1. Assume that the spot price of the British pound is $1.55, the 30-day annualized sterling interest...

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)

Strike Price
In finance, the strike price of an option is the fixed price at which the owner of the option can buy, or sell, the underlying security or commodity. Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most... Exchange Rate
The value of one currency for the purpose of conversion to another. Exchange Rate means on any day, for purposes of determining the Dollar Equivalent of any currency other than Dollars, the rate at which such currency may be exchanged into Dollars... Portfolio
A portfolio is a grouping of financial assets such as stocks, bonds, commodities, currencies and cash equivalents, as well as their fund counterparts, including mutual, exchange-traded and closed funds. A portfolio can also consist of non-publicly...