CURRENCY OPTIONS

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CURRENCY OPTIONS

• Currency Options give the buyer the opportunity,
  but not the obligation, to buy or sell an asset
  in the future at a pre-agreed price
• A buyer of an option can either exercise it or
  let it expire
• The specified price in the contract is called the
  strike or exercise price
• The party that sells the option contract is
  called the writer

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• Maturity date on the contract is called
  expiration date
• Exchange-traded options vs. OTC options
• A call option gives the buyer the right, but not
  the obligation, to buy the underlying financial
  asset or commodity
• A put option gives the buyer the right, but not
  the obligation, to sell the underlying financial
  asset or commodity

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• Two types of options depending on when they can
  be exercised European and American
• A buyer would pay no more for a European option
  than for an American option
• For foreign currencies, there are options on spot
  and options on futures
• An option can be in the money, out of the money,
  or at the money

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• Price of option is called option premium
• Option premium intrinsic value time value
• Intrinsic value spot rate - strike price
• Time value present value of expected payouts
  from the option, given that the option is
  exercised
• Contract sizes at the Philadelphia Exchange
  where only spot options are traded, sizes are
  half the size of corresponding futures contract

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• At the CME, options on futures are traded and
  sizes are corresponding futures sizes, except for
  
• E.g. American Call option on spot DM right to
  buy DM 1 million for 0.63 per DM from today
  until expiration on June 15
• E.g. European Put option on SFr futures right to
  sell SFr 10 million March 2001 futures for 0.76
  per SFr only on March 15

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• E.g. How much would a buyer pay for spot DM
  option with premium 0.26 cents?
• The buyer pays
• 0.0026/DM ? DM 62,500 162.50
• E.g. What is the intrinsic value of the 93 March
  spot Yen option if the spot rate today is
  0.009597?
• Intrinsic value 0.009597 - 0.009300 0.0279

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DETERMINANTS OF OPTION PRICES

• The following factors determine the price of an
  option
• ? Intrinsic value ()
• ? Volatility of the spot or futures rate ()
• ? Time to expiration ()
• ? Interest rate on currency of purchase (same as
  lower present value of exercise price positive
  for call, negative for put)

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• ? European or American option (ceteris paribus
  American option has higher value)
• ? Interest-rate differential (positive for put
  negative for call)
• Explanation higher interest rate on a currency
  implies expectation for depreciation
• E.g. higher rates in Japan imply yen depreciates
  against
• Initial spot rate 0.009/ new spot rate is
  0.0085/

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• A put option has higher value a call option has
  lower value

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UNDERSTANDING OPTION PRICING

• The price of a call or put option can be derived
  from a no-arbitrage condition
• For pricing a European call or put option, we can
  construct a replicating portfolio with the exact
  same payoff
• E.g. In the case of a European call option on FF,
  the option price must be equal to the initial
  value of the replicating portfolio

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• In this case, the portfolio will include
  borrowing and investing in FF
• E.g. Suppose in period 1 the spot rate is 1.4/
• Next period, the spot rate will be either
  1.5015/ or 1.2293/
• A European call option on spot with strike
  price 1.40/ will be worth 0.1015 if S? and
  zero if S?
• Suppose that rate is 9 and rate is 12

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• Consider the following borrowing and lending
  strategy
• ? Borrow 0.41903 in period 1 at the spot rate
• ? In period 2, if S?, we must repay 0.45849 if
  S? we must repay 0.45849
• ? In period 1, buy 0.330785 at the spot rate
  and deposit
• ? In period 2, we receive 0.56 if S? and
  0.45849 if S?

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• The net cash flows in period 2 from borrowing in
  and lending in are
• ? 0.10151 if S increases
• ? 0 if S decreases
• Notice that these are the same as in the case of
  the European call option on
• We can summarize the cash flows in a table

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• Through financial arbitrage, the price of the
  call option must be equal to the price of the
  replicating portfolio
• In the above example, the price of the option
  must be 0.04407/
• In a similar way, the price of a put option can
  be derived through a combination of lending and
  borrowing

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THE ASSUMPTIONS OF THE BLACK-SCHOLES MODEL

• The Black-Scholes model for pricing a European
  call option on a non-dividend-paying stock makes
  the following assumptions
• The rate of return on the stock follows a
  lognormal distribution. This means that the
  logarithm of 1 plus the return follows the normal
  distribution. The lognormal distribution is a
  convenient and realistic characterization of
  stock returns.

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• The risk-free rate and variance of the return on
  the stock are constant throughout the options
  life.
• There are no taxes or transaction costs.
• The stock pays no dividends.
• There are no riskless arbitrage opportunities.
• Investors can borrow or lend at the same
  risk-free rate of interest.

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THE BLACK-SCHOLES OPTION PRICING FORMULA

• The Black-Scholes formula is
• C SN(d1) - Ee-rTN(d2)
• where
• d1 ln(S/E) (r ?2/2)T/?(T)1/2
• d2 d1 - ?(T)1/2
• N(d1), N(d2) cumulative normal probabilities
• ?2 annualized variance of the continuously
  compounded return on the stock

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• r continuously compounded risk-free rate
• S stock price today
• E exercise price
• T time to expiration
• C price of a call option

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THE GARMAN-KOHLHAGEN MODEL

• The GK model is the best-known model for pricing
  foreign currency options
• The GK model is an extension of the Black-Scholes
  model to foreign currencies
• The GK model makes the following assumptions
• ? Currency fluctuations are best described by
  lognormal probability distribution (logs of
  percentage changes are normally distributed)

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• ? Interest rates for the two currencies are
  constant during the life of the option
• ? There are no transaction costs in trading
  options
• The GK model derives the following formula for
  pricing a European call option
• C Se-?TN(d1) - Ee-rTN(d2)

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• where
• d1 ln(Se-?T/E) r (?2/2)T/?(T)1/2
• d2 d1 - ?(T)1/2
• and every variable is defined in the same way as
  in the Black-Scholes model, except for ?, which
  symbolizes the foreign interest rate, ? which
  shows the standard deviation of the log of one
  plus the percentage change in the exchange rate
  and S, which is the spot exchange rate expressed
  in dollars per unit of foreign currency.

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EXAMPLE OF PRICING A EUROPEAN CALL OPTION WITH
THE GK MODEL

• Suppose that you are given the following
  information for a British pound 145 European
  call
• ? S 143.53
• ? E 145
• ? r .0532
• ? ? .0949
• ? ?2 .0225
• ? T .1342

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• To calculate the call price, we proceed as
  follows
• ? Compute d1
• d1 ln(143.53e-(.0949)(.1342)/145) (.0532
  (.0225)/2) .1342 / .15 ? (.1342)1/2 - 0.26
• ? Compute d2
• d2 - 0.26 - 0.15 ? (.1342)1/2 - 0.31
• ? Look up N(d1) from the table
• N (-0.26) 1 - 0.6026 0.3974

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• ? Look up N(d2) from the table
• N (-0.31) 1 - 0.6217 0.3783
• ? Plug into formula for C
• C 143.53e-(.0949)(.1342)(.3974) -
  145e-(.0532)(.1342)(.3783) 1.85