Stock Index Options

Chapter 9

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.– End-of-the-day exercise

• 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.– End-of-the-day exercise

Uses

• Speculation: – Bullish: long in call– Bearish: long in put– Stable market: short

• Portfolio Insurance

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 1.25(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 1.25(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

Fiduciary Call

• If the manager knew he was going to liquidate in September, he could sell the portfolio and invest the proceeds in a RF security and buy a call option.

• Assume the September option expires in six months, Rf = 6%, and the put-call parity holds:

P S C PV X

C P S PV X

C

( )

( )

( . ).

.50 1250

1250

10613 05

5

Fiduciary Call

• Strategy:

• Liquidate the portfolio for $50M.

• Invest $50M in RF security at 6% for 6 months.

• Buy 500 calls:

nV

X

Mcallsc 0 125

100 1250500.

$50

( )( )

Fiduciary CallFiduciary Call

S 1.25g Investment Call Total1000 -.25 $51.478M -.6525M $50.81125 -.125 $51.478M -.6525M 50.81250 0 $51.478M -.6525M 50.81375 .125 $51.478M 5.6975M 57.21500 .25 $51.478M 11.8475 63.3

gS

1250

1250

Call Max ST (500)( ) [ , ] (500)( )( . )100 1250 0 100 13 05

Pricing Index Options

• The B-S model and the BOPM are defined by a replicating portfolio. For an index option, the replicating portfolio is a stock portfolio consisting of all the stocks in the index or a proxy portfolio formed with fewer stocks that are highly correlated with the index (maybe a MF).

• Later we will define it in terms of a index futures position.

Proxy Portfolio

• A proxy portfolio can be viewed as a position in the index.

• For example, suppose you have a proxy portfolio worth Vo = $1.5M and the S&P 500 index is at So = 1250.

• The proxy portfolio would be the equivalent of buying n = $1.5M/1250 = 1200 index shares at $1250 per share.

BOPM

• Consider single-index BOPM with u = 1.0227, d = .9778, Rf = .005, So = 310, X= 310.

S

Cu

u

317 037

12 037

.

.

S

Cu

u

303118

0

.

S

H

B

C

0

0

0

0

310

8648

260 829

7 259

.

.

.*

Arbitrage Portfolio

If C C short in Call

and long in RP

RP H S in proxy portfolio

borrow B

M0 0

0 0

0

8649 310 09

83

*

.

: (. )( ) $268. .

$260. .

Black-Scholes Model for Index Options

• The most common way to price index option using the B-S model is to use the continuous dividend adjustment model where Sd is used instead of So.

• Note: BAW model may be a better model for pricing index options.

Dynamic Portfolio Insurance

• Because of the position limits on options, a portfolio insurance strategy formed with index put options may not be applicable for large portfolios.

• An alternative is to use a dynamic portfolio insurance strategy using a stock and bond portfolio which is managed over time using a binomial framework.

• Consider a portfolio which is highly correlated with a stock index that is currently at So = 150. Assume u = 1.1, d = 1/1.1 = .9091, Rf = .05, and n = 2.

• As show in the figure, with portfolio insurance one could obtain portfolio values of 3.025M if the market increases and 2.5M if its stays at 150 or declines.

S

V Muu

uu

1815

3025

.

.

S

V Mud

ud

150

2 5.

S

V Muu

uu

12397

2 066

.

.

S

V Md

d

136 3636

2 2727

.

.

S

V Mu

u

165

2 75.

S

V M0

0

150

2 5

.

With PI M2 5.

Dynamic Strategy Objective

• Objective: Construct a bond and stock portfolio and manage it in such a way that you will have possible CFs of 3.025M, 2.5M, and 2.5M given the three possible spot index values at the end of period 2.

Dynamic Strategy Methodology

• Define the portfolio as an investment in hypothetical shares of the stock index.

• Form a bond and stock portfolio consisting of a diversified stock portfolio and an a investment in RF security (Io)

nM

shares

M diversified portfoliois

equivalent to buying

index shares at per share

M

$2.,

$2.

,

$150 .

$2. ( , )( )

5

15016 667

5

16 667

5 16 667 150

3025. M

nS I0 0

n uS I rf( )0 0

n dS I rf( )0 0

2 5. M

2 5. M

Dynamic Strategy Methodology

• Period 1:• If the stock and bond portfolio is worth Vu =

$2.75M when Su = 165, then it could be converted to all stock and reach next period’s target of being either 3.025M or 2.5.

• If the stock and bond portfolio is worth Vd = 2.5M/1.05 = 2.381 when Sd = 136.3636, then the portfolio could be converted to all bonds and hit its target next period of 2.5.

Dynamic Strategy Methodology

• Mathematically, our problem is to solve for the n and Io where:

n uS I r V

n dS I r Vf u

f d

( )

( )0 0

0 0

n I V

n I Vu

d

( ) ( . )

( . ) ( . )

165 105

136 36 1050

0

Dynamic Strategy Methodology

• Solution:

nV V

uS dS

M Mshares

IuV dV

r u d

M MM

Portfolio Stock Portfolio n S M

Bond I M

Total M

M investment M insurance

u d

d u

f

*

*

*

*

. .

.,

( )

. ( . ) . ( . )

. ( . . )$0.

: ( ) $1.

$.

.

$2. .

0 0

0

0 0

0

2 75 2 381

165 136 3612 886

11 2 381 9091 2 75

105 11 90915940

12889 150 933

5940

2 527

5 027

3025. M

Stock Bond

M M

Total M

1933 5940

2 527

. .

.

2 5. M

2 5. M

Stock Bond

M M

Total

Convert to all stock

1933 11 5940 105

2 75

. ( . ) . ( . )

.

Stock Bond

M M

Total

Convert to all bonds

1933 9091 5940 105

2 381

. (. ) . ( . )

.

Foreign Currency Options

Chapter 10

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.

FC Put-Call ParityFC Put-Call Parity

• Conversion: • Conversion: Take E R dollars

Convert E R dollars

to R FC and invest at R

Go long in FC put

Go Short in FC call

FT

FT

FT

F

0

0

1

1

1

( ) ;

( )

( ) .

.

.

FC Put-Call ParityFC Put-Call Parity

• At T position will be worth X. • At T position will be worth X.

E X E X E X

Value of FC E E E

Long Put X E

Short Call E X

X X X

T T T

T T T

T

T

$

( )

1

0 0

0 0

E R P C X RFT

fT

0 0 01 1( ) ( )

FC BOPMFC BOPM

• BOPM for FC is similar to that for stock except that a foreign interest rate is included.

• The replicating portfolio consists of buying Ho units of FC at Eo and borrowing Bo dollars. The Ho units of FC can be invested for the period in a foreign RF security.

• BOPM for FC is similar to that for stock except that a foreign interest rate is included.

• The replicating portfolio consists of buying Ho units of FC at Eo and borrowing Bo dollars. The Ho units of FC can be invested for the period in a foreign RF security.

FC BOPMFC BOPM

• Single Period• Single Period

E

C

H E B

0

0

0 0 0

E uE

C Max uE X

H r uE B r

u

u

F us

0

0

0 0 0

0[ , ]

( )

E dE

C Max dE X

H r dE B r

d

d

F us

0

0

0 0 0

0[ , ]

( )

FC BOPMFC BOPM

H r uE B r C

H r dE B r CF us u

F us d

0 0 0

0 0 0

( )

( )

• Solve for Ho and Bo where:

HC C

E r u d

BC dE C uE

r uE dE

u d

F

u o d

us

00

00

0 0

( )

( ) ( )

( )

FC: B-S Model FC: B-S Model

• Use Ed instead of Eo:• Use Ed instead of Eo:

E E edR TF

0

FC Options UseFC Options Use

• Speculation on Exchange rates

• Hedging future FC payment or receipt positions

• Speculation on Exchange rates

• Hedging future FC payment or receipt positions

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[$. ]( ) ($. / )( )

Hedging Currency PositionsHedging Currency Positions

– Suppose a U.S. Company has a 625,000DM debt to be paid in September.

– The company expects the exchange rate to decrease from its current level of Eo =$.40/DM, suggesting lower dollar cost than $250,000, but it is not confident. To benefit from a decrease in Eo, while hedging against an Eo increase, the company buys 10 DM call contracts with X=$.40, C=$.01,size=62500DM.

– Suppose a U.S. Company has a 625,000DM debt to be paid in September.

– The company expects the exchange rate to decrease from its current level of Eo =$.40/DM, suggesting lower dollar cost than $250,000, but it is not confident. To benefit from a decrease in Eo, while hedging against an Eo increase, the company buys 10 DM call contracts with X=$.40, C=$.01,size=62500DM.

Hedge DM Cost at TE Call Profit E(625000) (3)-(2)

$.30/DM -6,250 $187,500 193750

.35 -6250 218,750 225000

.40 -6,250 250,000 256250

.45 25000 281,250 256250

.50 56250 312,500 256250

10 40 0 62500 10 01 62500Max E DM DM DMT[ $. , ] $. / ( )b g

Options Issued by Corporations

Chapter 11

WarrantWarrant

– A warrant is a call option issued by a Corporation.– 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.

– A warrant is a call option issued by a Corporation.– 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; the LM Co. sells 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; the LM Co. sells 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 her 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 her 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