OPTIONSOPTIONS

Concepts

Market

Concepts

Market

DefinitionDefinition

Option is a marketable security which gives the holder the right (but not the obligation) to buy an asset (stock, bond, currency, index, or futures contract) at a specific price (exercise price: X) on or possibly before a specific date (expiration date: T).

Option is a marketable security which gives the holder the right (but not the obligation) to buy an asset (stock, bond, currency, index, or futures contract) at a specific price (exercise price: X) on or possibly before a specific date (expiration date: T).

TermsTerms

– Holder: Buyer of the option; has the right to exercise; short position.

– Writer: Seller of the option; has the responsibility to fulfill the terms of the option if the holder exercises.

– Option Premium: Price of the option:• C = call premium

• P = put premium

– Holder: Buyer of the option; has the right to exercise; short position.

– Writer: Seller of the option; has the responsibility to fulfill the terms of the option if the holder exercises.

– Option Premium: Price of the option:• C = call premium

• P = put premium

TermsTerms

• American Option: can be exercised at any time on or before the expiration date.

• European Option: can be exercised only on the expiration date.

• Symbols: • T = Expiration when the option is exercised.

• o = current period

• t = any time between current and expiration

• American Option: can be exercised at any time on or before the expiration date.

• European Option: can be exercised only on the expiration date.

• Symbols: • T = Expiration when the option is exercised.

• o = current period

• t = any time between current and expiration

Fundamental StrategiesFundamental Strategies

• There are six fundamental strategies:– Call Purchase– Naked Call Write– Covered Call Write– Put Purchase– Naked Put Write– Covered Put Write

• There are six fundamental strategies:– Call Purchase– Naked Call Write– Covered Call Write– Put Purchase– Naked Put Write– Covered Put Write

Profit GraphProfit Graph

• Option Strategies can be evaluated in terms of a profit graph.

• A profit graph is a plot of the option position’s profit and stock price relation at expiration or when the option is exercised.

• Option Strategies can be evaluated in terms of a profit graph.

• A profit graph is a plot of the option position’s profit and stock price relation at expiration or when the option is exercised.

ST

ST

Call PurchaseCall Purchase

• Buy ABC 50 call for 3.

ST50

3

60

7

40

Naked Call WriteNaked Call Write

• Sell ABC 50 call for 3.

ST50

3

60

7

40

Covered Call WriteCovered Call Write

• Long 50 stock, short 50 call at 3.

ST50

3

60

7

40

Put PurchasePut Purchase

• Buy ABC 50 put for 3.

ST50

3

60

7

40

Naked Put WriteNaked Put Write

• Sell ABC 50 put for 3.

ST50

3

60

7

40

Covered Put WriteCovered Put Write

• Short ABC stock at 50, short put at 3.

ST50

3

60

7

40

Straddle PurchaseStraddle Purchase

• Buy 50 put for 3 and buy 50 call for 3:

ST50

4

60

6

40

Bull SpreadBull Spread

• Buy 50 call for 3 and sell 55 call for 1:

STST55

4

60

2

50

Simulated Long Stock PositionSimulated Long Stock Position

• Buy 50 call for 3, sell 50 put for 3:

STST50

3

60

7

40

7

10

10

Option Price Relations: CallsOption Price Relations: Calls

• The price of a call must be at least equal to its intrinsic value (IV).

• IV = Max[St-X,0]

• If C < IV, there is an arbitrage opportunity from buying the call, exercising, and then selling the stock in the market.

• In the absence of arbitrage:

• The price of a call must be at least equal to its intrinsic value (IV).

• IV = Max[St-X,0]

• If C < IV, there is an arbitrage opportunity from buying the call, exercising, and then selling the stock in the market.

• In the absence of arbitrage: C IVt C IVt

C = IV + TVPC = IV + TVP

• Call Price Curve

Ct

IV

StX

Call Curve.

Call FunctionCall Function

• Call Function:• Call Function:

C f S X Tt

( , , , )C f S X Tt

( , , , )

Note on VariabilityNote on Variability

– The call ( or put) price is positively related to the variability in the underlying stock. This is due to the limited loss feature of options.

– Greater volatility suggests a likelihood that the stock will increase substantially and a likelihood it will decrease substantially.

– However, given that a call’s losses are limited to just the premium, the extent of the price decrease is inconsequential.

– The call ( or put) price is positively related to the variability in the underlying stock. This is due to the limited loss feature of options.

– Greater volatility suggests a likelihood that the stock will increase substantially and a likelihood it will decrease substantially.

– However, given that a call’s losses are limited to just the premium, the extent of the price decrease is inconsequential.

Option Price Relations: PutsOption Price Relations: Puts

• The price of a put must be at least equal to its intrinsic value (IV).

• IV = Max[X-St,0]

• If P < IV, there is an arbitrage opportunity from buying the put, buying the stock, and exercising.

• In the absence of arbitrage:

• The price of a put must be at least equal to its intrinsic value (IV).

• IV = Max[X-St,0]

• If P < IV, there is an arbitrage opportunity from buying the put, buying the stock, and exercising.

• In the absence of arbitrage: P IVt P IVt

P = IV + TVPP = IV + TVP

• Put Price Curve:PtIV

StX

Put Curve.

Put FunctionPut Function

• Call Function:• Call Function:

P f S X Tt ( , , )P f S X Tt ( , , )

Put-Call Parity ModelPut-Call Parity Model

– Since the call and put derive their values from the underlying stock, the prices of the call, put, and stock are related. The relation governing their prices is know as the Put-Call Parity Model.

• The model is based on showing that a long stock, long put, and short call position (conversion) yields a riskless return equal to X at expiration and thus is equal to a price of RF security.

– Since the call and put derive their values from the underlying stock, the prices of the call, put, and stock are related. The relation governing their prices is know as the Put-Call Parity Model.

• The model is based on showing that a long stock, long put, and short call position (conversion) yields a riskless return equal to X at expiration and thus is equal to a price of RF security.

CF of Conversion at TCF of Conversion at T

Position S > X S = X S < X

Stock S S S

Call 0 0 -(S-X)

Put X-S 0 0

Total X X X

Position S > X S = X S < X

Stock S S S

Call 0 0 -(S-X)

Put X-S 0 0

Total X X X

Put-Call Parity ConditionPut-Call Parity Condition

• Since the conversion yields a riskless return equal to X at expiration, its value should be equal to the price of a riskless PDB with a face value of X and a maturity equal to T:

• Since the conversion yields a riskless return equal to X at expiration, its value should be equal to the price of a riskless PDB with a face value of X and a maturity equal to T:

S P CX

RPV X

fT0 0 0 1

( )

( )S P CX

RPV X

fT0 0 0 1

( )

( )

Option MarketOption Market

• Options are traded on organized exchanges.

• Noted Exchanges:– CBOE– PSE– PHLX

• The purpose of any exchange is to provided marketability.

• Options are traded on organized exchanges.

• Noted Exchanges:– CBOE– PSE– PHLX

• The purpose of any exchange is to provided marketability.

MarketabilityMarketability

• Option Exchanges provide marketabilty by:– Listings– Standardization

• Set: X, T, and Size

– Market Makers or Specialist– Option Clearing Corporation

• Option Exchanges provide marketabilty by:– Listings– Standardization

• Set: X, T, and Size

– Market Makers or Specialist– Option Clearing Corporation

Option Clearing CorporationOption Clearing Corporation

• Option Clearing Corporation (OCC) is a clearinghouse which guarantees the writer’s position on each contract and acts as an intermediary.

• As an intermediary, the OCC breaks up every contract after it has been established. By doing this, the OCC makes options more marketable.

• Option Clearing Corporation (OCC) is a clearinghouse which guarantees the writer’s position on each contract and acts as an intermediary.

• As an intermediary, the OCC breaks up every contract after it has been established. By doing this, the OCC makes options more marketable.

OCC ExampleOCC Example

• Suppose A buys an ABC 50 call from B for 3: A is long; B is short.

• After this contract is established, OCC breaks it up. – A’s right to exercise is now with the OCC, and

B’s responsibility is when the OCC exercises - which could be when someone (not just A) notifies the OCC they want to exercise.

• Suppose A buys an ABC 50 call from B for 3: A is long; B is short.

• After this contract is established, OCC breaks it up. – A’s right to exercise is now with the OCC, and

B’s responsibility is when the OCC exercises - which could be when someone (not just A) notifies the OCC they want to exercise.

OCC Record: Entry 1OCC Record: Entry 1

• A: Right to exercise

• B: Responsibility

• A: Right to exercise

• B: Responsibility

Offsetting Transaction for ‘A’Offsetting Transaction for ‘A’

• Suppose the price of ABC stock increases to $60, pushing the price of the ABC 50 call up to $12.

• Seeing the opportunity to profit, suppose A sells an ABC 50 call to C for $12.– OCC breaks up this contract.– OCC’s new entry of responsibility for A

cancels A’s right entry: A’s position is closed.

• Suppose the price of ABC stock increases to $60, pushing the price of the ABC 50 call up to $12.

• Seeing the opportunity to profit, suppose A sells an ABC 50 call to C for $12.– OCC breaks up this contract.– OCC’s new entry of responsibility for A

cancels A’s right entry: A’s position is closed.

OCC Record Entry 2OCC Record Entry 2

• A: Right to exercise

• B: Responsibility

• A: Responsibility

• C: Right to Exercise

Closed

Offsetting Transaction for BOffsetting Transaction for B

– Suppose when the price of ABC stock is at $60 and the price of the ABC 50 call is at $12, B begins to worry and decides to close his short position. He can do this by going long in the ABC 50 call.

– Suppose B buys an ABC 50 call from D.• OCC breaks up contract.

• OCC’s new entry of right for B cancels B’s responsibility entry: B’s position is closed.

– Suppose when the price of ABC stock is at $60 and the price of the ABC 50 call is at $12, B begins to worry and decides to close his short position. He can do this by going long in the ABC 50 call.

– Suppose B buys an ABC 50 call from D.• OCC breaks up contract.

• OCC’s new entry of right for B cancels B’s responsibility entry: B’s position is closed.

OCC Record Entry 3OCC Record Entry 3

• B: Responsibility

• C: Right

• B: Right

• D: Responsibility

Closed

Importance of the OCCImportance of the OCC

– By acting as an intermediary, the OCC makes it possible for option investors to close their positions.

– In this example:• ‘A’ was able to buy an ABC 50 call for 3, then

later close her position by selling another ABC 50 call for 12: Profit = 9.

• ‘B’ was able to sell an ABC 50 Call for 3, then later close his position by buying another ABC 50 call for 12: loss = 9.

– By acting as an intermediary, the OCC makes it possible for option investors to close their positions.

– In this example:• ‘A’ was able to buy an ABC 50 call for 3, then

later close her position by selling another ABC 50 call for 12: Profit = 9.

• ‘B’ was able to sell an ABC 50 Call for 3, then later close his position by buying another ABC 50 call for 12: loss = 9.

Types of TransactionsTypes of Transactions

• Opening

• Exercising

• Expiring

• Closing (or Offsetting)

• Opening

• Exercising

• Expiring

• Closing (or Offsetting)

Option CostsOption Costs

• Margin Requirements: – Initial Margin: Option writers are required to

deposit cash or risk-free securities with their brokers to secure their positions.

– Maintenance Margin: Writers are required to post additional cash of RF securities when stock prices move against them. See JG: 40-46.

• Other Costs: Commissions; bid-ask spread.

• Margin Requirements: – Initial Margin: Option writers are required to

deposit cash or risk-free securities with their brokers to secure their positions.

– Maintenance Margin: Writers are required to post additional cash of RF securities when stock prices move against them. See JG: 40-46.

• Other Costs: Commissions; bid-ask spread.

Other OptionsOther Options

• Stock Indices

• Warrants

• Rights

• Foreign Currency

• Debt Securities

• Futures

• Stock Indices

• Warrants

• Rights

• Foreign Currency

• Debt Securities

• Futures

Stock Index OptionStock Index Option

• Options on stock indices: S&P 500, S&P 100, and MMI.

• Features:– Cash Settlement: when you exercise, the

assigned writer pays you the difference between the closing (spot) index (S) and the exercise price; Call: S-X; Put: X-S.

– Multiplier: $100.

• Options on stock indices: S&P 500, S&P 100, and MMI.

• Features:– Cash Settlement: when you exercise, the

assigned writer pays you the difference between the closing (spot) index (S) and the exercise price; Call: S-X; Put: X-S.

– Multiplier: $100.

Portfolio InsurancePortfolio Insurance

• An important use of stock index options is portfolio insurance. This is a hedging strategy in which an equity portfolio manager protects the future value of her fund by buying index put options.– The put options provide downside protection

against a market decline, while allowing the fund to grow if the market increases.

• An important use of stock index options is portfolio insurance. This is a hedging strategy in which an equity portfolio manager protects the future value of her fund by buying index put options.– The put options provide downside protection

against a market decline, while allowing the fund to grow if the market increases.

Portfolio Insurance ExamplePortfolio Insurance Example

– Suppose a fund manager has a $50M portfolio that he may have to liquidate in September. The fund is well-diversified with a beta of 1.25. The current S&P 500 is at 1250, and there is a September S&P 500 put with an exercise price of 1250, multiplier of 100, and trading at 50.

– To form a portfolio insurance position, the manager would need to buy Beta(50M)/X =500 puts at a cost of $2.5M= (100)(500)(50).

– Suppose a fund manager has a $50M portfolio that he may have to liquidate in September. The fund is well-diversified with a beta of 1.25. The current S&P 500 is at 1250, and there is a September S&P 500 put with an exercise price of 1250, multiplier of 100, and trading at 50.

– To form a portfolio insurance position, the manager would need to buy Beta(50M)/X =500 puts at a cost of $2.5M= (100)(500)(50).

Put-Hedged Portfolio at TPut-Hedged Portfolio at T

S 1 . 2 5 g ( . ) 5 01 1 2 5 g P u t T o t a l1 0 0 0 - . 2 5 $ 3 7 . 5 M $ 1 0 M $ 4 7 . 51 1 2 5 - . 1 2 5 $ 4 3 . 7 5 M $ 3 . 7 5 4 7 . 5 01 2 5 0 0 $ 5 0 M - $ 2 . 5 4 7 . 5 01 3 7 5 . 1 2 5 $ 5 6 . 2 5 M - $ 2 . 5 5 3 . 7 51 5 0 0 . 2 5 $ 6 2 . 5 M - $ 2 . 5 6 0 . 0 0

gS

1250

1250 Put S M (500)( )[ ] $2.100 1250 5

Other Index UsesOther Index Uses

• Speculating on the market.

• Market Timing

• Speculating on Unsystematic Factors

• Dynamic Portfolio Insurance

• Speculating on the market.

• Market Timing

• Speculating on Unsystematic Factors

• Dynamic Portfolio Insurance

WarrantWarrant

– Warrant is a call option issued by a Co.– The contractual features of a warrant and a call

are the same.– The primary difference between a warrant and a

call is that the writer of the warrant is the issuing corporation.

– If a warrant is exercised, the corporation receives cash and creates new shares -- dilution effect.

– Warrant is a call option issued by a Co.– The contractual features of a warrant and a call

are the same.– The primary difference between a warrant and a

call is that the writer of the warrant is the issuing corporation.

– If a warrant is exercised, the corporation receives cash and creates new shares -- dilution effect.

Warrants ExampleWarrants Example

– Example: LM Co. is a $100,000 oil well company; all equity, with 100 shares outstanding and with each share worth $1000; has 4 shareholders, A, B, C, and D, each with 25 shares.

– Consider alternatives: Shareholder D sells a call option to investor E, giving her the right to buy 25 shares at X = $1100/share; LM Co. sell a warrant with the same features.

– Example: LM Co. is a $100,000 oil well company; all equity, with 100 shares outstanding and with each share worth $1000; has 4 shareholders, A, B, C, and D, each with 25 shares.

– Consider alternatives: Shareholder D sells a call option to investor E, giving her the right to buy 25 shares at X = $1100/share; LM Co. sell a warrant with the same features.

Oil Well Increases in Value Oil Well Increases in Value

• Oil Well increases in value to $120,000, causing LM stock to go to $1200.

• Call Exercised:– When Investor E exercises, shareholder D will

simply turn over his 25 shares for $1100/share.– Exercising has no impact on the value of LM.– IVc = $1200-$1100 = $100.

• Oil Well increases in value to $120,000, causing LM stock to go to $1200.

• Call Exercised:– When Investor E exercises, shareholder D will

simply turn over his 25 shares for $1100/share.– Exercising has no impact on the value of LM.– IVc = $1200-$1100 = $100.

Warrant ExercisedWarrant Exercised

– When E exercises warrant, the LM Co. will have to print 25 new shares (Nw) and sell them to E at X = $1100/share.

– The company will receive cash, but the number of shares will increase:

• 100 to 125 shares (dilution)

• ($1100) (25) = $27,500

– In this case, the exercise of the warrant lowers the value of the stock from $1200 to $1180.

– When E exercises warrant, the LM Co. will have to print 25 new shares (Nw) and sell them to E at X = $1100/share.

– The company will receive cash, but the number of shares will increase:

• 100 to 125 shares (dilution)

• ($1100) (25) = $27,500

– In this case, the exercise of the warrant lowers the value of the stock from $1200 to $1180.

Warrant ValueWarrant Value

Stock and Warrant Values:Stock and Warrant Values:

VLM

$120, $27,

$1180000 500

100 25

IVw $1180 $1100 $80

Warrant and Call RelationWarrant and Call Relation

• Relation: The value of the warrant is equal to the value of the call times the dilution factor:

• Relation: The value of the warrant is equal to the value of the call times the dilution factor:

IVn

n nIVw

wc

0

0

IVn

n nIVw

wc

0

0

RightsRights

– Definition: A right is call option issued by a corporation to existing shareholders, giving the holder the right to buy new shares at a specified price (subscription price).

– Corporations use rights to ensure they adhere to preemptive rights.

– A right is like a warrant: when it is exercised, new shares are created and the company receives cash.

– Definition: A right is call option issued by a corporation to existing shareholders, giving the holder the right to buy new shares at a specified price (subscription price).

– Corporations use rights to ensure they adhere to preemptive rights.

– A right is like a warrant: when it is exercised, new shares are created and the company receives cash.

Rights ExampleRights Example

ABC company is planning to raise $20M in equity to finance a capital acquisition. The company is worth $200M, has no debt, and has 1 million shares of stock outstanding, with each share currently trading at $200. ABC plans to finance its $20M investment with a rights offering in which it gives shareholders the right to buy new shares at $160, with each holder receiving one right for each share owned.

ABC company is planning to raise $20M in equity to finance a capital acquisition. The company is worth $200M, has no debt, and has 1 million shares of stock outstanding, with each share currently trading at $200. ABC plans to finance its $20M investment with a rights offering in which it gives shareholders the right to buy new shares at $160, with each holder receiving one right for each share owned.

Value of the RightValue of the Right

Number of new shares =

Number of Rights for 1 Share =

Stock Value

IV of Right

Number of new shares =

Number of Rights for 1 Share =

Stock Value

IV of Right

$20, ,

$160 /,

000 000125 000

Share

2 000 000

125 00016

, ,

,

$200 $20

,$195.

M M

M1 125 00056

[$196. $160] / $2.56 16 22

Foreign Currency OptionsForeign Currency Options

• Currency options are traded on the Philadelphia exchange (PHLX) and on exchanges in Toronto, Montreal, and Amsterdam. There is also a Dealer’s Market that is part of the Interbank FC market.

• PHLX offers trading in: BP, DM, JY, SF, FF, AD, and CD. See JG, pp. 305-307 for features of FC options.

• Currency options are traded on the Philadelphia exchange (PHLX) and on exchanges in Toronto, Montreal, and Amsterdam. There is also a Dealer’s Market that is part of the Interbank FC market.

• PHLX offers trading in: BP, DM, JY, SF, FF, AD, and CD. See JG, pp. 305-307 for features of FC options.

Hedging Currency PositionsHedging Currency Positions

– Suppose a U.S. Investment fund is expecting 625,000DM in September from its Eurobonds.

– The Fund expects the exchange rate to increase from its current level of Eo =$.40/DM, suggesting greater dollar revenues than $250,000, but it is not confident. To benefit from an increase in Eo, while hedging against an Eo decrease, the fund buys 10 DM put contracts with X=$.40, P=$.01,size=62500DM.

– Suppose a U.S. Investment fund is expecting 625,000DM in September from its Eurobonds.

– The Fund expects the exchange rate to increase from its current level of Eo =$.40/DM, suggesting greater dollar revenues than $250,000, but it is not confident. To benefit from an increase in Eo, while hedging against an Eo decrease, the fund buys 10 DM put contracts with X=$.40, P=$.01,size=62500DM.

Hedge DM Revenue at TE Put Profit E(625000) (2)+(3)

$.30/DM $56,250 $187,500 $243,750

.35 25,000 218,750 243,750

.40 -6,250 250,000 243,750

.45 -6,250 281,250 275,000

.50 -6,250 312,500 306,250

10 40 62500 10 01 62500Max E DM DM DM[$. ]( ) ($. / )( )