Options on stock indices, currencies, and futures

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Options on stock indices, currencies, and futures
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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

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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 SP 100 Index (OEX)
• The SP 500 Index (SPX)

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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

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Portfolio Insurance

• Consider a manager in charge of a
  well-diversified portfolio whose b 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

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Portfolio InsuranceExample

• Portfolio has a b 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?

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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
• 5880100 440,000
• the payoff from the options
• 5(900-880)100 10,000
• total value of the portfolio
• 440,00010,000 450,000

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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?

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When the portfolio beta is not 1.0Example

• Value of index in three months
  1040
• Return from change in index
  40/10004
• Dividends from index
  0.2541
• Total return from index
  415
• Risk-free interest rate
  0.25123
• Excess return from index
  5-32
• Expected excess return from portfolio
  224
• Expected return from portfolio
  347
• Dividends from portfolio
  0.2541
• Expected increase in value of portfolio
  7-16
• Expected value of portfolio
  500,0001.06 530,000

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Relationship between value of index and value of
portfolio for beta 2.0
The correct strike price for the 10 put option
contracts that are purchased is 960
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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

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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

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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

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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

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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

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Options On Stocks Paying Known Dividend
Yields

• Dividends cause stock prices to reduce on the
  ex-dividend date by the amount of the dividend
  payment
• The payment of a dividend yield at rate q
  therefore cause the growth 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 S0eqT today to ST

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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
• The stock starts at price S0 and provides a
  dividend yield at rate q
• The stock starts at price S0eqT and pays no
  dividends
• We can value European options by reducing the
  stock price to S0eqT and then behaving as
  though there is no dividend

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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

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Pricing Formulas

• By replacing S0 by S0eqT in Black-Sholes
  formulas ,we obtain that

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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

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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

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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

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Example

• A European call option on the SP 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

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Example continued

• The total dividend yield per annum is
• q (0.2 0.3)6 3
• one contract would cost 5,183

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Forward price

• Define F0 as the forward price of the index

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Implied dividend yields

• By

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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

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Using forward exchange rates

• Define F0 as the forward foreign exchange rate

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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

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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,K240(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)cents2,500
• Plus a long futures, if it closed out
  immediately
• 25,000X(251-250)cents250
• If the futures position is closed out immediately
• 25,000X (251-240)cents2,750

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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 KF0
• 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

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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

cKe-rTpF0 e-rT
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Example
European call option on spot silver for delivery
in six month
Use equation
cKe-rTpF0 e-rT
?PcKe-rT-F0 e-rT
P0.568.50e-0.1X0.5-8 e-0.1X0.5
1.04
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Bounds for futures options
cKe-rTpF0 e-rT
By
or
Similarly
or
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Bounds for futures options
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.
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Valuation by Binomial trees

• Consider the Portfolio long D
  futures short 1 call option
  
• Portfolio is riskless when 3D 4 -2D or
  D 0.8

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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
F0 ƒ
F0d ƒd
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A Generalization

• Consider the portfolio that is long ? futures
  and short 1 derivative
• The portfolio is riskless when

F0u D - F0 D ƒu
F0d D- F0D ƒd
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A Generalization

• Value of the portfolio at time T is F0u ? F0 ?
  ƒu
• Value of portfolio today is ƒ
• Hence ƒ F0u ? F0 ? ƒue-rT

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A Generalization

• Substituting for ? we obtain
• ƒ p ƒu (1 p )ƒd erT
• where

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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
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Blacks Formula

• The formulas for European options on futures are
  known as Blacks formulas

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Example

• European put futures option on crude oil,
• Fo20,K20, r0.09,T4/12,s0.25,ln(F0K)0,
• So that

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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

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