index, currency and futures options
DESCRIPTION
Index, Currency and Futures Options. Finance (Derivative Securities) 312 Tuesday, 24 October 2006 Readings: Chapters 13 & 14. Known Dividend Yield. Same probability distribution for stock price at time T if: Stock starts at price S 0 and provides a dividend yield = q - PowerPoint PPT PresentationTRANSCRIPT
Index, Currency Index, Currency and Futures and Futures
OptionsOptionsFinance (Derivative Securities) 312
Tuesday, 24 October 2006
Readings: Chapters 13 & 14
Known Dividend YieldKnown Dividend Yield
Same probability distribution for stock price at time T if:• Stock starts at price S0 and provides a
dividend yield = q• Stock starts at price S0e–qT and provides no
incomeReduce current stock price by dividend
yield, then value option as though stock pays no dividends
Option PricingOption Pricing
Lower Bound for Calls• c S0e–qT –Ke –rT
Lower Bound for Puts• p Ke–rT – S0e–qT
Put-call Parity• p + S0e–qT = c + Ke–rT
Black ScholesBlack Scholes
T
TqrKSd
T
TqrKSd
dNeSdNKep
dNKedNeScqTrT
rTqT
)2/2()/ln(
)2/2()/ln(
)()(
)()(
02
01
102
210
where
Binomial ModelBinomial Model
In a risk-neutral world the stock price grows at r – q rather than at r when there is a dividend yield q
The probability, p, of an up movement must therefore satisfy
pS0u + (1 – p)S0d = S0e(r-q)T
so that: pe d
u d
r q T
( )
Index OptionsIndex Options
Suppose that:• Current value of index is 930, dividend yields
of 0.2% and 0.3% expected in first and second months
• European call option with exercise price of 900 expires in two months
• Risk-free rate is 8%, volatility is 20% p.a.
What is the price of the option?
Index Options Index Options
Using Black Scholes:• d1 = 0.5444, d2 = 0.4628
• N(d1) = 0.7069, N(d2) = 0.6782
• c = 930 x 0.7069e–0.03(2/12) – 900 x 0.6782 e–0.082/12
= $51.83
Portfolio InsurancePortfolio Insurance
P
A
Suppose the value of the index is S0 and the strike price is K• If a portfolio has a of 1.0, the portfolio
insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars held
• If is not 1.0, the portfolio manager buys put options for each 100S0 dollars held
K is chosen to give the appropriate insurance level
Portfolio InsurancePortfolio Insurance
Suppose that:• Portfolio has a beta of 1.0, worth $500,000• Index currently stands at 1000• Risk-free rate is 12%, dividend yield is 4%,
volatility is 22% p.a.• Option contract is 100 times the index
What trade is necessary to provide insurance against the portfolio value falling below $450,000 in the next three months?
Portfolio InsurancePortfolio Insurance
Using Black Scholes, p = $6.48• Cost of insurance = 5 x 100 x 6.48 = $3,240
If index drops to 880:• Portfolio drops to $440,000• Option payoff = 5 x (900–880) x 100 =
$10,000
Portfolio InsurancePortfolio Insurance
What if beta was 2.0?• Choose K = 960 (Table 13.2)• p = $19.21• Since beta is 2.0, two put contracts required for each
$100,000• Cost of insurance = 10 x 100 x 19.21 = $19,210
If index drops to 880:• Portfolio drops to $370,000• Option payoff = 10 x (960–880) x 100 = $80,000• Cost of hedging is higher (more put options, higher K)
Currency OptionsCurrency Options
Denote foreign interest rate by rf
When a U.S. company buys one unit of the foreign currency it has an investment of S0 dollars
Return from investing at the foreign rate is rf S0 dollars
Foreign currency provides a “dividend yield” at rate rf
Currency Option PricingCurrency Option Pricing
Lower Bound for Calls• c S0e–rf T –Ke –rT
Lower Bound for Puts• p Ke–rT – S0e–rf T
Put-Call Parity• p + S0e–rf T = c + Ke–rT
Black ScholesBlack Scholes
T
Tf
rrKSd
T
Tf
rrKSd
dNeSdNKep
dNKedNeScTrrT
rTTr
f
f
)2/2()/ln(
)2/2()/ln(
)()(
)()(
0
2
0
1
102
210
where
Black ScholesBlack Scholes
F S e r r Tf
0 0 ( )
Tdd
T
TKFd
dNFdKNep
dKNdNFecrT
rT
12
20
1
102
210
2/)/ln(
)]()([
)]()([
Futures OptionsFutures Options
Call futures option allows holder to acquire:• Long position in futures • Cash amount equal to excess of futures price over
strike price at previous settlement
Put futures option enables holder to acquire:• Short position in futures • Cash amount equal to excess of strike price over
futures price at previous settlement
PayoffsPayoffs
If futures position is closed out immediately:• Payoff from call = F0 – K
• Payoff from put = K – F0
where F0 is futures price at time of exercise
Advantages over Spot Advantages over Spot OptionsOptions
Futures contract may be easier to trade than underlying asset
Exercise of the option does not lead to delivery of the underlying asset
Futures options and futures usually trade in adjacent pits at exchange
Futures options may entail lower transaction costs
Put-Call ParityPut-Call Parity
Strategy I: buy a European call on a futures contract and invest Ke-rT of cash
FT ≤ K FT > K
Buy Call 0 FT – K
Invest Ke–rT K K
Total K FT
Put-Call ParityPut-Call Parity
Strategy II: buy a European put futures option, enter a long futures contract, and invest F0e-rT
FT ≤ K FT > K
Long Futures FT – F0 FT – F0
Buy Put K – FT 0
Invest F0e-rT F0 F0
Total K FT
Put-Call ParityPut-Call Parity
If two portfolios provide the same return, they must cost the same to set up, otherwise an opportunity for arbitrage exists
c + Ke-rT = p + F0e-rT
Binomial PricingBinomial Pricing
Suppose that:• 1-month call option on futures has a strike
price of 29• In one month the futures price will be either
$33 or $28Futures Price = $33Option Price = $4
Futures Price = $28Option Price = $0
Futures price = $30Option Price = ?
Binomial PricingBinomial Pricing
Consider a portfolio:• Long futures, short 1 call futures option
Portfolio is riskless when 3– 4 = – 2 = 0.8
3– 4
–2
Binomial PricingBinomial Pricing
Riskless portfolio:• Long 0.8 futures, short 1 call futures option
Value of the portfolio in one month:• 3 x0.8 – 4 = –1.6
Value of portfolio today (r = 6%):• –1.6e–0.061/12) = –1.592
Value of futures is zero, so value of option must be $1.592
GeneralisationGeneralisation
A derivative lasts for time T and is dependent on a futures contract
F0 u ƒu
F0 d ƒd
F0
ƒ
GeneralisationGeneralisation
Consider the portfolio that is long futures
and short 1 derivative
The portfolio is riskless when:
F0u F0 – ƒu
F0d F0– ƒd
ƒu df
F u F d0 0
GeneralisationGeneralisation
Value of portfolio at time T:• F0u – F0 – ƒu
Value of portfolio today:• (F0u – F0 – ƒu)e–rT
Cost of portfolio today: • –f
Hence ƒ = – [F0u – F0 – ƒu]e–rT
Dividend YieldDividend Yield
Valuing futures is similar to valuing an option on a stock paying a continuous 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
Dividend YieldDividend Yield
Futures contracts require no initial investment
In a risk-neutral world the expected return should be zero
Expected growth rate of futures price is thereforezero
Futures price can therefore be treated like a stock paying a dividend yield of r
Black’s ModelBlack’s Model
TdT
TKFd
T
TKFd
dNFdNKep
dNKdNFecrT
rT
10
2
01
102
210
2/2)/ln(
2/2)/ln(
)()(
)()(
where
Futures Price v Spot Futures Price v Spot PricePrice
European Options• If a European call (put) futures option matures
before futures contract, and futures prices exceed spot prices, it is worth more (less) than the corresponding spot option
• When futures prices are lower than spot prices (inverted market) the reverse is true
Futures Price v Spot Futures Price v Spot PricePrice
American Options• If futures prices are higher than spot prices,
an American call (put) on futures is worth more (less) than a similar American call (put) on spot
• When futures prices are lower than spot prices (inverted market) the reverse is true