jacoby, stangeland and wajeeh, 20001 options a european call option gives the holder the right to...

Jacoby, Stangeland and Wajeeh, 2000

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OptionsA European Call Option

gives the holder the right to buy the underlying asset for a prescribed price (exercise/strike price), on a prescribed date (expiry date).

A European Put Option

gives the holder the right to sell the underlying asset for a prescribed price (exercise/strike price), on a prescribed date (expiry date).

American Options

exercise is permitted at any time during the life of the option (call or put).

Chapter 21


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Underlying Asset (S)The specific asset on which an option contract is based (e.g. stock, bond, real-estate, etc.).

For traded Stock Options: one call (put) option contract represents the right to buy (sell) 100 shares of the underlying stock.

Strike/Exercise Price (E)

The specified asset price at which the asset can be bought (sold) by the holder of a call (put) if s/he exercised his/her right.

Expiration Date (T)

The last day an option exists.


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Writer:Seller of an option (takes a short position in the option).

Holder:

Buyer of the option (takes a long position in the option).

Elements of an option contract: type (put or call) style (American or European) underlying asset (stock/bond/etc…) unit of trade exercise price expiration date


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Holding A European Call Option Contract- An Example

European style IBM corp. September 100 call:

entitles the buyer (holder) to purchase 100 shares of IBM common stock at $100 per share (E), at the options expiration date in September (T).

At the options expiration date (T):

For the Call Option Holder

If ST > E=$100:

Exercise the call option - pay $100 for an IBM stock with a market value of ST (e.g. ST=$105).

Payoff at T: ST - E = $105-$100=$5 > 0.

If ST Ÿ E=$100:

Can buy IBM stocks in the market for ST (e.g. ST=$90). Holder will not choose to exercise (option expires worthless).

Payoff at T: $0.


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Holding a European Call

Conclusion:

A call option holder will never lose at T (expiration), since his/her payoff is

never negative:

If ST Ÿ E=$100 If ST > E=$100

Call option value at T $0 ST - E = ST - $100

Payoff at T

0E=$100

ST


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For the Call Option Writer (Short Seller)

If ST > E = $100: Holder will exercise. Writer will deliver an IBM stock with a market value of ST ($105) to the holder, in return for E dollars ($100).

Payoff at T: E-ST = $100-$105= - $5 < 0.

If ST Ÿ E = $100: Holder will not exercise.

Payoff at T: $0.


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For the Call Option Writer (Short Seller)

Conclusion:

A call option writer will never gain at T (expiration), since his/her payoff is

never positive:

If ST Ÿ E=$100 If ST > E=$100

Call option value at T $0 E - ST = $100 - ST

Payoff at T

0E=$100

ST

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Holding A European Put Option Contract- An Example

European style IBM corp. September 100 put

entitles the buyer (holder) to sell 100 shares of IBM corp. common stock at $100 per share (E), at the option’s expiration date in September (T).

At the options expiration date (T):

For the Put Option Holder

If ST < E=$100:

Exercise the put option - receive $100 for an IBM stock with a market value of ST (e.g. ST=$90).

Payoff at T: E - ST = $100-$90=$10 > 0.

If ST E=$100:

Can sell IBM stocks in the market for ST (e.g. ST=$105). Holder

will not choose to exercise (option expires worthless).

Payoff at T: $0.


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Holding A European Put

Conclusion:

A put option holder will never lose at T, since his/her payoff is never

negative:

If ST E=$100 If ST < E=$100

Put option value at T $0 E - ST = 100 - ST

Payoff at T

0 ST

$100

E=$100


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For the Put Option Writer (Short Seller)

If ST<E = $100: Holder will exercise. Writer will pay E dollars

($100) in return for an IBM stock (worth ST = $90).

Payoff at T: ST - E = $90-$100 = -$10< 0.

If ST E = $100 : Holder will not exercise.

Payoff at T: $0.

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Conclusion:

A put option writer will never gain at T, since his/her payoff is never positive:

If ST E=$100 If ST < E=$100

Put option value at T $0 ST - E = ST - 100

Payoff

at T

0E=$100

ST

- $100

For the Put Option Writer (Short Seller)


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E=$85

Combinations of Options

You purchase a BCE stock, and simultaneously write (short sell) the July $85 European call option.

Your payoff diagram at expiration in July (T):

Payoff

at T

0 ST

Buy StockPayoff

at T

0 ST

Short Sell CallPayoff

at T

0 ST

Combination

E=$85

Same as Short Sell Put & Buy T-bill

$85

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The Put-Call Parity Relationship*You purchase the BCE stock, the July $85 put option, and short sell the July $85 call option (both options are European). Your payoff at expiration in July (T):

If ST = $100 If ST = $80

Stock (S)

Put (P)

Call(C)

Total (Certain) Payoff at T:

To calculate the PV of the certain payoff ($85=E) today, we use the risk-free rate:

fTrEeCPS

000

*Only for European Options


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You purchase the BCE stock, the July $85 European put option, and short sell the July $85 European call option

Your payoff at expiration in July (T):

The Put-Call Parity Portfolio

Payoff

at T

0 ST

Buy Stock

ST

Short Sell Call

ST

Combination

E=$85 E=$85

A Certain Payoff of

E=$85

E=$85

Buy Put

E=$85

$85

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Using Put-Call-Parity (PCP) to Replicate SecuritiesA synthetic security Definition - A portfolio of other securities which will pay the

same future cash flows as the security being replicated. Since payoffs at expiration (cash flows) are the same for the synthetic

security and the original security under all states of the world, their

current prices must be identical. Otherwise, if one is currently cheaper than the other, an arbitrage

opportunity will exist: buy (long) the cheaper security today for

the lower price, and simultaneously short sell the expensive

security for the higher price. This results in a positive initial

cash flow. This positive cash flow is an arbitrage profit (“free lunch”), since

at expiration, the cash flows from both positions will offset each other,

and the total cash flow at expiration will be zero.

From this point forward we notate: So = S, Po = P, and Co = C


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The Put-Call-Parity (PCP) Relationship:

This is a risk free T-bill that pays E dollars in T years

Recall that the PCP portfolio was created by:

Long one stock (+S), Long one put (+P), and Short one call (-C)

We saw that this is equivalent to:

Long a T-bill (+Ee-Trf)

Thus, we replicated a long position in a T-bill with: long stock, long put and short call.

For security replication purposes, use PCP with the following rule:

Long is “+”

Short is “-”

How Do We Replicate Securities?

CPSEe fTr

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A Synthetic Stock We first rearrange the PCP equation to isolate S:

According to the above replication rule:

Long one stock (+S) =

Long a T-bill (+Ee-Trf) & Long one call (+C) & Short one put (-P), The payoff (cash flow) at maturity:

ST < E ST > E

Long T-bill

Long Call

Short Put

Total Replicated Payoff: +ST +ST

Conclusion - holding the replicated portfolio is the same as holding the

stock

PCEeS fTr

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A Synthetic Call We first rearrange the PCP equation to isolate C:


Long one call (+C) =

Long one stock (+S) & Long one put (+P) & Short a T-bill (-Ee-Trf) The payoff (cash flow) at maturity:

ST < E ST > E

Long Stock

Long Put

Short T-bill

Total Replicated Payoff: $0 ST - E

Conclusion - holding the replicated portfolio is the same as holding a

call

fTrEePSC

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A Synthetic Put We first rearrange the PCP equation to isolate P:


Long one put (+P) =

Long a T-bill (+Ee-Trf) & Long one call (+C) & Short one stock (-S) The payoff (cash flow) at maturity:

ST < E ST > E

Long T-bill

Long Call

Short Stock

Total Replicated Payoff: E - ST $0

Conclusion - holding the replicated portfolio is the same as holding a

put

SCEeP fTr

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Bounding The Value of An American CallThe value of an American call can never be: below the difference b/w the stock price (S) and the exercise price (E).

If C < S - E: investors will pocket an arbitrage profit. Example: S = $100, E = $90, C = $8 => C = 8 < 10 = S – E Arbitrage Strategy:

Buy the call for $8, and exercise it immediately by paying the exercise price ($90)to get the stock (worth $100). This results in an immediate arbitrage profit (i.e. “free lunch”) of: -8-90+100= $2

Excess demand will force C to rise to $10 As long as there is time to expiration, we will have C > $10 = S - E

Above the value of the underlying stock (S) If it is, buy the stock directly

Boundary Conditions

Payoff at t

0E=$100

St

450

value of the American call will be here

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For an American Call:

C = C (S, E, T, , r)

(+) (-) (+) (+) (+)S - The higher the share price now, the higher the profit from exercising.

Thus the higher the option price will be.

E - The higher the exercise price now, the more it needs to be paid on

exercise. Thus, the lower the option value will be.

T - The more time there is to expiration, the higher the chance that the

stock price will be higher at T, and the higher the option value will be.

- The larger the volatility, the more probable a profitable outcome, thus

the higher C is.

r - The higher the interest rate, the P.V. of the future exercise price

decreases. The call price will increase.

For an American Put:

P = P(S, E, T, , r)

(-) (+) (+) (+) (-)

Determinants of American Option Pricing


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Determinants of American Option Pricing

Determinants of Relation to Relation to

Option Pricing Call Option Put Option

Stock price Positive Negative

Strike price Negative Positive

Risk-free rate Positive Negative

Volatility of the stock Positive Positive

Time to expiration date Positive Positive

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