© 2004 South-Western Publishing 1 Chapter 5 Option Pricing
Dec 21, 2015
© 2004 South-Western Publishing 1
Chapter 5
Option Pricing
2
Outline
IntroductionA brief history of options pricingArbitrage and option pricing Intuition into Black-Scholes
3
Introduction
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
4
Introduction (cont’d)
The Black-Scholes model:
tdd
t
trKS
d
dNKedSNC rt
12
2
1
21
and
2ln
where
)()(
5
A Brief History of Options Pricing: The Early Work
Charles Castelli wrote The Theory of Options in Stocks and Shares (1877)– Explained the hedging and speculation aspects
of options
Louis Bachelier wrote Theorie de la Speculation (1900)– The first research that sought to value derivative
assets
6
A Brief History of Options Pricing: The Middle Years
Rebirth of option pricing in the 1950s and 1960s– 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)
7
A Brief History of Options Pricing: The Present
The Black-Scholes option pricing model (BSOPM) was developed in 1973– An improved version of the Boness model– Most other option pricing models are modest
variations of the BSOPM
8
Arbitrage and Option Pricing
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
9
Arbitrage and Option Pricing (cont’d)
The effect of volatility Multiperiod binomial option pricing Option pricing with continuous
compounding Risk neutrality and implied branch
probabilities Extension to two periods
10
Arbitrage and Option Pricing (cont’d)
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
11
Introduction
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
12
Free Lunches
The apparent mispricing may be so small that it is not worth the effort– E.g., pennies on the sidewalk
Arbitrage opportunities may be out of reach because of an impediment– E.g., trade restrictions
13
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.
14
Free Lunches (cont’d)
Modern option pricing techniques are based on arbitrage principles– In a well-functioning marketplace, equivalent
assets should sell for the same price (law of one price)
– Put/call parity
15
The Theory of Put/Call Parity
Introduction Covered call and short put Covered call and long put No arbitrage relationships Variable definitions The put/call parity relationship
16
Introduction
For a given underlying asset, the following factors form an interrelated complex:– Call price– Put price– Stock price and– Interest rate
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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
18
Covered Call and Long Put
A riskless position results if you combine a covered call and a long put
Covered call Long put Riskless position
+ =
19
Covered Call and Long Put
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
20
No Arbitrage Relationships
The covered call and long put position has the following characteristics:– 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
21
No Arbitrage Relationships (cont’d)
If there is no arbitrage, then:
)1(
0)1(
0)1(
r
SrPC
r
SrPC
r
SrPCSS
22
No Arbitrage Relationships (cont’d)
If there is no arbitrage, then:
– The call premium should exceed the put premium by about the riskless rate of interest
– The difference will be greater as: The stock price increases Interest rates increase The time to expiration increases
rr
r
S
PC
)1(
23
Variable Definitions
C = call premium
P = put premium
S0 = current stock price
S1 = stock price at option expiration
K = option striking price
R = riskless interest rate
t = time until option expiration
24
The Put/Call Parity Relationship
We now know how the call prices, put prices, the stock price, and the riskless interest rate are related:
tr
KSPC
)1(0
25
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 = $75 Riskless interest rate = 5% Time until option expiration = 32 days
If there are no arbitrage opportunities, what is the equilibrium stock price?
26
The Put/Call Parity Relationship (cont’d)
Equilibrium Stock Price Example (cont’d)
Using the put/call parity relationship to solve for the stock price:
18.77$
)05.1(
00.75$00.1$50.3$
)1(
36532
0
tr
KPCS
27
The Put/Call Parity Relationship (cont’d)
To understand why the law of one price must hold, consider the following information:
C = $4.75P = $3
S0 = $50K = $50R = 6.00%t = 6 months
28
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
– The actual call price ($4.75) is too high or the put price ($3) is too low
29
The Put/Call Parity Relationship (cont’d)
To exploit the arbitrage, arbitrageurs would:– 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
30
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
Total $0.31 $0.31 $0.31
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The Binomial Option 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
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The Binomial Option Pricing Model (cont’d)
Possible states of the world:
$75
$50
$100
Today One Year Later
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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– 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?
34
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– Buy the stock and write N call options
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The Binomial Option Pricing Model (cont’d)
Possible portfolio values:
$75 – (N)($C)
$50
$100 - $25N
Today One Year Later
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The Binomial Option Pricing Model (cont’d)
We can solve for N such that the portfolio value in one year must be $50:
2
50$25$100$
N
N
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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%) The portfolio must be worth $45.45 today
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The Binomial Option Pricing Model (cont’d)
Assuming no arbitrage exists:
The option must sell for $14.77!
77.14$
45.45$275$
C
C
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The Binomial Option Pricing Model (cont’d)
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
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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:
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
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Put Pricing in the Presence of Call Options
Activity Cash Flow Today
Portfolio Value at Option Expiration
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
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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– Assume a value of $100
43
Binomial Put Pricing (cont’d)
Possible portfolio values:
$75
$50 + N($75 - $50)
$100
Today One Year Later
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Binomial Put Pricing (cont’d)
A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year
95.7$
91.90$275$
P
P
45
Binomial Pricing With Asymmetric Branches
The size of the up movement does not have to be equal to the size of the decline– E.g., the stock will either rise by $25 or fall by
$15
The logic remains the same:– First, determine the number of options
– Second, solve for the option price
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The Effect of Time
More time until expiration means a higher option value
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The Effect of Volatility
Higher volatility means a higher option price for both call and put options– Explains why options on Internet stocks have a
higher premium than those for retail firms
48
Multiperiod Binomial Option Pricing
In reality, prices change in the marketplace minute by minute and option values change accordingly
The logic of binomial pricing can be easily extended to a multiperiod setting using the recursive methods of solving for the option value
49
Option Pricing With Continuous Compounding
Continuous compounding is an assumption of the Black-Scholes model
Using continuous compounding to revalue the call option from the previous example:
88.14$
00.50$))(275($ 10.
C
eC
50
Risk Neutrality and Implied Branch Probabilities
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
51
Risk Neutrality and Implied Branch Probabilities (cont’d)
Assume the following:– 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%
52
Risk Neutrality and Implied Branch Probabilities (cont’d)
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%
53
Risk Neutrality and Implied Branch Probabilities (cont’d)
Define the following:– 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
54
Risk Neutrality and Implied Branch Probabilities (cont’d)
The average stock return is the weighted average of the two possible price movements:
%22.346578.01
%78.656667.03333.1
6667.01052.1
)(
down
up
up
rt
up
rtdownup
P
P
P
DU
DeP
eDPUP
55
Risk Neutrality and Implied Branch Probabilities (cont’d)
If the stock goes up, the call will have an intrinsic value of $100 - $75 = $25
If the stock goes down, the call will be worthless
The expected value of the call in one year is:
45.16$)0$3422.0()25$6578.0(
56
Risk Neutrality and Implied Branch Probabilities (cont’d)
Discounted back to today, the value of the call today is:
88.14$1052.1/45.16$
57
Extension to Two Periods
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 two-year European call shown on the next slide?
58
Extension to Two Periods (cont’d)
$75
$50
$100
Today One Year Later
$133.33 (UU)
$66.67 (UD = DU)
$33.33 (DD)
Two Years Later
59
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
72.34$1052.1/37.38$
37.38$)0$3422.0()33.58$6578.0(
60
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:
66.20$1052.1/84.22$
84.22$)0$3422.0()72.34$6578.0(
61
Extension to Two Periods (cont’d)
$20.66
$0
$34.72
Today One Year Later
$58.33 (UU)
$0 (UD = DU)
$0 (DD)
Two Years Later
62
Recombining Binomial Trees
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
63
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:
– The probability of an up movement is:
te
DU
DeP
t
up
64
Multiperiod Binomial Put Pricing
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
65
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?
66
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 $29.00Buy 1 put ($2.23)Buy 1 share ($75.00)Borrow $60e-(.10)(2) $49.12
$0.89
67
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.
68
American Versus European Option Pricing
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
69
European Put Pricing and Time Value
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
70
European Put Pricing and Time Value (cont’d)
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
71
Intuition Into Black-Scholes
Continuous time and multiple periods
72
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 Requires continuous time calculus (BSOPM)
The pricing logic remains:– A risk less investment should earn the riskless rate
of interest