Options
Chapter 2.5
Chapter 15
Learning Objectives
 Understand
key terms related to options
and options markets
 Compute
payoffs and profits to option
holders and writers
 Calculate
potential profits from various
options strategies
 Describe
the put-call parity relationship
2
Derivative

A derivative is a security who’s value is
dependent on another assets
 Base
Asset Ex: commodity prices, bond and stock
prices, or market index values
 Derivatives are contingent claims

Their payoffs depend on the value of another securities.
3
Options: Calls and Puts
Options: holder has the right, but not the
obligation, to buy or sell a given quantity of an
asset on (or before) a given date, at a price
agreed upon today (strike or exercise price)
 Call: Owner has the right, but not the
obligation, to BUY an asset at the strike price
 Put: Owner has the right, but not the
obligation, to SELL an asset at the strike

4
Rights and Obligations
Buyer:
Gets all the Rights
Seller:
Gets all the
Obligations
Calls
Right to Buy the
asset
Is obliged to Sell the
asset
Puts
Right to Sell the
asset
Is obliged to Buy the
asset
5
Option Value

The value of an option at expiration is a function of
the stock price and the exercise price.
Example: Exercise price of $80
Stock Price
$60
70
80
90
100
110
Call Value
0
0
0
10
20
30
Put Value
20
10
0
0
0
0
Options Terminology

Exercising the Option
 Using

Strike Price or Exercise Price
 The

market price
Expiration Date
 The

price specified by the option
Spot Price
 The

the option
option’s maturity date
European & American options
 Europeans can only be exercised at expiration.
 Americans can be exercised at any time up to
expiration. BUT NEVER ARE
7
Options Terminology

In-the-Money: Exercising the option results in a profit
 Call:

Call value decreases as the strike price increases
 Put:


exercise price < market price
exercise price > market price
Put value increases as the strike price increases
At-the-Money: Exercising the option results in 0 profit
 exercise
price = market price
 Out-of-the-Money: Exercising would generate a loss
 Call: market price < exercise price
 Put: market price > exercise price
8
The Option Contract

The purchase price of the option is called the
premium
 Stock
options cover 100 shares & premium is on a per
share basis



Sellers (or Writers) of options receive premium.
Call: If the Holder (or Buyer) exercises the
option, the Writer must sell the underlying asset
Put: If the Holder (or Buyer) exercises the option,
the Writer must buy the underlying asset.
9
Call Payoffs

Holder of the option
 Pays
the premium at time=0
 Has the right to exercise the option at time=T

Notation
 Stock

Price at T = ST
T is maturity, t is any other time
 Exercise
Price = X
Don’t Exercise
Call Payoff (Intrinsic Value)
0
Exercise
ST - X
10
Call Holder Payoff and Profit

Value of the Call at T: CT = Max [ST – X, 0]
CT
Profit
ST >X
ST – X
ST - X - Ct
ST < X
0
-Ct
11
Payoff and Profit to Call Option at Expiration
What is the strike price?
What is the premium?
12
What is the value of a Call at t?
At maturity CT = Max [ST – X, 0]
 Would you be willing to sell for St-X at t?

 Hint:

What could happened to the stock price?
Time value is the premium a rational investor
will pay above an options intrinsic value
 This
base on the likelihood that the stock price will
move making the option more valuable
 Time Value = Option Price – Intrinsic Value
13
Time Value

A call with an exercise prices $100, is selling
for $16. If the current stock price is $110, what
is the calls intrinsic value & time value?
14
Call Writer Payoff and Profit
Exercise Cost
Profit
ST >X
-(ST – X)
-(ST - X) + Ct
ST < X
0
Ct
15
Payoff and Profit to Call Writers at Expiration
16
Calls: A Zero Sum Game
Call Option
Decision
Option Payoff
(holder)
Option Profit
(holder)
Option Payoff
(writer)
Option Profit
(writer)
If ST < X
If ST > X
No exercise
Exercise
0
ST – X
-C
(ST – X) – C
0
- (ST – X)
+C
C - (ST – X)
17
Zero Sum Profit Diagram
Option payoffs ($)
60
Exercise price = $50; Option Price = $10
Buy a call
40
20
10
20
40
50
60
80
100
12
–10
–20
–40
Stock price ($)
Write a call
Profit and Loss on a Call
A February 2013 call on IBM with an exercise
price of $195 was selling on January 18, 2013,
for $3.65.
 The option expires on the third Friday of the
month, or February 15, 2013.
 If the price of IBM on Feb 15, 2013 is $194,
what is the call worth?

19
Profit and Loss on a Call (cont.)

Suppose IBM sells for $197 at expiration
 Remember:

strike = $195, premium = $3.65
What is the value of the option?
 Call
Intrinsic value = stock price-exercise price
Will the option be exercised?
 What is the Profit/Loss on this investment?

 Profit

= Final value – Original investment
What must the price of IBM be for the option
to break-even?
20
Call Option Problem
At time=0 you buy a call option on IBM for
$3.00. The option gives you the right to buy 100
shares of IBM stock at time=T at $65
 What
is the payoff to you if ST = $70?
 What
is the payoff for the writer if ST = $70?
 What
is the payoff to you if ST = $60?
 What
is the payoff for the writer if ST = $60?
21
Puts Payoffs

Holder of the option
 Pays
the premium at time=0
 Has the right to exercise the option at time=T

Notation
 Stock
Price at T = ST
 Exercise Price = X
Don’t Exercise
Put Payoff (Intrinsic Value)
0
Exercise
X - ST
22
Put Holder Payoff and Profit

Value of the Put at T: PT = Max [X - ST , 0]
Payoff
Profit
ST <X
X - ST
X - ST - Pt
ST > X
0
-Pt
23
Payoff and Profit to Put Option at Expiration
24
Put Writer Payoff and Profit
Exercise Cost
Profit
ST <X
-(X - ST)
-(X - ST) + Pt
ST > X
0
Pt
25
Puts: Another Zero Sum Game
Put Option
Decision
Option Payoff
(holder)
Option Profit
(holder)
Option Payoff
(writer)
Option Profit
(writer)
ST < X
If ST > X
Exercise
No Exercise
X – ST
0
(X-ST) - P
-P
- (X-ST)
0
+P - (X-ST)
+P
26
Profit and Loss on a Put

Consider a February 2013 put on IBM with an
exercise price of $195, selling on January 18,
2013 for $5.00.
 Option
holder can sell a share of IBM for $195 at
any time until February 15.

If IBM sells for $196, what is the put worth?
27
Profit and Loss on a Put
Suppose IBM’s price at expiration is $188.
 What is the value of the option?
Put value = Exercise price- Stock price
 Will the option be exercised?
 What is the Profit/Loss on this investment?
Profit = Final value – Original investment
 What is the HPR?

28
Put Option Problem
At time=0 you buy a put option on ITT stock for
$2.00. The option gives you the right to sell 100
shares of ITT stock at time=T at $50
 What
is the payoff to you if ST = $55?
 What
is the payoff to the put seller if ST = $55?
 What
is the payoff to you if ST = $45?
 What
is the payoff to the put seller if ST = $45?
29
Option versus Stock Investments
Could a call option strategy be preferable to a
direct stock purchase?
 Suppose you think a stock, currently selling for
$100, will appreciate.
 A 6-month call costs $10

 Calls

cover 100 shares
You have $10,000 to invest.
30
Option versus Stock Investment
Investment
Strategy
Investment
Equity only
Buy stock @ $100 (100 shares)
$10,000
Options only
Buy calls @ $10 (1,000 options)
$10,000
Options +
T-Bills
Buy calls @ $10 (100 options)
Buy T-bills @ 3% Yield
$1,000
$9,000
31
Strategy Payoffs
32
Option Versus Stock Investment
33
Strategy Conclusions

The all-option portfolio, B, responds more than
proportionately to changes in stock value
 Options

are inherently levered.
Portfolio C, T-bills plus calls, shows the insurance
value of options.
C
‘s T-bill position cannot be worth less than
$9270.
 Some return potential is sacrificed to limit
downside risk.
34
Combining Options

Puts and calls can serve as the building blocks
for more complex option contracts
 Can

be used to manage risk
This is financial engineering
 Allows
you to tailor the risk-return profiles to meet
your client’s desires

Ex: Protective Puts: Underlying asset and put
are combined to guarantee a minimum
valuation
 Put
is insurance against stock price declines.
35
Protective Put Strategy
Limits the downside risk of an investor that owns the
stock
Protective Put payoffs
Payoffs
$40
Buy a put with an exercise
price of $50, for $10
Buy the $0
stock, for $40
$40
$40
$50
Stock Price
36
Payoff Calculations
• Remember: P = $10, St = $40, X = $50
At Maturity (T)
Position
Today (t)
ST < X
ST > X
Buy a Put
Buy the
Stock
Total
37
Protective Put Strategy (Profit)
Portfolio
Payoffs
Protective Put payoffs
$50
$0
$50
Stock Price
38
Protective Put at Expiration
39
Covered Calls
Purchase stock and write calls against it.
 Call writer gives up any stock value above X
in return for the initial premium.
 Imposes “Sell Discipline”
 Forces you to sell a stock that you planned on
once it hits a certain price

40
Covered Call
Buy the
stock, for $40
Imposes sell discipline
Payoffs
Covered Call payoffs
$10
Stock price ($)
$40
–$30
$50
Sell an at the money call $10
–$40
41
Payoff Calculations
At Maturity (T)
Position
Today (t)
ST < E
ST > E
Sell a Call
Buy the
Stock
Total
42
Covered Call Position at Expiration
43
Straddle
The straddle is a bet on volatility.
 Long straddle: Buy call and put with same
exercise price and maturity.

A
non-directional bet on large price change
 To make a profit, the change in stock price must
exceed the cost of both options.

Short straddle: Sell a call and a put with the
same exercise and maturity
 Bet
that price will be relatively constant
44
Long Straddle A way to profit on big price
movements without knowing the direction
Option payoffs ($)
The stock is @ $50, and we expect a big move
Buy an at the money
call for $10
40
30
The L.S. makes money as long as
the price moves more than $20
Stock price ($)
–10
30
40
60 70
Buy an at the money
put for $10
–20
$50
45
Payoff Calculations
At Maturity (T)
Position
Today (t)
ST < E
ST > E
Buy a Put
Buy a Call
Total
46
Straddle Position at Expiration
47
Short Straddle The opposite of a long straddle
Option payoffs ($)
We don’t want big price swings
The S.S. makes money as long as the price
doesn’t move more than $20
20
Sell an at the money put for $10
10
Stock price ($)
30
40
$50
60
70
–30
–40
Sell an at the money call $10
48
Payoff Calculations
At Maturity (T)
Position
Today (t)
ST < E
ST > E
Sell a Put
Sell a Call
Total
49
Spread

A spread is a combination of two or more calls
(or two or more puts) on the same stock with
differing exercise prices or times to maturity.
 Some
options are bought, whereas others are sold
(written)

A bullish spread is a way to profit from
moderate stock price increases
 EX.
Buy a call with a strike of $20 and sell a call
with a strike price of $25
50
Bullish Spread Position at Expiration
51
Replicating Portfolio Intuition
We can value a derivative by valuing a
portfolio of securities, that we can value, that
has the same payoffs
 Two portfolios with the same payoffs must
cost the same


The Law of One Price: Two securities with the
same payoffs needs to have the same price
52
Binomial Option Tree Pricing
1.
2.
3.
We are going to assume that the price of the
stock in one period is either Up, or Down
We form a portfolio of the stock and bonds
that has the same payoffs
Since our portfolio has the same payoffs as
the option, they must cost the same

The Law of One Price
53
Payoff Characteristics

Spread
 This
is the difference between the payoffs in the
Up State and the Down State

Amount
 The
amount of the payoffs
54
Replicating a Call

Consider a call with an:
X
= $55
 S0 = $55
 rf = 10%
 ST will be either $75 or $35
 What is the value of the call today?
55
Payoffs
S0
ST
C1
$75
$55
$35
Spreads
56
Replicating the Spread: Delta

Delta: This is the number of shares needed to
replicate an option
 Book
calls this the hedge ratio
 Delta
= (option spread) / (stock spread)
 Delta =
57
Replicating Amounts
If we buy the stock, it has value in both the Up
and Down state of the world
 The option is only valuable in the Up State
 How do we replicate the 0 payoff in the Down
state?


How much?
58
Payoffs to Replicating
S0
S1
C1
$75
$20
$35
$0
Our .5 Repay
Stock Loan
Total
$55
59
Price of the Call Today

How did we replicate the call?
We bought 0.5 shares of stock
 Today that would costs $____
We borrowed $_____________?
Today this portfolio costs?
What does the call costs us?
60
Replicating a Call: Given

Consider an at the money call:
X
= $25
 St = $25
 rf = 5%
 ST will be either 15% more or less than the current
price
 What is the value of the call today?
61
Payoffs to Replicating: Given
S0
S1
C1
Our .5 Repay
Stock Loan
Total
$28.75
$3.75
$14.38
-$10.63
$3.75
$21.25
$0
$10.63
-$10.63
$0
$25
Spread
7.50
3.75
Delta = 3.75/7.5 = 0.5
Call = -25 * 0.5 + 10.63/1.05; the call costs $2.38
62
Replicating a Put
Consider a put with an:
 E = $55
 St = $55
 rf = 10%
 ST will be either $75 or $35.
 What is the value of the put today?

63
Payoffs
S0
S1
P1
$75
$55
$35
Spreads
64
Put Delta
Delta = (option spread) / (stock spread)
Delta =
We will _____ ______ shares of stock
 If you short then we lose money if the price
increases, options “never” lose money.
How do we replicate this?

65
Payoffs to Replicating
S0
S1
P1
$75
$0
$35
$20
Short 0.5 Receive Total
Stock
Loan
$55
66
Price of the Put Today

How did we replicate the Put?
Shorted 0.5 shares of stock, making ____
We lent _______
Today this portfolio costs?
What does the put costs us?
67
What if the option covers multiple
periods?

Just work backwards
 Regardless
of how many periods you are given the
approach I showed you will work.
 Determine the possible prices at maturity, then use
those prices to determine the price for the period
before that. Repeat until you get to today.
68
A Two Period Call

Consider a two period call:
X
= $55
 St = $55
 rf = 10%
 Each period the price either increases or decreases
by $10.
 What is the value of the call today?
69
S0
S1
C1
S2
C2
$55
Spreads
Upper
Spreads
Bottom
70
What is the Price of C1u & C1b?

C1b =
 Delta

=
C1u =
 Delta
=
How many shares must we buy? When?
 How much must we borrow?

71
Moving Back One Period
S0
S1
C1u,C1b
$65
$15
$45
$0
$20
$15
$55
Spreads
72
What is the Price of C0?
C0 =
 Delta
= ____ / ____ = ____
 How much will we borrow?
73
Put Call Parity

We can also use replicating portfolios to value
puts and calls in relation to one another
 Put
Call Parity is the relation between a put and
call with the same exercise price (E) and maturity

The payoffs from buying a call and selling a
put is the same as the payoffs from buying the
stock and borrowing the PV of the exercise
price → Must cost the same
P –C = PV (X) - St
74
PV(X)

There are two ways to determine the PV(X)
time: PV (X) = X/(1+r)t
 Continuous time: PV (X) = Xe-rT
 Discrete

This is what is used in practice
75
Portfolios
Portfolio 1: Buy a call and Write a put
 Portfolio 2: Buy the stock but Borrow PV (X)

 Levered

Equity positions
If the payoffs are the same the price must be
the same
-C+P = -S0 + PV(X)
C-P = S0 – PV(X)
76
Payoff-Pattern of Long Call–Short Put Position
77
Proof by Counter Example

Assume that:
Stock Price = 110
Call Price = 14
Put Price = 5
Risk Free = 5%
Maturity = 6 months Strike Price = 105

Portfolio 1 costs
 -14

+ 5 = -9
Portfolio 2 costs:
 -110

+ 105e-5 = -7.59
Two different costs for the sample payoffs →
ABRITRAGE
78
Arbitrage Strategy Payoff
79
Put Call Parity Example

What is the value of a put with an exercise
price of $51, if the stock is currently trading at
$49. The price of the corresponding call option
is $4.65. According to put-call parity, if the
effective annual risk-free rate of interest is 4%
and there is 1 year till expiration, what should
be the value of the put?
It doesn’t matter if compounding is monthly
or quarterly
 FYI:
80
The Black-Scholes Model
The Black-Scholes Model follows the same logic we just did
but breaks time down into nanoseconds
-Allows us to value options in the real world just as we
have done in the 2-state world.
C0  S  N(d1 )  Ee  rT  N(d 2 )
Where
C0 = the value of a European option at time t = 0
r = the risk-free interest
rate.
2
σ
N(d) = Probability that a
ln(S / E )  (r  )T
standardized, normally
2
d1 
distributed, random
 T
d 2  d1   T
variable will be less than
or equal to d.
81