Introduction to options
Some basic concepts
FIN 819: lecture 6
Today’s plan
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Briefly review the case discussed last week.
Review what we have learned so far
Introduction of options
• Definition of options
• Position diagrams
• No arbitrage argument
• Put-call parity
• Application of put-call parity
• How does stock return volatility affect option values?
FIN 819: lecture 6
What have we learned so far?

So far we have discussed or reviewed some
fundamental concepts and ideas about valuation
• Present value concept
• Discounted cash flow approach and the NPV rule
• Free-cash flow calculation
• Cost of capital (WACC)
• CAPM
• Levered and unlevered betas
• Two cases related to these concepts
FIN 819: lecture 7
Introduction to options
What is an option in Finance?
 An option is a right or opportunity to do something at
a specified price or cost on or before some specified
date.
 An option, is a contract
 Options are everywhere.
•
•
IBM offers its CEO a bonus that is related to the stock price
(stock options)
IBM postpones investment in a positive-NPV project, even if
IBM has capital for taking this investment.
FIN 819: lecture 7
Options
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Financial options
Real options
It depends on what is the underlying
asset the option is written in
FIN 819: lecture 7
Options in financial assets
What is an option written in financial assets?
 A financial option is a right to sell or buy
some financial asset at a specified price or
cost on or before some specified date.
• (ex: option written on IBM or Dell)
 A financial option can be regarded as a
contract between sellers and buyers
 The financial asset specified in the financial
option contract is often called the underlying
financial asset.
FIN 819: lecture 7
Real Options


Real options
• Options that are written on real assets are called
real options
• For example, the option to set up a factory is
called a real option
In the next two lectures, we focus on financial
options, since it is easier to price them.
FIN 819: lecture 7
Now we focus on (financial)
options…
Suppose the financial asset is common
stock or stock
There are two basic types of options,
• Call option
• the right to buy a share of stock at a specified
price before or on some date.
• Put option
• the right to sell a share of stock at a specified
price before or on some date.
FIN 819: lecture 7
Strike Price, Expiration Date

The specified price is called the strike
price or exercise price. (the price you
would like to buy/sell the underlying
stock)

The specified date is called the maturity
date or expiration date. (the date by
which you want to buy/sell the stock)
FIN 819: lecture 7
Exercising an Option
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An option is exercised when the buyer of the option
decides to buy or sell the stock at the specified price,
at which time the seller must sell or buy the stock at
the specified price
Oh yes, according to expiration terms
FIN 819: lecture 7
American vs. European Option

Call options
•
•

•
Buy the stock on the specified date
American call option
•
Buy the stock on or before the specified date
Put options
•
•
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European call option
European put option
•
Sell the stock on the specified date
American put option
•
Sell the stock on or before the specified date
Key Difference?
•
•
European (1 date),
American (many dates up until expiration date)
FIN 819: lecture 7
Option Obligations
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Options are rights (to the buyer), and are obligations (to
the seller)
This means that:
•
•
the buyer of an option may or may not exercise the option.
However, the seller of the option must sell or buy the
underlying assets if the buyer decides to exercise the
option.
Call option
Put option
Buyer
Right to buy asset
Right to sell asset
Seller
Obligation to sell asset
Obligation to buy asset
FIN 819: lecture 7
Payoff or cash flows from
options at expiration date

The payoff of a call option with a strike price
K at the expiration date T is
max( S (T )  K ,0)
• Where S(T) is the stock price at time T

The payoff of a put option with a strike price K at the
expiration date T is
max( K  S (T ),0)
•
Where S(T) is the stock price at time T
FIN 819: lecture 7
Example on payoffs
Suppose that you have bought one European put and a
European call on stock ABC with the same strike price
of $55. The payoffs of your options certainly depend on
the price of ABC on expiration
Stock Price
$30
40
50
60
70
80
Call
0
0
0
5
15
25
Put
25
15
5
0
0
0
FIN 819: lecture 7
Option payoff at expiration
Call option $ payoff
Call option value (graphic) given a $55 exercise price.
$20
55
Share Price
FIN 819: lecture 7
75
Option payoff
Put option value
Put option value (graphic) given a $55 exercise price.
$5
50 55
Share Price
FIN 819: lecture 7
Option payoff
Call option $ payoff
Call option payoff (to seller) given a $55 exercise price.
55
Share Price
FIN 819: lecture 7
Option payoff
Put option $ payoff
Put option payoff (to seller) given a $55 exercise price.
55
Share Price
FIN 819: lecture 7
Some examples

Please draw position diagrams for the
following investment:
• Buy a call and put with the same strike price
•
•
and maturity (straddle)
Buy two calls with different strike prices (K1
and K2) and sell two calls with a strike price
that equals the average strike price of the two
calls you bought. (butterfly)
Buy a stock and a put (protective put)
FIN 819: lecture 7
Option payoff
Position Value
Straddle - Long call and long put
Straddle
Share Price
FIN 819: lecture 7
Option Value
Position Value
Butterfly
K1
Share Price
FIN 819: lecture 7
K2
Option payoff
Position Value
Protective Put - Long stock and long put
Protective Put
Share Price
FIN 819: lecture 7
Form your desired portfolios
Suppose you have access to risk-free
securities, stocks, calls and puts. Can you form
a portfolio now to have the following payoffs at
time T?
Position Value

K
K
K1
FIN 819: lecture 7
Share Price
No arbitrage concept or one
price rule

If two securities have the exactly the
same payoff or cash flows in every state
of future, these two securities should
have the same price; otherwise there is
an arbitrage opportunity or money
making opportunity.
FIN 819: lecture 7
Example
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Two treasury bonds A and B both have
the maturity of 10 years and coupon rate
of 6%. Certainly, they should have the
same price; otherwise suppose that A is
more expensive than B. You can make
money by buying B and short selling A.
You can make money PA-PB.
Do you like this beautiful idea?
FIN 819: lecture 7
Put-Call Parity

Let P(K,T) and C(K,T) be the prices of a
European put and a call with strike
prices of K and maturity of T. Then we
have
C( K , T )  S0  P( K , T )  KR
C( K , T )  KRf T  S0  P( K , T )
Where
R f 1  r f
FIN 819: lecture 7
T
f
or
Let’s show put-call parity
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We can first use position diagrams to
show put-call parity
We can also simply use the payoffs in
the future to show put-call parity
This exercise is a good way of getting
used to the ideas of the single price rule
or no arbitrage argument.
FIN 819: lecture 7
Position diagram
Position Value
Payoff of investing PV(K) in risk-free security and
buying a call
K
Share Price
FIN 819: lecture 7
Position diagram
Position Value
Payoff of long stock and long put
K
Share Price
FIN 819: lecture 7
The conclusion

Since both portfolios in the previous two
slides give you exactly the same payoff,
they must have the same price. That is,
C( K , T )  KRf T  S0  P( K , T )
FIN 819: lecture 7
Another way of showing put-call
parity
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Consider the following portfolio:
• Buy the stock
• Buy a European put option
• Borrow the present value of the strike price KR f T
• Where Rf= 1+ rf
The cost of this portfolio is
The payoff of this portfolio
• If ST >= K, the payoff is ST-K
• If ST < K, the payoff is 0.
FIN 819: lecture 7
S0  P  KRf T
Portfolio (continues)
Position
Stock
Borrowing
Put
Total
Final payoff of the portfolio
Portfolio
S0
 KR f T
P
S 0  P  KR f T
ST<K
ST
ST>K
-K
-K
K - ST
0
FIN 819: lecture 7
ST
0
ST - K
The conclusion

Since the portfolio and a call option have
exactly the same payoff, their prices
should be the same. That is,
C( K , T )  S0  P( K , T )  KR
FIN 819: lecture 7
T
f
Applications of option concepts
and put-call parity
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One important application of option
concepts and put-call parity is the
valuation of corporate bonds.
For example, suppose that a firm has
issued $K million zero-coupon bonds
maturing at time T. Let the market value
of the firm asset at time t be V(t).
FIN 819: lecture 7
Applications of option concepts
and put-call parity (continue)
Position Value
Payoff of equity
K
Market value of asset
FIN 819: lecture 7
Applications of option concepts
and put-call parity (continue)
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So based on the position payoff diagram
in the previous slide, we can see that
the value of equity is just the value of a
call option with strike price K.
Then bond value =Asset value –equity
value (value of call: C(K,T)
Using the put-call parity, we have
Bond value=PV(K)- P(K,T) (value of put )
FIN 819: lecture 7
Applications of option concepts
and put-call parity (continue)
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Who will bear the default cost: equity
holders or debt holders?
The value of risky corporate bonds is
equal to the value of the safe corporate
bonds minus the cost of default.
When will the firm default?
• At time T, if the value of asset is less than K,
the firm will default. P(K,T) is the cost of this
default to bond holders.
FIN 819: lecture 7
The value of option


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Since an option is a right to buy or sell
securities, its price is non-negative.
If the option price is negative, what will
happen?
If the option value must be non-negative, can
you use what you have learned to value a call
option or put option by considering the
following two things
•
•
Expected cash flows
The risk of options
FIN 819: lecture 7
Volatility and option value
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For options, the larger the volatility of the
underlying asset, the larger the value of
the option
For stocks, the larger the volatility of the
underlying asset, the smaller the value of
stocks
Why ?
FIN 819: lecture 7