Question

1. Assume that the spot price of the British pound is \$1.55, the 30-day annualized sterling interest rate is 10%, the 30-day annualized U.S. interest rate is 8.5%, and the annualized standard deviation of the dollar:pound exchange rate is 17%. Calculate the value of a 30-day PHLX call option on the pound at a strike price of \$1.57.
2. Suppose the spot price of the yen is \$0.0109, the threemonth annualized yen interest rate is 3%, the three-month annualized dollar rate is 6%, and the annualized standard deviation of the dollar:yen exchange rate is 13.5%. What is the value of a three-month PHLX call option on the Japanese yen at a strike price of \$0.0099/¥?
Option pricing stems from application of the most productive idea in all of finance-arbitrage. The idea underlying arbitrage pricing of a new asset is simple: Create a portfolio of assets with known market prices that exactly duplicates the distribution of payoffs of the new asset. The price of the new asset must equal the cost of purchasing the mimicking portfolio.
Otherwise, arbitrageurs would earn riskless profits. This is the technique used by Fischer Black and Myron Scholes in developing the Black-Scholes option pricing model.4 In order to develop a closed-form solution for the pricing of a currency option, we must make some assumptions about the statistical properties of the spot and forward exchange rates. Assuming that both these exchange rates are lognormally distributed (i.e., that their natural logarithm follows a normal distribution), one can duplicate the price of a European call option exactly, over a short time interval, with a portfolio of domestic and foreign bonds. This portfolio can be represented as

C(t) = aS(t)B*(t, T) + bB(t, T) .... (8A.1)

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