Lecture 7 The Fundamentals of Options Primary Text Edwards and Ma: Chapter 18

Lecture 7The 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.

OptionsTerminology

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.

OptionsTerminology

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.

OptionsExchange-Traded Options

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.

OptionsExchange-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.

OptionsHow Options Work

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.

Call Put

Strike Exp. Last sale Change Open Int. Last Sale Change Open Int.

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

20.00 May 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

OptionsListing of Home Depot Option Prices: CBOE, 24 March 2009

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.


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.


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

OptionsProperties of Option Pricing

Intrinsic value: 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.

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


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

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


Call Put

Strike Exp. Premium Int. Val. Time Val. Premium Int. Val. Time Val.

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

20.00 May 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

OptionsComponents of Option Premiums

HD Stock Closing price on 24 March 2009: $22.95 per share

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.


The relationship between time value and strike price: 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-the-money.


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)

Investing with Options

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

Call Option Payoff Diagram

Stock Price at Expiration Date

Cal

l Op

tion

Pay

off

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

Put Option Payoff Diagram

Stock Price at Expiration

Put O

ption

Pay

off

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


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 Strike Price Options Long Call Short Callat Expiration of the Option Premium profits/losses profits/losses

0 100 10 -10 1020 100 10 -10 1040 100 10 -10 1060 100 10 -10 1080 100 10 -10 10

100 100 10 -10 10120 100 10 10 -10140 100 10 30 -30160 100 10 50 -50180 100 10 70 -70200 100 10 90 -90

0 20 40 60 80 100 120 140 160 180 200-100

-80

-60

-40

-20

0

20

40

60

80

100

Long Call

Short Call


Pro

fits

or

Los

ses

Profit/Loss Long and Short Put Options.

Stock Price Strike Price Options Long Put Short Putat Expiration of the Option Premium Profits/losses Profits/losses

0 100 10 90 -9020 100 10 70 -7040 100 10 50 -5060 100 10 30 -3080 100 10 10 -10

100 100 10 -10 10120 100 10 -10 10140 100 10 -10 10160 100 10 -10 10180 100 10 -10 10200 100 10 -10 10

0 20 40 60 80 100 120 140 160 180 200-100

-80

-60

-40

-20

0

20

40

60

80

100

Long Put

Short Put


Pro

fits

or

Los

ses

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:

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.


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

Strategy 2: Buy calls (with strike price $100) on 10,000 share of the stock

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).


100100000,100

000,100000,1%

T

T MPMP

RR

100)0,100(10100000,100

000,100)0,100(000,10%

T

T MPMaxMPMax

RR

)0,100(5.5100100000

100000)0,100(100005.190000%

T

T MPMaxMPMax

RR


Portfolio rate of returns from alternative investment strategies

Stock Price Strike Price Callat Expiration of the Option Option Value Strategy 1: Stock Strategy 2: Call Strategy 3: Mix

0 100 0 -100 -100 -5.520 100 0 -80 -100 -5.540 100 0 -60 -100 -5.560 100 0 -40 -100 -5.580 100 0 -20 -100 -5.5

100 100 0 0 -100 -5.5120 100 20 20 100 14.5140 100 40 40 300 34.5160 100 60 60 500 54.5180 100 80 80 700 74.5200 100 100 100 900 94.5

Rate of Return on Portfolio (%)

0 20 40 60 80 100 120 140 160-100

-50

0

50

100

150

Rate of Returns Diagrams for Alternative Bullish Stock Strategies

100% Stock 100% Options 10% Options

100% risk-Free


Rat

e of

Ret

urn

on

Por

tfol

io (

%)

State of the Riskless Strategy 1: Strategy 2: Strategy 3:

Economy 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%

Rate of Return on Portfolio

• 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%.

Which of the three strategies is the best for you?

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 StrategyPayoff Structure for Stock plus Put Strategy

Position If MP T < SP=$100 If MP T > SP=$100

Stock MP T MP T

Put $100 - MP T 0

Stock plus Put $100 MP T

Value of Position at Maturity

0

40

80

120

160

200

0 20 40 60 80 100 120 140 160 180 200

Pay

off


Payoff Diagram for Stock plus Put Strategy

Stock Put Stock plus Put

Payoff Structure for Bond plus Call StrategyPayoff Structure for Bond plus Call

Position If MP T < SP=$100 If MP T > SP=$100

Bond $100 $100

Call 0 MP T - $100

Bond plus Call $100 MP T

Value of Position at Maturity

0

40

80

120

160

200

0 20 40 60 80 100 120 140 160 180 200

Pay

off


Payoff Diagram for Bond plus Call.

Bond Call Bond plus Call

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:

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: Common Stock

Cr

SPPMP

T

)1(

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:

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: Short Call Synthetic

Tr

SPMPPC

)1(

Rearranging the put-call parity relationship we have:

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: Long Call Synthetic

Tr

SPMPPC

)1(

The put-call parity relationship can also be rearranged as :

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: Short Put Synthetic

Tr

SPMPCP

)1(

The put-call parity relationship can also be rearranged as :

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.

The Put-Call Parity Pricing Relationship: Long Put Synthetic

Tr

SPMPCP

)1(

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 (market premium for the call is higher than the calculated call premium)

Immediate

Transactions Cash Flow If MP T < $100 If MP T > $100

Sell (Write) a Call

Buy (Long) a Put

Sell the bond with FV of $100

Buy a share of the Stock

Net Cash Flows

Profit/Loss

Cash Flow at Maturity Date

Buy Replicating Portfolio (Synthetic Long Call)

Call-Put Arbitrage

Call-Put Arbitrage (market premium for the call is higher than the calculated call premium)

Immediate


Sell (Write) a Call $18 0 − (MP T − $100)

Buy (Long) a Put − $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

Profit/Loss $0.59 $0.59


Buy Replicating Portfolio (Synthetic Long Call)

Call-Put Arbitrage

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


Sell (Write) a Put

Buy (Long) a Call

Short a share of the Stock

Buy the Bond with FV of $100

Net Cash Flows

Profit/Loss


Buy Replicating Portfolio (Synthetic Long Put)

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


Sell (Write) a Put $11 − ($100 − MPT ) 0

Buy (Long) a Call − $17.41 0 MP T − $100

Short a share of the Stock $100 − MP T − MP T

Buy the Bond with FV of $100 − $92.59 $100 $100

Net Cash Flows $1 $0 $0

Profit/Loss $1.00 $1.00


Buy Replicating Portfolio (Synthetic Long Put)

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.

If MP = SP/(1+r)T => C = P If MP > SP/(1+r)T => C > P If MP < SP/(1+r)T => C < P

PCr

SPMP

T

)1(

Option Pricing: The Black-Scholes Model

Black and Scholes derive the following equations for pricing European call options on non-dividend-paying stocks:

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

rTeSPdNMPdNC )()( 21

T

TrSPMPd

)2/()/ln( 2

1Tdd 12

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:

Option Pricing: The Black-Scholes Model

rTeSPdNMPdNP )](1[]1)([ 21

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.

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

1

)(1

2

n

XXsd

N

t t

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 = × monthly standard deviation

For weekly data: σ = × weekly standard deviation

For daily data: σ = × daily standard deviation

12

52

252

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.

Pricing Factors Call Premium (C) Put Premium (P)

Stock market price (MP t ) (↑) Increase (↑) Decrease (↓)

Strike Price (SP t ) (↑) Decrease (↓) Increase (↑)

Time to Expiration (T − t ) (↑) Increase (↑) Increase (↑)

Interest Rate (r ) (↑) Increase (↑) Decrease (↓)

Volatility (σ ) (↑) Increase (↑) Increase (↑)

holding other factors constant

Effect of an increase in each pricing factor on the option value,

Documents

Lecture 7 The Fundamentals of Options Primary Text Edwards and Ma: Chapter 18