Options and Derivatives
For 9.220, Term 1, 2002/03
02_Lecture17 & 18.ppt
Student Version
Outline
1.
2.
3.
4.
5.
6.
Introduction
Option Definitions
Option Payoffs
Intuitive Option Valuation
Put-Call Parity
Summary and Conclusions
Introduction
 We now turn our attention to options.
 Almost every decision can be framed in terms
of an option. The more we understand options,
the more we can understand the valuation
implications of decisions, flexibility and
information.
 An understanding of options will also help us
understand more about what affects the values
of equity and debt.
 We will start our analysis with financial options,
and then later turn to real options and
corporate securities as options.
Option Definitions
 Derivative Security:
 a derivative security is a financial
security that is a claim on another
security or underlying asset.
 Examples include options, warrants,
rights, futures, swaps, convertibles, etc.
 The value of the derivative is dependent
on the value of the underlying asset.
Option
 If you have an option, then you have the
right to do something. I.e., you can make a
decision or take some action.
 The option owner has a choice to make.
 Usually the choice can be made over time after
more information is known.
 Having an option is valuable; options never have
a negative value, because, at worst, the option
owner can discard the option and not take any
action.
Call Option
 The owner of a call option on an asset has the RIGHT
TO BUY the asset (e.g., share of stock) for a prespecified price (called the exercise or strike price) during
some time period (that ends with the expiration date or
maturity date).
 The call option owner is said to be “long” in the option.
The owner possesses the right to do something (the
right to buy).
 The call option issuer (called the call option writer) is
said to be “short” in the option. If the call option owner
exercises his/her option, then the call option writer
has an obligation to sell the asset and must fulfill the
terms of the contract.
Put Option
 The owner of the put option has the RIGHT
TO SELL an asset (e.g., share of stock) at a
prespecified price (the exercise or strike
price) during some time period.
 As with call options, The option owner is said to
be “long” in the option. The put owner possesses
the right to do something (the right to sell).
 The put option issuer (called the put option
writer) is said to be “short” in the option. If the
put option owner exercises his/her option, then
the put option writer has an obligation to buy
the asset and must fulfill the terms of the
contract.
Option Writer
 The option writer is the person who created (or wrote)
the option and then sold it to someone else. As the
writer did not previously own the option, the act of
writing and selling the option is equivalent to shortselling the option.
 The option writer has sold an option or right to the
option owner.
 Thus the writer has taken on an obligation to act on
the instruction of the option owner as specified by the
option contract.
 When selling the option to the buyer, the writer
receives as compensation the “option premium”= the
price the buyer pays for the option.
American and European Options
 American options can be exercises at
any time up to and including the
expiration date.
 European options can only be
exercised on the expiration date
 All else equal, an American option must
be worth at least as much as a European
Option
I’m in the money 
 An option is said to be “in the money”
if exercising it would produce a
positive payoff.
 An “at the money” option would
generate a zero payoff if exercised.
 An “out of the money” option would
generate a negative payoff if
exercised. (Thus an out of the money
option would never be exercised.)
Option Payoffs at Expiration
 It is useful to examine the payoffs of
options when they are about to
expire. At this point in time, the
owner is forced to make the decision
to exercise or to abandon the option.
The option owner will exercise if the
option is in the money when it is
about to expire.
 Consider Call and Put Options…
Long Call Option Payoff at Expiration
(Option to Buy for Exercise Price = E)
Stock Should call be
Price exercised? (E=$50)
$0
$25
$50
$55
$75
Payoff
Long Call Option Payoff at Expiration
 If the call is in-the-money, it is worth ST - E.
 If the call is out-of-the-money, it is worthless.
 Therefore the value of the call when it is about to
expire is as follows:
CaT = CeT = Max[ST - E, 0]
 This assumes investors are rational and will not
exercise to get a negative payoff.
 Where
ST is the value of the stock at expiry (time T)
E is the exercise price.
CaT is the value of an American call at expiry
CeT is the value of a European call at expiry
Long Call Option Payoff at Expiration
60
Option payoffs ($)
40
20
0
0
10
20
30
40
50
60
70
80
90
100
Stock price ($)
-20
-40
-60
Exercise price = $50
Self Study – fill in the blank cells
Long Call Option Payoff at Expiration
(Option to Buy for Exercise Price = E)
Stock Should call be
Price exercised? (E=$80)
$0
$20
$40
$60
$80
$100
$120
$140
$160
Payoff
Short Call Option Payoff at Expiration
Stock Will Owner exercise
Price the Call? (E=$50)
$0
$25
$50
$55
$75
Payoff to Writer
Short Call Option Payoff at Expiration
60
Option payoffs ($)
40
20
0
-20
0
10
20
30
40
Stock price ($)
-40
-60
Exercise price = $50
50
60
70
80
90
100
Short Call Payoff at Expiration
 The short call payoff is just the
negative of the long call payoff.
 The payoff of the short call when it is
about to expire is as follows:
-CaT = -CeT = -Max[ST - E, 0]
= Min[E-ST, 0]
Self Study – fill in the blank cells
Short Call Option Payoff at Expiration
(Option to Buy for Exercise Price = E)
Stock Will Owner exercise
Price the Call? (E=$80)
$0
$20
$40
$60
$80
$100
$120
$140
$160
Payoff to Writer
Long Put Option Payoff at Expiration
(Option to Sell for Exercise Price = E)
Stock Should put be
Price exercised? (E=$50)
$0
$25
$50
$55
$75
Payoff
Long Put Payoff at Expiration
 At expiry, an American put option is worth
the same as a European option with the
same characteristics.
 If the put is in-the-money, it is worth E –
ST if exercised.
 If the put is out-of-the-money, it is
worthless. Thus the long put payoff when
the put is about to expire is as follows:
PaT = PeT = Max[E - ST, 0]
 Where
PaT is the value of American Put at expiry
PeT is the value of European Put at expiry
Long Put Option Payoff at Expiration
60
Option payoffs ($)
40
20
0
0
10
20
30
40
50
60
70 80
90
100
Stock price ($)
-20
-40
-60
Exercise price = $50
Self Study – fill in the blank cells
Long Put Option Payoff at Expiration
(Option to Sell for Exercise Price = E)
Stock Should put be
Price exercised? (E=$80)
$0
$20
$40
$60
$80
$100
$120
$140
$160
Payoff
Short Put Option Payoff at Expiration
Stock Will put owner
Price exercise? (E=$50)
$0
$25
$50
$55
$75
Payoff to Writer
Short Put Option Payoff at Expiration
60
Option payoffs ($)
40
20
0
0
10
20
30
40
50
60
70 80
90
100
Stock price ($)
-20
-40
-60
Exercise price = $50
Short Put Payoff at Expiration
The short put payoff is just the
negative of the long put payoff.
Thus the short put payoff when
the put is about to expire is as
follows:
-PaT = -PeT = -Max[E - ST, 0]
= Min[ST-E, 0]
Self Study – fill in the blank cells
Short Put Option Payoff at Expiration
(Option to Sell for Exercise Price = E)
Stock Will Owner exercise
Price the Put? (E=$80)
$0
$20
$40
$60
$80
$100
$120
$140
$160
Payoff to Writer
Call Option Values with time left
until expiration.
 In the previous slides, we saw the payoffs
for options at the time of expiration.
 If the option was expiring, these payoffs for
the “long positions” would represent the
value of the options at expiration.
 If there is significant time left before the
option expires, then the value of the option
is different than the previous payoff
amounts.
 The following slides examine call options
and explain their valuation when there is
time left before expiration.
Boundaries on an American call
option’s value
 In the best outcome, the result of
owning a call option is that you will
eventually own the underlying asset.
 If the underlying asset is the share of
stock, then the most the call option
could be worth is the current market
price of the share of stock.
 This gives us an upper bound on the
value of the call option.
Upper Bound on the Value of an
American Call
$200
$175
$125
$100
$75
$50
$25
$ Current Market Price of Stock
$200
$175
$150
$125
$100
$75
$50
$25
$0
$0
$ Value
$150
Lower Bound on the Value of an
American Call
 Can an American call option sell for a price
less than the payoff from exercising?
 Consider, for example, if E=100, St=$120,
and Cat=$5 then
Upper and Lower Bounds on the
American Call Value
$200
$175
$125
$100
$75
$50
$25
$ Current Market Price of Stock
$200
$175
$150
$125
$100
$75
$50
$25
$0
$0
$ Value
$150
What is the actual value of the
American Call?
 The actual value of the American Call
depends on how much time there is
until it expires, the underlying stock’s
total risk (σ), the risk-free interest
rate over the life of the call, and
obviously the stock price and exercise
price.
Actual Value of the American Call
$200
$175
$125
$100
$75
$50
$25
$ Current Market Price of Stock
$200
$175
$150
$125
$100
$75
$50
$25
$0
$0
$ Value
$150
Understanding the Value of the
American Call
Next, consider
this area
$200
$175
$125
$100
$75
$25
1st
Consider
this area
$150
$125
$100
$75
$50
$25
$0
$0
$ Current Market Price of Stock
Finally,
consider this
area
$200
$50
$175
$ Value
$150
Change in Call Option Value as the
Time until Expiration Decreases
$125
$75
$50
$25
$ Current Market Price of Stock
$200
$175
$150
$125
$100
$75
$50
$25
$0
$0
$ Value
$100
Change in Call Option Value as the Risk
of the Underlying Asset Decreases
$125
$75
$50
$25
$ Current Market Price of Stock
$200
$175
$150
$125
$100
$75
$50
$25
$0
$0
$ Value
$100
Change in Call Option Value as the
Risk-free Interest Rate Decreases
$125
$75
$50
$25
$ Current Market Price of Stock
$200
$175
$150
$125
$100
$75
$50
$25
$0
$0
$ Value
$100
Summary of Factors that Affect the
American Call Option Value
Factor
Effect on value of the
American call option
A stock price increase
Value rises
An increase in the exercise
price
Value decreases
An increase in the stock’s risk Value increases
A decrease in the time left
until expiration
Value decreases
A decrease in the risk-free
rate
Value decreases
Summary of Factors that Affect the
American Put Option Value
Factor
Effect on value of the
American put option
A stock price increase
Value decreases
An increase in the exercise
price
Value increases
An increase in the stock’s
risk
Value increases
(same as for call)
A decrease in the time left
until expiration
Value decreases
(same as for call)
A decrease in the risk-free
rate
Value increases (PV of exercise
price received is higher)
Questions for Discussion
1. Assuming there are no dividends paid,
would it ever be optimal to exercise an
American call option early (i.e., before the
expiration date)?

What does this imply about the values of
American and European call options?
2. Assuming there are no dividends paid,
would it ever be optimal to exercise an
American put option early?

What does this imply about the values of
American and European put options?
Put-call Parity
 Given the following securities: a call option, a put
option, the underlying stock, and risk-free borrowing
or lending; it is possible to replicate the payoffs from
one security by holding a combination of the other
three.
 To prevent arbitrage, it must hold that the value of
the replicated security equals the sum of the values of
the components used to replicate it.
 This leads to a relationship between put and call
options (which holds precisely for European options).
 The relationship is called Put-Call Parity
Synthetic Security
 Definition: A synthetic security is a
portfolio of other securities which will
have the same cash flows as the
“original” security being copied.
 To determine the relationship
between put and call options, we can
construct a synthetic call and equate
its value to a real call option.
Construction of a Synthetic European Call:
initial transactions at date t
Initial Transactions:
 buy 1 share of stock (long the stock)
-St
 borrow the present value of the
exercise price (E) at the risk free rate
(or short the risk-free asset)
+(PV of E)
 buy (long) the put option with the same
underlying stock, exercise price and
expiration date as the call option
Initial net cash flow (will be an outflow):
-Pet
-St + (PV of E) - Pet
 where St is the stock price at time t
 Pet is the price at time t of the European put option
Synthetic European Call:
transactions on the expiration date (T)
Final cash flows given the different relevant states of nature
(which depend on whether ST is less than or greater than E):
ST < E
ST ≥ E
+ST
+ST
-E
-E
 liquidate the long put option position
(discard or exercise depending upon
which is optimal)
+(E-ST)
0
Net cash flow at the expiration date T:
0
+(ST-E)
 liquidate the long stock position
(sell the stock)
 liquidate the short-risk free asset
position (repay the loan)
Potential arbitrage strategies:
 If the “real” call is more expensive than the synthetic call, we
could go long a synthetic call and write a real call. If the real
call is less expensive, we could short a synthetic call and buy a
real call. These strategies create a positive cash flow at date t.
 Under either strategy, the final net cash flows at date T (or
final payoffs) are guaranteed to be zero under all relevant
states of nature.
 The above strategies would yield a positive initial cash inflow
followed by zero cash flows with no risk. All investors would
try to exploit the arbitrage opportunity. As they did so, the
prices of the synthetic and real calls would adjust (until they
became equal) and the arbitrage opportunity would disappear.
 The concept of no-arbitrage can be used to show the
relationship between the prices of European put and call
options by equating the price of a synthetic call to the price of
a real call.
Put-Call Parity Condition
Cash flows from purchasing
(or constructing) the option
Equating the cash flows and
rearranging terms, we get
the following relationships:
Real Call
Synthetic Call
-Cet
-St + (PV of E) - Pet
-Cet=-St + (PV of E) - Pet
Cet=St - (PV of E) + Pet
Pet = Cet-St + (PV of E)
Where Cet is the cost of the European call option at time t
St is the stock price at time t
Pet is the cost of the European put option at time t
PV of E is the present value at time t of the exercise price
Note: The put-call parity condition holds precisely for European options.
However, because it may be optimal to exercise an American put early, the
value of the American put will be higher than what is determined above.
Self Study – fill in the blank cells
Construction of a Synthetic European Put:
initial transactions at date t
Initial Transactions: fill in the empty cells
 short 1 share of stock
 Invest the present value of the exercise
price (E) at the risk free rate (or long
the risk-free asset)
 Buy a call option on the stock with
same exercise price and same
expiration date
Initial net cash flow (will be an outflow):
 where St is the stock price at time t
 Cet is the price at time t of the European call option
Self Study – fill in the blank cells
Synthetic European Put:
transactions on the expiration date (T)
Final cash flows given the different relevant states of nature
(which depend on whether ST is less than or greater than E):
ST < E
ST ≥ E
E-ST
0
 liquidate the short stock position
(buy the stock)
 liquidate the long risk-free asset
position (collect the proceeds from
the investment)
 liquidate the long call option position
(discard or exercise depending upon
which is optimal)
Net cash flow at the expiration date T:
Put-Call Parity Example
 Suppose a call option on Macrosoft is
currently selling for $9.50. The call has
E=$75 and 6 months until expiration. The
risk-free rate of interest is 4% per year
effective and Macrosoft is currently trading
for $76 per share.
 What is the value of a European put option
on Macrosoft (with the same terms as the
call: E=$75 and 6 months until expiration)?
 If a European put with E=$80 and 6
months until expiration is selling for $9.95,
what is the equivalent call value?
Self Study – calculate the bolded cells
Put-Call Parity Examples
Cet
Pet
S
E
$9.30
$5.01
$92
$90
$7.54
$7.25
$88
$90
$9.84
$4.80
$83
$80
$6.83
$7.79
$77
$80
$11.30
$4.51
$75.00 $70
$5.03
$8.25
$65
$70.00
Notes:
•Rf=3.5% per
year effective
•Expiration is in
9 months for
each option
Summary and Conclusions
 The two main types of financial options are calls (option to buy)
and puts (option to sell).
 The factors that affect the value of options are the underlying
asset price, the volatility of the underlying asset returns, the
exercise price, the risk-free rate, and the time until expiration of
the option.
 We can construct synthetic options, and from these we can relate
the prices of put and call options (the Put-Call Parity Condition).
 As all options have value, understanding what impacts the value
of an option is important and will help us understand other
finance concepts that contain option characteristics.


Including the option to continue, expand, contract or abandon a
capital budgeting project.
Special option-like features that are part of other financial
securities
 Including call and conversion features of bonds and preferred stock,
the limited liability feature of common stock, etc.