Lecture 7
The Fundamentals of Options
Primary Text
Edwards and Ma: Chapter 18
Options




An option is a contract that gives its holder a right but not an
obligation to purchase or sell a specific asset (e.g., commodity futures
or security) at a specific price on or before a specified date in the
future.
To acquire this right, the buyer of the right (i.e., the option buyer or
holder) pays a premium to the seller of the right (i.e., the option seller
or writer).
If the option holder chooses to exercise her right to buy or sell the asset
at the specified price, the option writer has an obligation to deliver or
take delivery of the underlying asset. The potential loss to an option
writer is unlimited (?).
In contrast, if the option holder chooses not to exercise her right, but to
allow the option to expire, her loss is limited to the premium paid.
Options
Terminology

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

Option Holder (Buyer) – An individual (or firm) who pays the
premium to acquire the right.
Option Writer (Seller) – An individual (or firm) who sells the
right in exchange for a premium.
Premium – the market value of the option, in effect the price of
the insurance.
Strike Price – The fixed price specified in an option contract is
called the option’s strike price or exercise price.
Expiration Date – The date after which an option can no longer
be exercised is called its expiration date or maturity date.
Options
Terminology




Call Option – An option (a right but not an obligation) to buy a
specified asset at a set price on or before a specified date in the
future.
Put Option – An option (a right but not an obligation) to sell a
specified asset at a set price on or before a specified date in the
future.
American-type Option – An American-type option can be
exercised at any time prior to the contract’s expiration date, at the
holder’s discretion.
European-type Option – A European-type option can only be
exercised on the contract’s expiration date.
Options
Exchange-Traded Options

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Exchange-traded options contracts are standardized and traded on
organized (and government designated) exchanges.
An exchange-traded option specifies a uniform underlying asset, one
of a limited number of strike prices, and one of a limited number of
expiration dates.
Strike price intervals and expiration dates are determined by the
exchange.
Performance on options contracts is guaranteed by a clearing
corporation that interposes itself as a third party to all option
contracts.
Thus, contract standardization and a clearing corporation
guarantee provide the fundamental structure for exchange-traded
options.
Options
Exchange-Traded Options

Once an exchange-traded option contract is purchased, contract
obligation may be fulfilled in one of these three ways:



The option holder exercises her right on or before the
expiration date – the option writer must then adhere to the terms
of the option contract, and accept the other side of the position.
The option writer keeps the premium.
The option holder allows the option to expire unexercised – the
premium is retained by the option writer, and the writer’s
obligation is discharged.
Either or both the option holder and/or writer executes an
offsetting transaction in the option market, eliminating all
future obligations. In this case, the rights or obligations under the
original contract are transferred to a new option holder or writer.
Options
How Options Work
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In addition to its type (call or put) and the name of the
underlying asset/security, an option is identified by its strike
price and expiration date.
For exchange traded options, the strike price and expiration date
are determined by the rules of the exchange.
On the Chicago Board Options Exchange (CBOE), a single call
option contract gives its holder the right to buy 100 shares of
the underlying stock and is of the American-type.
Original maturities of CBOE options vary from three months to
three years, and they all expire on the third Friday of the
month in which they mature.
Options
Listing of Home Depot Option Prices: CBOE, 24 March 2009
Call
Put
Strike
Exp.
Last sale
Change
20.00
April
3.60
0.20
9,055
0.35
0.03
16,036
22.50
1.49
-0.11
15,003
1.00
0.11
5,970
25.00
0.41
-0.05
13,125
2.14
-0.31
1,714
27.50
0.06
0.01
444
4.60
0.00
259
3.90
0.00
9971
0.72
0.04
16,614
22.50
1.95
-0.10
23,199
1.50
0.00
21,346
25.00
0.87
-0.05
35,601
2.62
-0.10
9,322
27.50
0.31
0.03
20,317
5.50
0.00
3495
20.00
May
Open Int. Last Sale
Change
Open Int.
Options
Listing of Home Depot Option Prices: CBOE, 24 March 2009
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The HD April 09 call option with a strike price of $20.00 per share
was last traded at a price (premium) of $3.60 per share.
A buyer of this option, therefore, would have to make an immediate
payment of $3.60 per share (or $360 per contract) to the writer of the
option.
The buyer of the call option would have the right (but not the
obligation) to purchase 100 shares of HD at $20 until April 17 (the
third Friday in April).
If exercised, the holder’s net cost per share of the HD stock would be
$23.60 per share.
If the holder let the option expire without exercising the right, her net
loss would be $360.
Options
Listing of Home Depot Option Prices: CBOE, 24 March 2009
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The HD April 09 put option with a strike price of $20.00 per
share was last traded at a price (premium) of $0.35 per share.
A holder of this option, therefore, would have to make an immediate
payment of $0.35 per share (or $35 per contract) to the writer of the
option.
The holder of the put option would have the right (but not the
obligation) to sell 100 shares of HD at $20 until April 17 (the third
Friday in April).
The holder’s net revenue per share of the HD stock would be $19.65
per share.
If the holder let the option expire without exercising the right, her
net loss would be $35.
Options
Properties of Option Pricing

The level of the strike price and the value of option: Call
options with lower strike prices are more valuable and the put
options with higher strike prices are more valuable to the
holders.
 Call:
Strike Price ↑ => Premium ↓
 Put:
Strike Price ↑ => Premium ↑

Intrinsic value versus time value: Option premiums have two
components


Intrinsic value
Time value
Options
Properties of Option Pricing

Intrinsic value:
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
If the current stock price is above the strike price of a call (or
below the strike price of a put), the option has intrinsic value.
An option with intrinsic value is said to be in-the-money.
If the current stock price is equal to or below the strike price of a
call (or equal to or above the strike price of a put), the option has
no intrinsic value.
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
An option with no intrinsic value is said to be at-the-money if
current market price of the stock is equal to the strike price
An option with no intrinsic value is said to be out-of-the-money if
current market price of the stock is below the strike price of a call
and above the strike price of a put
Options
Properties of Option Pricing

Market Scenario
Call
Put
Market price > Strike Price
In-the-Money
Out-of-the-Money
Market price = Strike Price
At-the-Money
At-the-Money
Market price < Strike Price
Out-of-the-Money
In-the-Money
Time value: Why, then, the options that are out-of-the-money
have positive premiums?
 Because, it still has a time value. The difference between an
option’s price (premium) and its intrinsic value is called the
option’s time value.

Time Value = Premium – Intrinsic Value
Options
Properties of Option Pricing

Intrinsic value and Time Value:

Call:
Intrinsic Value = Market Price – Strike Price
Time Value = Premium – Intrinsic Value
Premium = Intrinsic Value + Time Value

Put:
Intrinsic Value = Strike Price – Market Price
Time Value = Premium – Intrinsic Value
Premium = Intrinsic Value + Time Value
Options
Components of Option Premiums
HD Stock Closing price on 24 March 2009: $22.95 per share
Call
Put
Strike
Exp.
Premium
20.00
April
3.60
2.95
0.65
0.35
0.00
0.35
22.50
1.49
0.45
1.04
1.00
0.00
1.00
25.00
0.41
0.00
0.41
2.14
2.05
0.09
27.50
0.06
0.00
0.06
4.60
4.55
0.05
3.90
2.95
0.95
0.72
0.00
0.72
22.50
1.95
0.45
1.50
1.50
0.00
1.50
25.00
0.87
0.00
0.87
2.62
2.05
0.57
27.50
0.31
0.00
0.31
5.50
4.55
0.95
20.00
May
Int. Val. Time Val. Premium Int. Val.
Time Val.
Options
Properties of Option Pricing

The relationship between time value and time to expiration: the
longer the time remaining until an option’s expiration, the higher the
premium tends to be, everything else being equal.


The difference between the premiums of two options with the same strike
price but different expiration date is the same as the difference between the
time values of the two options.
 For example, the premium for the HD May 2009 call option with the
strike price of 20.00 is ($3.90) higher than the premium for the HD
April 2009 call option ($3.60) with the same strike price. The
difference between the time value of these two options is also $0.30
per share.
This is because a longer time provides more opportunity for the price of
the underlying asset to move to a level where the option is in-the-money,
and where the purchase and sale of the asset at the specified strike price
will be profitable.
Options
Properties of Option Pricing

The relationship between time value and strike price:
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The magnitude of an option’s time value reflects the potential of the
option to gain intrinsic value during its life.
A deep out-of-the-money option has little potential to gain intrinsic
value because to do so asset prices will have to change substantially.
Therefore, it will have little time value.
Similarly, a deep in-the-money option is as likely to lose intrinsic
value as to gain it, as a consequence also has little time value.
In general, time value is at the maximum when an option is at-themoney.
Investing with Options
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
Options make it possible for investors to modify their risk exposure to
the underlying asset
 Denote the exercise (strike) price of an option on date t by SPt and
the market price of the underlying stock by MPt
 At expiration the strike price of the call option and the market price
of the underlying stock can be denoted by SPT, and MPT,
respectively.
At expiration the payoff from a call option is the larger number
between its intrinsic value (time value is zero at expiration) and zero.
Max (MPT −SPT, 0)

At expiration the payoff from a put option is the larger number
between its intrinsic value (time value is zero at expiration) and zero.
Max (SPT −MPT, 0)
Stock Price
Strike Price
Call
Put
at Expiration
of the Option
Option Value
Option Value
0
100
0
100
20
100
0
80
40
100
0
60
60
100
0
40
80
100
0
20
100
100
0
0
120
100
20
0
140
100
40
0
160
100
60
0
180
100
80
0
200
100
100
0
Put Option Payoff Diagram
Call Option Payoff Diagram
120
100
80
Put Option Payoff
Call Option Payoff
100
60
40
80
60
40
20
20
0
0
0
20
40
60 80 100 120 140 160 180 200
Stock Price at Expiration Date
0
20
40
60 80 100 120 140 160 180 200
Stock Price at Expiration
Investing with Options

Denote call and put premiums by C and P respectively.

Profit/Loss of the Call Holder = Max (MPT −SPT, 0) − C

Profit/Loss of the Call Writer = C − Max (MPT −SPT, 0)

Profit/Loss of the Put Holder = Max (SPT −MPT, 0) − P

Profit/Loss of the Put Writer = P − Max (SPT −MPT, 0)
Profit/Loss Long and short Call Options.
Stock Price
at Expiration
0
20
40
60
80
100
120
140
160
180
200
Strike Price
of the Option
100
100
100
100
100
100
100
100
100
100
100
Profits or Losses
100
80
60
40
20
0
-20
-40
-60
-80
-100
Options
Premium
10
10
10
10
10
10
10
10
10
10
10
Long Call
profits/losses
-10
-10
-10
-10
-10
-10
10
30
50
70
90
Short Call
profits/losses
10
10
10
10
10
10
-10
-30
-50
-70
-90
Long Call
Short Call
0
20
40
60
80 100 120 140
Stock Price at Expiration
160
180
200
Profit/Loss Long and Short Put Options.
Stock Price
at Expiration
0
20
40
60
80
100
120
140
160
180
200
Strike Price
of the Option
100
100
100
100
100
100
100
100
100
100
100
Options
Premium
10
10
10
10
10
10
10
10
10
10
10
Long Put
Profits/losses
90
70
50
30
10
-10
-10
-10
-10
-10
-10
Short Put
Profits/losses
-90
-70
-50
-30
-10
10
10
10
10
10
10
100
80
Profits or Losses
60
Long Put
40
20
0
-20
-40
Short Put
-60
-80
-100
0
20
40
60
80
100
120
140
Stock Price at Expiration
160
180
200
Investing with Options
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In addition to using options to modify risk exposure, buying or selling
options is an alternative way to take a position in the market for a
trader who does not own the underlying asset.
For example, suppose that you have $100,000 to invest and you expect
that the price of a stock is going to increase (bullish).
Assume that the riskless interest rate is 5% per year and the stock pays
no dividends.
Compare your portfolio’s rate of return for three alternative
investment strategies over a one-year holding period:
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

Invest the entire $100,000 in the stock.
Invest the entire $100,000 in call options of the stock.
Invest 10% ($10,000) in call options of the stock and the rest ($90,000) in
the risk-free asset.
Investing with Options
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
Assume that the current price of the stock is $100 per share, and the
premium for the call option with $100 strike price is $10 per share.
Under the three alternative strategies, the investment scenarios and
corresponding rate of returns are

Strategy 1: Buy 1,000 share of the stock
RR % 

Strategy 2: Buy calls (with strike price $100) on 10,000 share of the stock
RR % 

1,000  MPT  100,000
100  MPT  100
100,000
10,000  Max ( MPT  100, 0)  100,000
100  10  Max ( MPT  100, 0)  100
100,000
Strategy 3: Buy calls (with strike price $100) on 1,000 share of the stock
and invest $90,000 in risk free asset (with 5% interest rate).
RR % 
90000 1.05  1000  Max ( MPT  100, 0)  100000
100  5.5  Max ( MPT  100, 0)
100000
Investing with Options
Portfolio rate of returns from alternative investment strategies
Stock Price
at Expiration
0
20
40
60
80
100
120
140
160
180
200
Strike Price
of the Option
100
100
100
100
100
100
100
100
100
100
100
Call
Rate of Return on Portfolio (%)
Option Value Strategy 1: Stock Strategy 2: Call Strategy 3: Mix
0
-100
-100
-5.5
0
-80
-100
-5.5
0
-60
-100
-5.5
0
-40
-100
-5.5
0
-20
-100
-5.5
0
0
-100
-5.5
20
20
100
14.5
40
40
300
34.5
60
60
500
54.5
80
80
700
74.5
100
100
900
94.5
Rate of Returns Diagrams for Alternative Bullish Stock Strategies
150
100% Stock
100% Options
10% Options
100% risk-Free
Rate of Return on Portfolio (%)
100
50
0
-50
-100
0
20
40
60
80
Stock Price at Expiration
100
120
140
• Which of the three strategies is the best for you?
It depends on your expectation about the stock price and your risk tolerance
• Suppose that you have scenario forecasts for three possible states of the
economy. You believe that
• there is a probability of 0.2 (20% chance) that there will be a boom and
stock price will rise by 50% during the year,
• a probability of 0.6 (60% chance) that the economy will be normal and
market will rise by 10%, and
• a probability of 0.2 (20% chance) that there will be a recession and the stock
will fall by 30%.
Rate of Return on Portfolio
State of the
Economy
Riskless
Strategy 1: Strategy 2: Strategy 3:
Probability Investment 100% Stock 100% Call
10% Call
Boom (+50%)
0.2
5%
50%
400%
44.50%
Normal (+10%)
0.6
5%
10%
0
4.50%
Recession (-30%)
0.2
5%
-30%
-100%
-5.50%
Which of the three strategies is the best for you?
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



Normal situation – Strategy 1 performs the best
Boom – Strategy 2 performs the best
Recession – Strategy 3 performs the best
Thus, none of the strategies dominates the other.
Depending on an investor’s risk tolerance, he or she might
choose any one of them.
Indeed, a very highly risk-averse investor might prefer the
strategy of investing all 100,000 in the risk-free asset to earn
5% for sure.
The Put-Call Parity Pricing Relationship:
Common Stock



We have just seen that a strategy of investing some of your
money in the riskless asset and some in a call option can
provide a portfolio with a guaranteed minimum value and an
upside slope equal to that of investing in the underlying stock.
There are at least two other ways of creating that same pattern
of payoffs:
 Buy a share of stock and a European put option and
 buy a pure discount bond and a European call option.
Consider a share of a stock with market price $100, and
European call and put options with strike price $100 and
premium $10.
Payoff Structure for Stock plus Put Strategy
Payoff Structure for Stock plus Put Strategy
Value of Position at Maturity
Position
If MP T < SP=$100 If MP T > SP=$100
Stock
Put
Stock plus Put
MP T
MP T
$100 - MP T
0
$100
MP T
Payoff Diagram for Stock plus Put Strategy
200
Stock
Put
Stock plus Put
Payoff
160
120
80
40
0
0
20
40
60
80
100
120
140
Stock Price at Expiration
160
180
200
Payoff Structure for Bond plus Call Strategy
Payoff Structure for Bond plus Call
Value of Position at Maturity
Position
If MP T < SP=$100 If MP T > SP=$100
Bond
$100
$100
0
MP T - $100
$100
MP T
Call
Bond plus Call
Payoff Diagram for Bond plus Call.
200
Bond
Call
Bond plus Call
Payoff
160
120
80
40
0
0
20
40
60
80
100
120
140
Stock Price at Expiration
160
180
200
The Put-Call Parity Pricing Relationship:
Common Stock




Thus, a portfolio consisting of a stock plus a European put
option (with strike price SP) is equivalent to a pure discount
default-free bond (with face value SP) plus a European call
option (with strike price SP).
So, by the Law of One Price, they must have the same price.
The following equation expresses the pricing relation:
SP
MP  P 
C
T
(1  r )
The equation is known as the put-call parity relationship.
The relationship allows one to determine the price of any one
of the four securities from the values of the other three.
The Put-Call Parity Pricing Relationship:
Short Call Synthetic



The put-call parity relationship can also be used as a recipe for synthesizing
any one of the four from the other three.
Use the sign of each variable to determine long or short
 A plus sign indicates cash inflow – thus, short
 A minus sign indicates cash outflow – thus, long
Rearranging the put-call parity relationship we have:
SP
C  P  MP 
(1  r ) T

That is the characteristics of a short call option can be broken into three
components:
 Short the put option with the same strike price
 Short the underlying stock at the prevailing market price
 Long the bond with the face value equal to the SP of the options
The Put-Call Parity Pricing Relationship:
Long Call Synthetic

Rearranging the put-call parity relationship we have:
SP
 C   P  MP 
(1  r ) T

That is the characteristics of a long call option can be broken
into three components:
 Long a put option with the same strike price of the call
 long the stock at the prevailing market price
 Short the bond with face value equal to the SP of the
options
The Put-Call Parity Pricing Relationship:
Short Put Synthetic

The put-call parity relationship can also be rearranged as :
SP
P  C  MP 
(1  r )T

That is the characteristics of a short put option can be broken
into three components:
 Short a call option with the same strike price
 Long the underlying stock at the prevailing market price
 Short the bond with the FV equal to the SP of the options
The Put-Call Parity Pricing Relationship:
Long Put Synthetic

The put-call parity relationship can also be rearranged as :
SP
 P  C  MP 
(1  r ) T

That is the characteristics of a long put option can be broken
into three components:
 Long the call option with the same strike price
 Short the underlying stock at the prevailing market price
 Long the bond with the FV equal to the SP of the options

Similarly, long or short stock or bond can also be synthesized using the
put-call parity relationship – following the same principle.
Call-Put Arbitrage






The equations for call and put premiums can also be regarded as
formulas for converting a put into call and vice versa.
For example, suppose that
MP = $100, SP = $100, P = $10, T = 1
year, and r = 0.08
Then the price of the call option, C, would have to be 17.41
C = 10 +100 – 100/1.08 = 17.41
If the price of the call is too high or too low relative to the price of the
put, and there are no barriers to arbitrage, arbitragers can make a
certain profit.
For example, if C is $18 and there are no barriers to arbitrage, an
arbitrager can lock in a riskless profit by selling a call with strike price
$100 and simultaneously buying a put with the same strike price and
expiration date, borrowing the capital at the risk-free interest rate, and
buying the underlying stock.
Call-Put Arbitrage
Call-Put Arbitrage (market premium for the call is higher than the calculated call premium)
Immediate
Transactions
Cash Flow
Cash Flow at Maturity Date
If MP T < $100
If MP T > $100
Sell (Write) a Call
Buy Replicating Portfolio (Synthetic Long Call)
Buy (Long) a Put
Sell the bond with FV of $100
Buy a share of the Stock
Net Cash Flows
Profit/Loss
Call-Put Arbitrage
Call-Put Arbitrage (market premium for the call is higher than the calculated call premium)
Immediate
Transactions
Sell (Write) a Call
Cash Flow at Maturity Date
Cash Flow
If MP T < $100
If MP T > $100
$18
0
− (MP T − $100)
Buy Replicating Portfolio (Synthetic Long Call)
− $10
$100 − MP T
0
Sell the bond with FV of $100
$92.59
− $100
− $100
Buy a share of the Stock
− $100
MP T
MP T
Net Cash Flows
$0.59
0
0
$0.59
$0.59
Buy (Long) a Put
Profit/Loss
Put-Call Arbitrage
If the price of the put option is high relative to call prices, an arbitrager
can lock in a riskless profit by selling the overpriced put and buying a call,
selling the underlying stock, and lending the proceed at the risk-free
interest rate (8%).
Put-Call Arbitrage (Market premium for a put is higher than the calculated put premium).
Immediate
Transactions
Cash Flow
Cash Flow at Maturity Date
If MP T < $100
If MP T > $100
Sell (Write) a Put
Buy Replicating Portfolio (Synthetic Long Put)
Buy (Long) a Call
Short a share of the Stock
Buy the Bond with FV of $100
Net Cash Flows
Profit/Loss
Put-Call Arbitrage
If the price of the put option is high relative to call prices, an arbitrager
can lock in a riskless profit by selling the overpriced put and buying a call,
selling the underlying stock, and lending the proceed at the risk-free
interest rate (8%).
Put-Call Arbitrage (Market premium for a put is higher than the calculated put premium).
Immediate
Transactions
Sell (Write) a Put
Cash Flow at Maturity Date
Cash Flow
If MP T < $100
If MP T > $100
$11
− ($100 − MPT )
0
Buy Replicating Portfolio (Synthetic Long Put)
Buy (Long) a Call
Short a share of the Stock
Buy the Bond with FV of $100
Net Cash Flows
Profit/Loss
− $17.41
0
MP T − $100
$100
− MP T
− MP T
− $92.59
$100
$100
$1
$0
$0
$1.00
$1.00
The Put-Call Parity Pricing Relationship

We can gain some additional insight into the nature of the
relationship among puts, calls, stocks, and bonds by
rearranging the terms in the put-call parity relationship.
SP
MP 
 CP
T
(1  r )



If MP = SP/(1+r)T
If MP > SP/(1+r)T
If MP < SP/(1+r)T
=>
=>
=>
C=P
C>P
C<P
Option Pricing: The Black-Scholes Model

Black and Scholes derive the following equations for pricing European call
options on non-dividend-paying stocks:
C  N (d1 )  MP  N (d 2 )  SP  e  rT
ln( MP / SP)  (r   2 / 2)  T
d1 
 T







d 2  d1   T
C = price (premium) of the call
MP = current market price of the stock
SP = current strike price of the call
r = riskless interest rate
T = time to maturity of the option in years
σ = standard deviation of the annualized rate of return on the stock
N(d)= probability that a random draw from a standard normal distribution will be
less than d
Option Pricing: The Black-Scholes Model


The formulation of the model is the construction of a
hypothetical risk-free portfolio, consisting of long call options
and short positions in the underlying stock, on which an
investor earns the riskless rate of interest.
We can derive the formula from the price of a put option by
substituting C in the put-call parity condition:
P  [ N (d1 )  1]  MP  [1  N (d 2 )]  SP  e  rT
Estimation of Price Volatility



Price volatility refers to the degree of volatility of price change – the
percentage changes in prices.
A commonly used measure of this volatility is the standard deviation of
previous daily, weekly, or even monthly percentage changes in prices.
The percentage price change is often calculated as the difference between
the natural logarithms of the current and previous prices.
2
(
X

X
)
t 1 t
N
sd 




n 1
N = Number of observations
Xt = ln(MPt/MPt-1), i.e., the percentage price change
MPt = stock price at the end of period t
X-bar = mean of Xt
Estimation of Price Volatility




Depending on which data interval is used to calculate the
standard deviation of prices, annualized price volatility (σ)
is obtained by multiplying the calculated standard deviation
by the square root of the number of periods:
For monthly data: σ = 12 × monthly standard deviation
For weekly data: σ = 52 × weekly standard deviation
For daily data:
σ = 252 × daily standard deviation
Fundamental Determinants of Option Prices

In general, prior to expiration the value of an option depends
upon six variables:






the current value of the underlying asset or stock (MPt)
the options strike price (SPt)
the time remaining until the option expires (T − t)
the current level of the risk-free interest (r)
the anticipated volatility of the price of the underlying asset or
stock (σ)
Cash dividend yield (d)
Fundamental Determinants of Option Prices
Determinants of Options Premiums - effect of an increase in each factor.
Effect of an increase in each pricing factor on the option value,
holding other factors constant
Pricing Factors
Call Premium (C)
Stock market price (MP t )
(↑)
Put Premium (P)
Increase (↑)
Decrease (↓)
Strike Price (SP t )
(↑)
Decrease (↓)
Increase (↑)
Time to Expiration (T − t )
(↑)
Increase (↑)
Increase (↑)
Interest Rate (r )
(↑)
Increase (↑)
Decrease (↓)
Increase (↑)
Increase (↑)
Volatility (σ )
(↑)