Transcript FI4000 Fundamentals of Valuation

FINC4101
Investment Analysis
Instructor: Dr. Leng Ling
Topic: Introduction to options
1
Learning objectives
1. Define call and put options.
2. Understand the various features of options: exercise
price, option premium, option exercise, American vs.
European.
3. Describe how options trading is organized.
4. List the various types of option contracts.
5. Compute the payoff and profit of call option holder, call
option writer, put option holder and put option writer.
6. Describe the composition of various option strategies.
7. Compute the payoff and profit of various option
strategies.
2
Concept Map
Foreign
Exchange
Portfolio
Theory
Asset
Pricing
FI400
Equity
Derivatives
Market
Efficiency
Fixed Income
3
Derivative security
A derivative transaction involves no actual transfer of
ownership of the underlying assets at the time the
contract is initiated. A derivative represents an
agreement to transfer ownership of underlying assets at
a specific place, price, and time specified in the contract.
Its value (or price) depends on the value of the
underlying assets.
The underlying assets: stocks, bonds, interest rates,
foreign
exchanges,
index,
commodities,
some
derivatives, etc.
4
What is an option?
A derivative security that gives the
holder the right to buy / sell
an asset (the “underlying”)
at a specified price (“exercise price”)
on or before the option expiration date.
5
Two types of options:
Call vs. Put options
 Call
option
 Gives holder the right to buy an asset at
a specified exercise price on or before a
specified expiration date.
 Put
option
 Gives holder the right to sell an asset at a
specified exercise price on or before a
specified expiration date.
6
Exercise price
 Exercise
price
For a call option, it is the price set for buying
the underlying asset.
 For a put option it is the price set for selling
the underlying asset.

 Exercise
price is also called the strike
price.
7
Option premium





Options are financial assets. If you want an option, you
have to buy it from an option seller (counterparty).
The purchase price or cost of an option is the option
premium.
The option seller earns the option premium.
The option premium is an immediate expense for the
buyer and an immediate return for the seller, whether or
not the owner (buyer) ever exercises the option.
In option markets, to sell an option is to “write” an option.
An option seller is also called an “option writer”.
8
Examples

At March 1, XYZ stock’s spot price = $110. A
trader buys a call option on XYZ at strike
(exercise) price = $100/share. The right lasts
until August 15, and the price (option premium)
of this call option is $15/share.

At March 1, ABC stock’s spot price = $100. A
trader buys a put option to on ABC at strike
(exercise) price = $120/share. The right lasts
until August 15, and the price (option premium)
of this put option is $22/share.
9
The long and short of it…
 If

you buy an option, then you are
“long the option” or “long option” or you have a
“long position”.
 If

you sell an option, then you are
“short the option” or “short option” or you have
a “short position”.
Example: if you buy a call option, you are “long call”.
10
Buyer (Long)
Seller (Short)
Call
- Right to buy the underlying
(i.e. to exercise the option)
- Pays the premium
- Obligation to sell the underlying,
if buyer exercises the option
- Receives the premium
Put
- Right to sell the underlying
(i.e. to exercise the option)
- Pays the premium
- Obligation to buy the underlying,
if buyer exercises the option
- Receives the premium
11
Options trading (1)

1.
2.
Option contracts are traded in two types
of markets:
Over-the-counter (OTC) markets
Exchanges, such as:




Chicago Board Options Exchange (CBOE)
Chicago Mercantile Exchange (CME)
International Securities Exchange
Option Clearing Corporation (OCC)
12
Options trading (2)
1.
2.
OTC
Option contract can
be customized to
needs of trader.
Difficult to trade.
Secondary market
illiquid.
1.
2.
Exchanges
Option contracts are
standardized by
maturity dates and
exercise price.
Easy to trade.
Secondary market is
liquid.
13
Options on IBM June 7, 2004
Source: Wall Street Journal Online Edition, June 8, 2004.
14
Underlying asset
 Individual
stocks
 Stock market indexes

S&P 100, S&P 500, DJIA, Nikkei 225, FTSE
100 etc.
 Futures
 Foreign
currency
 Treasury bonds, Treasury notes
And many others.
15
Option exercise (1)
To “exercise a call option” means to use the
option to buy the underlying asset at the
exercise price.
To “exercise a put option” means to use the
option to sell the underlying asset at the
exercise price.
16
Option exercise (2)
Question: When do you exercise an option?
Answer: Simple. Only when it’s optimal to do so.
That is, when you are better off exercising the
option.
“Buy low, sell high”
Question: What if exercising the option does not
make me better off?
Answer: Simple. Don’t exercise. After all, it’s just an
option.
17
American vs. European options

American option: Holder has the right to exercise
the option on or before the expiration date.

European option: Holder has the right to
exercise the option only on the expiration date.

Most traded options in the US are Americanstyle. Exceptions: foreign currency options,
some stock index options.
18
Payoff of Long Bond Position at
Expiration
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
-5
-10
-15
-20
-25
-30
19
Payoffs of a Call Option
Long Call at $20
Short Call at $20
30
30
25
25
20
20
15
15
10
10
5
5
0
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
5
10
15
20
25
30
35
40
20
Profit/Loss of a Call Option
Long Call at $20
Short Call at $20
30
30
25
25
20
20
15
15
10
10
5
5
0
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
5
10
15
20
25
30
35
40
21
Profit/Loss of Long and Short on
Call Option
30
30
25
25
20
20
15
15
10
10
5
5
0
0
0
5
10
15
20
25
30
35
0
40
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
5
10
15
20
25
30
35
40
22
Payoffs of a Put Option
Long Put at $20
Short Put at $20
30
30
25
25
20
20
15
15
10
10
5
5
0
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
5
10
15
20
25
30
35
40
23
Profit/Loss of a Put Option
Long Put at $20
Short Put at $20
30
30
25
25
20
20
15
15
10
10
5
5
0
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
5
10
15
20
25
30
35
40
24
Profit/Loss of Long and Short on
Put Option
30
30
25
25
20
20
15
15
10
10
5
5
0
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
5
10
15
20
25
30
35
40
25
Call Option’s Payoff/Profit at Expiration

Payoff for a Long Call:
ST  X if ST  X
0
if ST  X

Profit for a Long Call: payoff - option premium

Payoff for a Short Call:
 (ST  X ) if ST  X
0

if ST  X
Profit for a Short Call: option premium + payoff
26
Put Option’s Payoff/Profit at Expiration

X  ST if ST  X
Payoff for a Long Put:
0
if ST  X

Profit for a Long Put: payoff – option premium

Payoff for a Short Put:
 ( X  ST ) if ST  X
0

if ST  X
Profit for a Short Put: option premium + payoff
27
Example

A trader short a Call at X=20 with a premium of
$5. At maturity, the stock price is 30. What is the
profit/loss to this trader?
Profit/Loss = 5 + [-(30-20)] = 5 -10 = -5

A trader long a Put at X=30 with a premium of
$5. At maturity, the stock price is 15. What is the
profit/loss to this trader?
Profit/Loss = (30-15) - 5 = 15 - 5 = 10
28
Call option:
Payoff & Profit at expiration (1)

Consider a call option on a share of IBM
stock with an exercise price of $80 per
share. Suppose this call option expires on
July 16, 2004. Suppose today is the
expiration date. The call option price
(premium) was $5.
29
Call option:
Payoff & Profit at expiration (2)
1. Are you better off exercising the option?
2. What is the payoff from the option exercise?
3. What is the profit from the option exercise?
Answer these questions if IBM’s stock price is
(a) 95, (b) 76 and (c) 81.
 What is the breakeven point for this call option?
 Breakeven point is the stock price at which
profit is zero.
30
Payoff & profit diagram of call
option holder at expiration
14
12
cost of option
Payoff/ profit
10
8
Payoff
6
4
Profit
2
0
Stock price at expiration
-2
Break even point
-4
-6
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
call payoff
0
0
0
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
call profit
-5
-5
-5
-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
31
Payoff and profit of
call option writer
 Compute
payoff, profit and breakeven
point, if the stock price at expiration is
 (a) 95, (b) 76 and (c) 81.
32
Payoff & profit diagram of call
option writer at expiration
10
Profit
5
Break even point
Payoff/ profit
option premium
0
Stock price at expiration
Payoff
-5
-10
-15
-20
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
call w riter payoff
0
0
0
0
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
-14
-15
call w riter profit
5
5
5
5
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
33
Call Review

Which of the following statements about the value (i.e.,
payoff) of a call option at expiration is false?
a. A short position in a call option will result in a loss if the
stock price exceeds the exercise price.
b. The value of a long position equals zero or the stock
price minus the exercise price, whichever is higher.
c. The value of a long position equals zero or the exercise
price minus the stock price, whichever is higher.
d. A short position in a call option has a zero value for all
stock prices equal to or less than the exercise price.
34
Put option:
Payoff & Profit at expiration (1)
 Consider
a put option on a share of IBM
stock with an exercise price of $80 per
share. Suppose this put option expires on
July 16, 2004. Suppose today is the
expiration date. The put option price
(premium) was $3.
35
Put option:
Payoff & Profit at expiration (2)
1. Are you better off exercising the option?
2. What is the payoff from the option exercise?
3. What is the profit from the option exercise?
Answer these questions if IBM’s stock price is
(a) 89, (b) 70 and (c) 79.
 What is the breakeven point for this put option?
 Breakeven point is the stock price at which
profit is zero.
36
Payoff & profit diagram of put
option holder at expiration
10
8
Payoff/ profit
6
Payoff
4
2
Break even point
Stock price at expiration
0
option premium
-2
-4
Profit
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
put payoff
10
9
8
7
6
5
4
3
2
1
0
0
0
0
0
0
0
0
0
0
put profit
7
6
5
4
3
2
1
0
-1
-2
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
37
Payoff and profit of
put option writer
 Compute
payoff, profit and breakeven
point, if the stock price at expiration is
(a) 89, (b) 70 and (c) 79.
38
Payoff & profit diagram of put
option writer at expiration
4
Profit
Break even point
2
option premium
Payoff/ profit
0
Stock price at expiration
-2
Payoff
-4
-6
-8
-10
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
put w riter payoff
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
0
0
0
0
0
0
0
0
0
put w riter profit
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
3
3
3
3
3
3
3
3
3
39
Put Review

Consider a put option written on ABC Inc.’s stock. The
put option’s exercise price is $80. Which of the following
statements about the value (payoff) of the put option at
expiration is true?
a. The value of the short position in the put is $4 if the stock
price is $76.
b. The value of the long position in the put is -$4 if the stock
price is $76.
c. The long put has value when the stock price is below the
$80 exercise price.
d. The value of the short position in the put is zero for stock
prices equaling or exceeding $76.
40
In-class Practice
You purchased one IBM July 100 call
contract for a premium of $4.00. Assuming
that the stock price on the expiration date is
$105. What is the payoff and net profit/loss?
What is the break even point? Draw the
payoff and net profit/loss lines on the
diagram.
41
Practice 9
 Chapter
15: 4, 5, 6.
42
Moneyness (1)—intrinsic value

An option (call or put) is:
1. In the money (ITM) if exercising it
produces a positive payoff to the holder
2. At the money (ATM) if the asset price and
exercise price are equal.
3. Out of the money (OTM) if exercising it
produces a negative payoff to the holder.
43
Moneyness (2)
ST < X
Call option
Put option
ST = X
Out of the At the money
money
In the
money
At the money
ST > X
In the
money
Out of the
money
44
Moneyness questions (1)
Consider two call options written on ABC Inc.’s
stock. The first call, C1, has an exercise price of
$50. The second call, C2, has an exercise price
of $70. Both calls have the same expiration
date. Today is the expiration date. C1 is in the
money while C2 is out of the money. Which of
the following is true about ST, the stock price on
the expiration date?
a. ST > $50
b. ST > $70
c. $70 > ST > $50
d. ST < $50

45
Moneyness questions (2)
Consider two put options written on XYZ Inc.’s
stock. The first put, P1, has an exercise price of
$20. The second put, P2, has an exercise price
of $35. Both puts have the same expiration date.
Today is the expiration date. P1 is out of the
money while P2 is in the money. Which of the
following is true about ST, the stock price on the
expiration date?
a. ST < $20
b. ST < $35
c. $20 < ST < $35
d. ST > $35

46
Option vs. Stock Investment (1)

Compared to a stock investment, options offer
1) Leverage

Pure option investment magnifies gains and losses
compared to pure stock investment.
2) Insurance

Combining options with T-bills (money market fund)
limits losses compared to pure stock investment.
Consider the following…
47
Option vs. Stock Investment (2)

Suppose you think Wal-Mart stock is going to appreciate
substantially in value in the next year. The stock’s
current price, S0, is $100, and the call option expiring in
one year has an exercise price, X, of $100 and is selling
at a price, C, of $10. With $10,000 to invest, you are
considering three alternatives:
a) Invest all $10,000 in the stock, buying 100 shares.
b) Invest all $10,000 in 1,000 options (10 contracts)
c) Buy 100 options (one contract) for $1,000 and invest the
remaining $9,000 in a money market fund paying 4%
interest annually.
48
Option vs. Stock Investment (3)

1.
2.
3.
4.
Compute the rate of return for each
alternative for four stock prices one year
from now:
$80
$100
$110
$120
49
Option vs. Stock Investment (4)
Price of stock 1 year from now
Stock price:
a) All stocks
(100 shares)
b) All options
(1,000 shares)
c) Money market
fund + 100 options
$80
$100
$110
$120
-20%
0%
10%
20%
0%
100%
-100% -100%
-6.4%
-6.4% 3.6% 13.6%
50
Rate of return to strategies
51
Option strategies
 An

 We
1.
2.
3.
option strategy is:
A portfolio of options (and possibly the
underlying asset) designed to produce a
particular payoff pattern.
look at the following strategies:
Protective put (Insurance on portfolio)
Covered call
Straddle
52
Protective put
 Portfolio
consisting of a put option and the
underlying asset.
 Guarantees
that minimum portfolio value
(payoff) is equal to the put’s exercise
price.
53
Protective put:
Payoff & profit at expiration
S0 = initial asset price, and
P = put option premium.
Cost of the position = asset price + put premium
= S0 + P
Payoff of stock
Payoff of put
Total payoff
Profit
ST ≤ X
ST
X – ST
X
X – (S0+P)
ST > X
ST
0
ST
ST – (S0 + P)
54
Payoff & profit of protective
put position at expiration
55
Protective put problem

1.
2.
You establish a protective put position on ABC
stock today by buying 100 shares of ABC stock
at $40 per share and buying a 3-month put
option contract on the same stock. Each put
option has a strike price of $40 and a premium
of $8.
At the end of 3 months, compute the profit from
the position if ABC’s stock price is $30.
What is the breakeven stock price for this
strategy?
56
Portfolio Insurance
Long 1 stock (portfolio) and Long a Put
Option on the stock (portfolio), S=20,
X=20, P=5.
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
-5
-10
-15
-20
-25
-30
57
Covered call

Writing a call option on an asset together with
buying the asset.

Written option is “covered” because the potential
obligation to deliver the stock is covered by the
stock held.

Strategy produces immediate cash flows through
the sale of the call options.
58
Covered call:
Payoff & profit at expiration
S0 = initial asset price, and
C = call option premium.
Cost of the position = asset price - call premium
= S0 - C
Payoff of stock
- Payoff of call
Total payoff
Profit
ST ≤ X
ST
0
ST
ST – (S0 – C)
ST > X
ST
– (ST – X)
X
X – (S0 – C)
59
Payoff & profit of covered call
position at expiration
60
Covered call problem

1.
2.
You establish a covered call position on XYZ
stock today by buying 100 shares of XYZ stock
at $16 per share and writing a 3-month call
option contract on the same stock. Each call
option has a strike price of $17 and a premium
of $0.25.
At the end of 3 months, compute the profit from
the position if XYZ’s stock price is $14.
What is the breakeven stock price for this
strategy?
61
Straddle
A
combination of a call and put, each with
the same exercise price (X) and expiration
date (T).
 Rationale
You believe a stock will move a lot in price but
are uncertain about the direction of the move.
 Straddle allows you to benefit from a price
move in either direction.

62
Straddle:
Payoff & profit at expiration
C = call option premium
P = put option premium
Cost of the position = call premium + put premium
=C+P
Payoff of call
+ Payoff of put
Total payoff
Profit
ST < X
0
X – ST
X – ST
X – ST – (C + P)
ST ≥ X
ST – X
0
ST – X
ST – X – (C + P)
63
Payoff & profit of
straddle position at expiration
64
Straddle problems

You establish a straddle position on ABC Inc.’s stock by
buying a three-month call option and a three-month put
option. Both options have an exercise price of $50.
ABC’s current stock price is $50 per share. The call
option premium is $5 and the put option premium is $3.
1. Compute the straddle’s profit on the expiration date if
ABC’s stock price on expiration date is $60.
2. How far would ABC’s stock price have to fall for you to
make a profit on your initial investment?
65
Which strategy? (1)

You are the portfolio manager of Nohope Equity Fund. One of your
stock holdings is Refin Corp. Your equity analyst tells you that
Refin’s stock price is not expected to rise substantially within the
foreseeable future. At the same time, you need to raise cash right
now to meet fund redemptions. What would be a simple options
strategy to exploit your conviction about the stock price’s future
movements and allow you to earn immediate income?
a.
b.
c.
d.
e.
Long call.
Long put.
Protective put.
Covered call.
Long straddle.
66
Which strategy? (2)

a.
b.
c.
d.
e.
PUTT Corporation’s common stock has been trading in
a narrow price range for the past month, and you are
convinced it is going to break far out of that range in the
next three months. You don’t know whether it will go up
or down, however. What would be a simple options
strategy to exploit your conviction about the stock price’s
future movements?
Long call.
Long put.
Protective put.
Covered call.
Long straddle.
67
Practice 10
Chapter 15: 7,8,13,16,17,22,23, 27.
Hints: to graph the payoff diagram of strategies,
you need to firstly draw the payoff tables as
demonstrated in slide 54, 59, 63. Based on the
tables, you can draw the diagram. See the
following example, which is the solution to 16
(a).
68
Position
S < X1
X1 < S < X2
X2 < S < X3
X3 < S
Long call (X1)
0
S – X1
S – X1
S – X1
Short 2 calls (X2)
0
0
–2(S – X2)
–2(S – X2)
Long call (X3)
0
0
0
S – X3
Total
0
S – X1
2X2 – X1 – S
(X2–X1 ) – (X3–X2) = 0
69
Determinants of option value

1.
2.
3.
4.
5.
6.
The values of call and put options are affected
by:
Underlying asset price
Exercise price
Volatility of the asset price
Option’s time to expiration
Riskfree interest rate
Cash payouts from underlying asset, e.g.,
dividend.
70
Determinants of call option value
If this variable increases
Value of call option
Asset price, S
Increases
Exercise price, X
Decreases
Volatility, 
Increases
Time to expiration, T
Increases
Interest rate, rf
Increases
Cash payouts e.g., dividend
Decreases
71
Determinants of put option value
If this variable increases
Value of put option
Asset price, S
Decreases
Exercise price, X
Increases
Volatility, 
Increases
Time to expiration, T
Increases/ Uncertain
Interest rate, rf
Decreases
Cash payouts e.g., dividend
Increases
72
Binomial option pricing model (16.2)

Proposed by Cox, Ross and Rubinstein


Replication principle:



“Option Pricing: A Simplified Approach”, Journal of
Financial Economics, 1979, 7, 229-263.
Two portfolios producing the exact same future
payoffs must have the same value.
Otherwise, there will be opportunities for riskless
arbitrage.
Use this model to price European call options.
73
Single-Period Binomial Model (1)




S=100, it will move to either 110 or 90 in one year
X=100
If the investors borrow money, the interest rate=6% for
one year.
What is the price of the European call option?
74
Single-Period Binomial Model (2)
Try to find a synthetic portfolio including
stocks and bonds, which will replicate the
payoff of a Call option.
Solve: 10  N 110 B  (1  0.06)
0  N  90  B  (1  0.06)
N  0.5
B  42.4528
75
Single-Period Binomial Model (3)



S=100, it will move to either 110 or 90 in one year
X=100, r=6%
Form a synthetic portfolio: short position in a bond (sell a
bond to borrow money) at $42.4528 and long position in
½ share of stock
ST=110
ST=90
after 1 year
Synthetic portfolio
stock
bond
Net payoff
Call
Call
55
-45
10
45
-45
0
10
0
76
Single-Period Binomial Model (4)
Since the payoff (value) for the synthetic
portfolio is exactly the same as that for the
Call option in all circumstances, the price
(initial value) of the portfolio must be the
same as that of the Call.
C0  N  S0  B  0.5 100 42.4528 7.5472 7.55
77
Another way to Price a call option (1)

Compute the price of a call option written on the
stock of ABC Inc. The stock is currently selling
for S0 = $100. The stock price will either
increase by a factor of u = 2 to $200 or fall by a
factor of d = 0.5 to $50 by year end. The call
option has a strike price of $125 and a time to
expiration of one year. The risk-free interest rate
is 8% p.a.
78
Another way to Price a call option (2)
Terms:
Su = u x S0 = year end stock price if price rises to $200
Sd = d x S0 = year end stock price if price falls to $50
C0 = call option price
Cu = call option payoff if stock price is Su
Cd = call option payoff if stock price is Sd
1. Call option payoffs
Cu = 200 – 125 = 75
Cd = 0 (not optimal to exercise)
79
Another way to Price a call option (3)
2) Hedge ratio, H
Cu - Cd
H =
uS 0 - dS 0
H = (75 – 0)/(200 – 50) = 75 /150 = 0.5/1
3) Form a portfolio that is short 1 call and long 0.5 shares
of ABC stock.
80
Another way to Price a call option (4)
4) Compute the end-of-period payoff.
Payoff in 1 year for each possible
stock price
Write 1 call
Buy 0.5 shares
Total
Sd = $50
Su = $200
0
0.5 x 50 = 25
25
-(200 – 125) = -75
0.5 x 200 = 100
25
Year end payoff is certain!
81
Another way to Price a call option (5)
5) Compute present value of $25 with a one-year risk-free
interest rate of 8%.
PV = 25/1.08 = $23.1481
6) Set value of hedged position equal to present value of
the certain payoff and solve for call option’s value.
0.5S0 – C0 = 23.1481
50 – C0
= 23.1481
C0
= 26.8519
= 26.85
82
Binomial option pricing model
 Basic

idea:
You can form a portfolio consisting of stock
and call options that produces a certain (norisk) payoff in the future.
 This
is also called the hedged position.
Discounting this payoff at the risk-free rate
gives the portfolio value today.
 Using this present value and the current stock
price, you solve for the call option price.

83
Another pricing problem

1.
2.
You are attempting to value a call option with
an exercise price of $100 and one year to
expiration. The underlying stock pays no
dividends, its current price is $100, and you
believe it has a 50% chance of increasing to
$120 and a 50% chance of decreasing to $80.
The riskfree interest rate is 10% p.a.
What is the hedge ratio?
Calculate the call option’s value using the
Binomial pricing model. Verify that the call
option price is $13.64.
84
Price a call option using hedge ratio
Cu  C d
A
H

uS 0  dS 0 B
Implication: form a portfolio that is long A shares of stock
and short B calls
85
Price a put option using hedge ratio
Pu  Pd
A
H

uS 0  dS 0
B
Implication: form a portfolio that is long A shares of stock
and long B puts
Example: Chapter 16, 8
86
Multi-period Binomial Tree
87
Put-Call Parity Relationship (1)
 Links
the prices of European call and put
options.
 Given:
European call option price
 underlying asset price
 Riskfree rate


The Put-Call Parity Relationship produces
the put price.
88
Put-Call Parity Relationship (2)
Assumptions:
1.
Options are European options.
2.
Both call and put options are written on the
same underlying asset.
3.
Underlying asset does not pay any cash flow
(e.g., dividends) before option expiration.
4.
Continuous compounding.
89
Put-Call Parity Relationship (3)
 The
relationship says that:

A portfolio that is long 1 call and short 1 put
has the same payoffs at expiration as…

A portfolio made up of the underlying asset
plus a borrowing position. The riskfree interest
rate is r.
90
Put-Call Parity Relationship (4)
 Suppose
you buy a call option and write a
put option, each with the same exercise
price, X, and the same expiration date, T.
At expiration, the portfolio payoff:
ST ≤ X
ST > X
Long call
Short put
0
-( X – ST)
ST – X
0
Total
ST – X
ST – X
91
Put-Call Parity Relationship (5)
Compare this payoff to that of a portfolio made
up of:
 1 share of stock
 borrowing equal to the present value of the
exercise price, Xe-rT. At maturity, repay X.
 At expiration, the portfolio payoff:

Long stock
Loan repayment
Total
ST ≤ X
ST
ST > X
ST
-X
ST – X
-X
ST – X
92
Put-Call Parity Relationship (3)
Call
option
price
Put
option
price
Current
stock
price
Present value of
exercise price
C - P = S 0 - PV (X )
Using continuous compounding, we have
C - P = S0 - Xe
r = Riskfree rate (per annum basis)
- rT
Time to
expiration
in years 93
Pricing a put option (1)
 Put
option price, P
P = C - (S 0 - Xe
- rT
)
94
Practice 11
Chapter 16: 5,8,9,35,36
95