Options for the sale or purchase of foreign currency have basically the same character as options on stocks and the methodology of Black and Scholes can be applied to the valuation of such options. Options on foreign currencies are not exactly the same as stock options. In foreign currency options there is an ambiguity about whether a particular option should be called a call or a put. A call is the option to buy a security at a specified price within a specified period of time. A put is the right to sell a security at a specified price within a specified period of time. When an American has the right to buy British pounds at a specified exchange rate in dollars this option would be called a call, but it can equally well be considered the right to sell British pounds for dollars. Likewise a foreign currency put can just as well be considered a call. This ambiguity actually exists in the matter of stock options. If someone has a call option to buy a share of Microsoft stock for $100 it could be considered a put option to sell $100 for one share of Microsoft stock.
A more essential difference between foreign currency options and stock options is that stock options may or may not pay a dividend and if a dividend is paid on a stock it is paid at a specific time. There is no payment corresponding to a dividend paid on foreign currency per se, but purchasers of foreign currency could invest their holdings in the foreign country's money market and earn interest. This interest on foreign money market holdings can be considered to accrue continuously. Thus foreign currency options correspond to options on stock that pay dividends continuously.
Robert C. Merton extended the Black-Scholes methodology to the valuation of options on a stock that pays dividends continuously and the valuation formulas are:
and N(z) is the area under the standard normal distribution function from -∞ to z
and
where
σ = standard deviation of the rate of return on holding the
security.
All parameters are presumed to be constant over the life of the option. Only the current security price S is stochastic.
Biger and Hull stretch this formula a bit to make it fit a foreign currency option by taking S to be the spot market value of one unit of the foreign currency; i.e. the direct quote exchange rate. The rate of accrual of dividends is taken to be r* the dollar value of the interest earned in the foreign money market. This is where the stretch comes in. Even if the foreign risk-free interest rate is presumed to be constant the dollar value of that interest fluctuates with the exchange rate S and has the same trend as S. While this is a bit of a stretch it is a reasonable assumption to get a handle on the problem.
There is nice specialization of the Black-Scholes type formula when the information available from the forward foreign currency exchange rate is taken into account. Let F be the direct quote exchange rate for the time of expiration of the option, T time units in the future. Interest rate parity requires that:
where r and r* are the domestic and foreign risk-free interest rates, and thus
When this latter relationship is incorporated into the formula for the value of European call the result is:
The corresponding formula for the valuation of a European put is:
It is notable that both the spot rate S and the foreign interest rate r* have disappeared from the formulas. The explanation is that given the relationship
For an implementation of the above formulas see Foreign Currency Option Value Computer.
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