the same – riskless investments should
earn riskless rates of return
Input factors are the same
–
–
–
–
–

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Stock price
Strike price
Time until expiration
Interest rates
Stock volatility
Using developing computer power in the early 70’s, the model
moves to continuous time calculus – outcomes now not
limited to only two, many different time intervals - in theory
there are an infinite number of future outcomes
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:
–
45
Requires continuous time calculus (BSOPM)
A riskless investment should earn the riskless
rate of interest