Options on stock indices, currencies, and futures
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Options on stock indices, currencies, and futures
description
Options on stock indices, currencies, and futures. Options On Stock Indices. Contracts are on 100 times index; they are settled in cash - PowerPoint PPT Presentation
Transcript of Options on stock indices, currencies, and futures
Options on stock indices, currencies, and futures
Options On Stock Indices
Contracts are on 100 times index; they are settled in cash
On exercise of the option ,the holder of a call option receives (S-K)*100 in cash and the writer of the option pays this amount in cash ;the holder of a put option receives
(K-S)*100 in cash and the writer of the option pays this amount in cash
S :the value of the index
K :the strike price
The most popular underlying indices in the U.S.
The Dow Jones Index (DJX) The Nasdaq 100 Index (NDX) The Russell 2000 Index (RUT) The S&P 100 Index (OEX) The S&P 500 Index (SPX)
LEAPS
Leaps are options on stock indices that last up to 3 years
They have December expiration dates Leaps also trade on some individual stocks ,they have
January expiration dates
Portfolio Insurance
Consider a manager in charge of a well-diversified portfolio whose is 1.0
The dividend yield from the portfolio is the same as the dividend yield from the index
The percentage changes in the value of the portfolio can be expected to be approximately the same as the percentage changes in the value of the index
Portfolio InsuranceExample Portfolio has a of 1.0 It is currently worth $500,000 The index currently stands at 1000 What trade is necessary to provide insurance against
the portfolio value falling below $450,000?
Portfolio InsuranceExample Buy 5 three-month put option contracts on the index
with a strike price of $900 The index drops to 880 in three months the portfolio is worth about
5*880*100 = $440,000 the payoff from the options
5*(900-880)*100 = $10,000 total value of the portfolio
440,000+10,000 = $450,000
When the portfolio beta is not 1.0Example Portfolio has a beta of 2.0 It is currently worth $500,000 and index stands at
1000 The risk-free rate is 12% per annum The dividend yield on both the portfolio and the index
is 4% How many put option contracts should be purchased
for portfolio insurance?
When the portfolio beta is not 1.0Example Value of index in three months 1040 Return from change in index 40/1000=4% Dividends from index 0.25*4=1% Total return from index 4+1=5% Risk-free interest rate 0.25*12=3% Excess return from index 5-3=2% Expected excess return from portfolio 2*2=4% Expected return from portfolio 3+4=7% Dividends from portfolio 0.25*4=1% Expected increase in value of portfolio 7-1=6% Expected value of portfolio $ 500,000*1.06= $530,000
Relationship between value of index and value of portfolio for beta = 2.0
Value of Index in 3 months
Expected Portfolio Value in 3 months ($)
1,080 570,000 1,040 530,000 1,000 490,000 960 450,000 920 410,000 880 370,000
The correct strike price for the 10 put option contracts that are purchased is 960
Currency Options
Currency options trade on the Philadelphia Exchange (PHLX)
There also exists an active over-the-counter (OTC) market
Currency options are used by corporations to buy insurance when they have an FX exposure
Currency options Example An example of a European call option
Buy $1,000,000 euros with USD at an exchange rate of 1.2000 USD per euro ,if the exchange rate at the maturity of the option is 1.2500 ,the payoff is
1,000,000*(1.2500-1.2000) = $50,000
Currency options Example
An example of a European put option
Sell $10,000,000 Australian for USD at an exchange rate of 0.7000 USD per Australian ,if the exchange rate at the maturity of the option is 0.6700 ,the payoff is
10,000,000*(0.7000-0.6700) = $300,000
Range forwardsShort range-forward contract Buy a European put option with a strike price of K1 and
sell a European call option with a strike price of K2
Range forwardsLong range-forward contract Sell a European put option with a strike price of K1 and
buy a European call option with a strike price of K2
Options On Stocks Paying Known Dividend Yields Dividends cause stock prices to reduce on the ex-dividend date b
y the amount of the dividend payment The payment of a dividend yield at rate q therefore cause the gro
wth rate in the stock price to be less than it would otherwise be by an amount q
With a dividend yield of q ,the stock price grow from S0 today to ST
Without dividends it would grow from S0 today to
STeqT
Alternatively ,without dividends it would grow from S0e–qT today to
ST
European Options on StocksProviding a Dividend Yield We get the same probability distribution for the stock
price at time T in each of the following cases:1. The stock starts at price S0 and provides a dividend yield at
rate q
2. The stock starts at price S0e–qT and pays no dividends
We can value European options by reducing the stock price to S0e–qT and then behaving as though there is no dividend
Put-call parity
Put-call parity for an option on a stock paying a dividend yield at rate q
For American options, the put-call parity relationship is
qTrT eSpKec 0
rTqT KeSPCKeS 00
Pricing Formulas
By replacing S0 by S0e–qT in Black-Sholes formulas ,we obtain that
T
TqrKSd
T
TqrKSd
qTK
S
K
eS
dNeSdNKep
dNKedNeSc
qT
qTrT
rTqT
)2/2()/ln(
)2/2()/ln( where
ln ln since
)()(
)()(
02
01
00
102
210
Risk-neutral valuation
In a risk-neutral world ,the total return must be r ,the dividends provide a return of q ,the expected growth rate in the stock price must be r – q
The risk-neutral process for the stock price
SdzSdtqrdS
Risk-neutral valuation continued
The expected growth rate in the stock price is
r –q ,the expected stock price at time T is S0e(r-q)T
Expected payoff for a call option in a risk-neutral world as
Where d1 and d2 are defined as above Discounting at rate r for the T
)()( 210)( dKNdNSe Tqr
)()( 210 dNKedNeSc rTqT
Valuation of European Stock Index Options We can use the formula for an option on a stock
paying a dividend yield
S0 : the value of index
q : average dividend yield
: the volatility of the index
)()(
)()(
102
210
dNeSdNKep
dNKedNeScqTrT
rTqT
Example
A European call option on the S&P 500 that is two months from maturity
S0 = 930, K = 900, r = 0.08
= 0.2, T = 2/12
Dividend yields of 0.2% and 0.3% are expected in the first month and the second month
Example continued
The total dividend yield per annum is q = (0.2% + 0.3%)*6 = 3%
one contract would cost $5,183
83.516782.09007069.0930
6782.0)(
7069.0)(
4628.012/22.0
12/2)2/22.003.008.0()900/930ln(
5444.012/22.0
12/2)2/22.003.008.0()900/930ln(
12/208.012/203.0
2
1
2
1
eec
dN
dN
d
d
Forward price
Define F0 as the forward price of the index 00
TqreSF
T
TKFd
T
TKFd
dNeFdNKep
dNKedNeFcrTrT
rTrT
)2/2()/ln(
)2/2()/ln( where
)()(
)()(
02
01
102
210
Implied dividend yields
By qTrT eSpKec 0
0
0
0
0
ln1
ln
S
Kepc
Tq
S
KepcqT
S
Kepce
KepceS
rT
rT
rTqT
rTqT
Valuation of European Currency Options
A foreign currency is analogous to a stock paying a known dividend yield
The owner of foreign receives a yield equal to the risk-free interest rate, rf
With q replaced by rf , we can get call price, c, and put price, p
T
TrrKSd
T
TrrKSd
dNeSdNKep
dNKedNeSc
f
f
TrrT
rTTr
f
f
)2/2()/ln(
)2/2()/ln( where
)()(
)()(
02
01
102
210
Using forward exchange rates Define F0 as the forward foreign exchange rate
T
TKFd
T
TKFd
dNFdKNep
dKNdNFecrT
rT
)2/2()/ln(
)2/2()/ln( where
)()(
)()(
02
01
102
210
00
Trr feSF
American Options
The parameter determining the size of up movements, u, the parameter determining the size of down movements, d
The probability of an up movement is
In the case of options on an index
In the case of options on a currency
tqrea
trr fea
du
dap
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Nature of Futures Options
A call futures is the right to enter into a long futures
contracts at a certain price.
A put futures is the right to enter into a short futures
contracts at a certain price.
Most are American; Be exercised any time during the
life .
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Nature of Futures Options
When a call futures option is exercised the holder acquires :
If the futures position is closed out immediately:
Payoff from call = F0 – K
where F0 is futures price at time of exercise
1. A long position in the futures
2. A cash amount equal to the excess of
the futures price over the strike price
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Example
Today is 8/15, One September futures call option on copper,K=240(cents/pound), One contract is on 25,000 pounds of copper.
Futures price for delivery in Sep is currently 251cents 8/14 (the last settlement) futures price is 250
IF option exercised, investor receive cash:
25,000X(250-240)cents=$2,500
Plus a long futures, if it closed out immediately:
25,000X(251-250)cents=$250
If the futures position is closed out immediately
25,000X (251-240)cents=$2,750
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Nature of Futures Options When a put futures option is exercised the holder acquires :
If the futures position is closed out immediately:
Payoff from put =K–F0
where F0 is futures price at time of exercise
1. A short position in the futures
2. A cash amount equal to the excess of
the strike price over the futures price
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Reasons for the popularity on futures option
Liquid and easier to trade
From futures exchange, price is known immediately.
Normally settled in cash
Futures and futures options are traded in the same exchange.
Lower cost than spot options
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European spot and futures options
Payoff from call option with strike price K on spot price of an asset:
Payoff from call option with strike price K on futures price of an asset:
When futures contracts matures at the same time as the option:
)0,Smax( T K
)0,Fmax( T K
TSTF
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Put-Call Parity for futures options Consider the following two portfolios:
A. European call plus Ke-rT of cash
B. European put plus long futures plus cash equal to F0e-rT
They must be worth the same at time T so that
c+Ke-rT=p+F0 e-rT
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Example
%10
,50.8$,56.0$,8$F0
fR
KcEuropean call option on spot silver for delivery in six month
c+Ke-rT=p+F0 e-rT
Use equation
→P=c+Ke-rT-F0 e-rT
P=0.56+8.50e-0.1X0.5-8 e-0.1X0.5
=1.04
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Bounds for futures options
rTrT eKec 0F
PeKe rTrT 0F
rTeKc )F( 0
rTeKP )F( 0
c+Ke-rT=p+F0 e-rT
Similarly
or
or
By
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Bounds for futures options
Kc 0F
0FKP
Because American futures can be exercised at any time,
we must have
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Valuation by Binomial trees
A 1-month call option on futures has a strike price of 29.
Futures Price = $33Option Price = $4
Futures Price = $28Option Price = $0
Futures price = $30Option Price=?
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Valuation by Binomial trees
• Consider the Portfolio: long futuresshort 1 call option
• Portfolio is riskless when 3– 4 = -2 or = 0.8
3– 4
-2
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Valuation by Binomial trees
• The riskless portfolio is:
long 0.8 futures
short 1 call option• The value of the portfolio in 1 month
is -1.6• The value of the portfolio today is
-1.6e – 0.06/12 = -1.592
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A Generalization
• A derivative lasts for time T and
is dependent on a futures price
F0u ƒu
F0d ƒd
F0
ƒ
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A Generalization
dFuF
ff du
00
F0u F0 – ƒu
F0d F0– ƒd
• Consider the portfolio that is long Δ futures and short 1 derivative
• The portfolio is riskless when
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A Generalization
• Value of the portfolio at time T is F0u Δ –F0 Δ – ƒu
• Value of portfolio today is – ƒ
• Hence ƒ = – [F0u Δ –F0 Δ – ƒu]e-rT
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A Generalization
• Substituting for Δ we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
pd
u d
1
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Drift of a futures price in a risk-neutral word
Valuing European Futures Options We can use the formula for an option on a stock
paying a dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r ) Setting q = r ensures that the expected growth of F
in a risk-neutral world is zero
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Growth Rates For Futures Prices
• A futures contract requires no initial investment• In a risk-neutral world the expected return
should be zero• The expected growth rate of the futures price is
therefore zero• The futures price can therefore be treated like a
stock paying a dividend yield of r
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Results
T
TqrKSd
T
TqrKSd
dNeSdNKep
dNKedNeScqTrT
rTqT
)2/2()/ln(
)2/2()/ln(
)()(
)()(
02
01
102
210
where
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Black’s Formula
• The formulas for European options on futures are known as Black’s formulas
TdT
TKFd
T
TKFd
dNFdNKep
dNKdNFecrT
rT
10
2
01
102
210
2/2)/ln(
2/2)/ln( where
)( )(
)( )(
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Example
European put futures option on crude oil,…Fo=20,K=20, r=0.09,T=4/12,σ=0.25,ln(F0=K)=0,So that
12.1$4712.0205288.020
5288.0)(
4712.0)(
07216.02
07216.02
12/409.0
2
1
2
1
Xep
dN
dN
Td
Td
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American futures options vs spot options
• If futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spot
• When futures prices are lower than spot prices (inverted market) the reverse is true
