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Stock Index Options. Chapter 9. Stock 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. - PowerPoint PPT Presentation
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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
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