lecture 8 - fundamentals of options

8/12/2019 Lecture 8 - Fundamentals of Options

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Lecture 8The Fundamentals of Options

Primary Text

Edwards and Ma: Chapter 18


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Options An option is a contract that gives its holder a rightbut not an

obligationto 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 premiumto the seller of the right (i.e., the option seller

orwriter).

If the option holder chooses to exercise her right to buy or sell the assetat 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.


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Options

Terminology

Option Holder (Buyer)An individual (or firm) who pays the

premium to acquire the right.

Option Writer (Seller)An individual (or firm) who sells theright in exchange for a premium.

Premiumthe market value of the option, in effect the price of

the insurance.

Strike PriceThe fixed price specified in an option contract iscalled the options strike price or exercise price.

Expiration DateThe date after which an option can no longer

be exercised is called its expiration date or maturity date.


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Options

Terminology

Call OptionAn 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 OptionAn 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 OptionAn American-type option can be

exercised at any time prior to the contracts expiration date, at theholders discretion.

European-type OptionA European-type option can only be

exercised on the contracts expiration date.


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Options

Exchange-Traded Options

Exchange-traded options contracts are standardizedand 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 ofexpiration dates.

Strike price intervals and expiration dates are determined by the

exchange.

Performance on options contracts is guaranteed by a clearingcorporationthat interposes itself as a third party to all option

contracts.

Thus, contract standardizationand a clearing corporation

guaranteeprovide the fundamental structure for exchange-traded

options.


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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 datethe 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 unexercisedthe

premium is retained by the option writer, and the writersobligation 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.


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Options

How Options Work

In addition to its type(call or put) and the nameof the

underlying asset/security, an option is identified by its strike

price andexpiration 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.


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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

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 $20.00per 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 theoption.

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 holders 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



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The HD April 09 put option with a strike price of $20.00per

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 theoption.

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 holders 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



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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


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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 ifcurrent market price of the stock is equal to the strike price

An option with no intrinsic value is said to be out-of-the-moneyif

current market price of the stock is below the strike price of a call

and above the strike price of a put



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Market Scenario Call Put

Market price>Strike Price I n-the-Money Out-of-the-Money

Market price=Strike Price At-the-Money At-the-MoneyMarket price


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Intrinsic value and Time Value:

Call: Intrinsic Value = Market PriceStrike PriceTime Value = PremiumIntrinsic Value

Premium = Intrinsic Value + Time Value

Put: Intrinsic Value = Strike PriceMarket Price

Time Value = PremiumIntrinsic Value

Premium = Intrinsic Value + Time Value



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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

Options

Components of Option PremiumsHD Stock Closing price on 24 March 2009: $22.95 per share


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The relationship between time value and strike price:

The magnitude of an options time value reflects the potential of the

option to gain intrinsic value during its life.

A deep out-of-the-moneyoption has little potential to gain intrinsicvalue because to do so asset prices will have to change substantially.

Therefore, it will have little time value.

Similarly, a deep in-the-moneyoption 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.



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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 tby SPtand

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 numberbetween its intrinsic value (time value is zero at expiration) and zero.

Max(MPTSPT, 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


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Denote call and put premiums by Cand Prespectively.

Profit/Loss of the Call Holder =Max(MPT

SPT

, 0) C

Profit/Loss of the Call Writer = C Max(MPTSPT, 0)

Profit/Loss of the Put Holder = Max(SPT MPT, 0) P

Profit/Loss of the Put Writer = P Max(SPT MPT, 0)


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Profit/Loss Long and short Call Options.

Stock Price Strike Price Options Long Call Short Call

at Expiration of the Option Premium profits/losses profits/losses

0 100 10 -10 10

20 100 10 -10 1040 100 10 -10 10

60 100 10 -10 10

80 100 10 -10 10

100 100 10 -10 10

120 100 10 10 -10

140 100 10 30 -30

160 100 10 50 -50

180 100 10 70 -70

200 100 10 90 -90

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 20 40 60 80 100 120 140 160 180 200

ProfitsorLosses

Stock Price at Expiration

Long Call

Short Call


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Profit/Loss Long and Short Put Options.

Stock Price Strike Price Options Long Put Short Put

at Expiration of the Option Premium Profits/losses Profits/losses

0 100 10 90 -90

20 100 10 70 -70

40 100 10 50 -5060 100 10 30 -30

80 100 10 10 -10

100 100 10 -10 10

120 100 10 -10 10

140 100 10 -10 10

160 100 10 -10 10

180 100 10 -10 10200 100 10 -10 10

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 20 40 60 80 100 120 140 160 180 200

ProfitsorL

osses


Long Put

Short Put


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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 expectthat 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 portfolios rate of return for three alternativeinvestment 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.



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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

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

TMP

MPRR

100)0,100(10100000,100

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

T

TMPMax

MPMaxRR

)0,100(5.5100100000

100000)0,100(100005.190000

%

TT

MPMax

MPMax

RR


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Investing with OptionsPortfolio rate of returns from alternative investment strategies

Stock Price Strike Price Call

at Expiration of the Option Option Value Strategy 1: Stock Strategy 2: Call Strategy 3: Mix

0 100 0 -100 -100 -5.5

20 100 0 -80 -100 -5.5

40 100 0 -60 -100 -5.5

60 100 0 -40 -100 -5.5

80 100 0 -20 -100 -5.5

100 100 0 0 -100 -5.5120 100 20 20 100 14.5

140 100 40 40 300 34.5

160 100 60 60 500 54.5

180 100 80 80 700 74.5

200 100 100 100 900 94.5

Rate of Return on Portfolio (%)


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-100

-50

0

50

100

150

0 20 40 60 80 100 120 140

RateofReturnon

Portfolio(%)


Rate of Returns Diagrams for Alternative Bullish Stock Strategies

100% Stock 100% Options

10% Options 100% risk-Free


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Which of the three strategies is the best for you?

Normal situationStrategy 1performs the best

BoomStrategy 2performs the best

RecessionStrategy 3performs the best Thus, none of the strategies dominates the other.

Depending on an investors risk tolerance, he or she might

choose any one of them.

Indeed, a very highly risk-averse investor might prefer thestrategy of investing all 100,000 in the risk-free asset to earn

5% for sure.


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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.


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Payoff Structure for Stock plus Put Strategy

Payoff 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

Payoff


Payoff Diagram for Stock plus Put Strategy

Stock Put Stock plus Put

ff S f C S


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Payoff Structure for Bond plus Call Strategy

Payoff 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

Payo

ff


Payoff Diagram for Bond plus Call.

Bond Call Bond plus Call


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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.


Common Stock

Cr

SP

PMP T )1(


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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 inflowthus, short A minus sign indicates cash outflowthus, 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


Short Call Synthetic

Tr

SPMPPC

)1(


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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


Long Call Synthetic

Tr

SPMPPC

)1(


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

That is the characteristics of a short put option can be broken


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


Short Put Synthetic

Tr

SPMPCP

)1(


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

That is the characteristics of a long put option can be broken


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 relationshipfollowing the same principle.


Long Put Synthetic

Tr

SPMPCP

)1(


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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 +100100/1.08 = 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 Cis $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.


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

Immediate

Transactions Cash Flow If MPT< $100 If MPT> $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


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

Immediate

Transactions Cash Flow If MPT< $100 If MPT> $100

Sell (Write) a Call $18 0 (MPT $100)

Buy (Long) a Put $10 $100 MPT 0

Sell the bond with FV of $100 $92.59 $100 $100

Buy a share of the Stock $100 MPT MPT

Net Cash Flows $0.59 0 0

Profit/Loss $0.59 $0.59

Cash Flow at Maturity Date

Buy Replicating Portfolio (Synthetic Long Call)

Call-Put Arbitrage


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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 C


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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

= 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

1

Tdd 12


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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 Cin the put-call parity condition:

Option Pricing: The Black-Scholes Model

rTeSPdNMPdNP

)](1[]1)([ 21


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Estimation of Price Volatility

Price volatility refers to the degree of volatility of price changethe

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


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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 ()

Cash dividend yield (d)


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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 (MPt) Increase () Decrease ()

Strike Price (SPt) () Decrease () Increase ()

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

Interest Rate (r) () Increase () Decrease ()

holding other factors constant

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

lecture 8 - fundamentals of options

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