In the interbank foreign exchange market, options are not quoted with prices. They are quoted indirectly with implied volatilities. The convention for converting volatilities to prices is the Garman and Kohlhagen (1983) option pricing formula. Mathematically, the formula is identical to Merton's (1973) formula for options on dividend-paying stocks. Only the term q, which did represent a stock's dividend yield, now represents the foreign currency's continuously compounded risk-free rate. Like the Merton formula, the Garman and Kohlhagen formula applies only to European options. Generally, OTC currency options are European.
Values for a call price c or put price p are:
Here, log denotes the natural logarithm, and:
s = the current exchange rate (domestic currency per unit of foreign currency)
x = the strike exchange rate
r = the continuously compounded domestic risk free interest rate
q = the continuously compounded foreign risk free interest rate
t = the time in years until the expiration of the option
σ = the implied volatility for the underlying exchange rate
Φ = the standard normal cumulative distribution function.
Consider a put GBP call USD option on GBP10MM for which we want a USD value. The option is struck at 1.65 USD/GBP and expires in .09 years. The current exchange rate is 1.62 USD/GBP. Assume 18% (that is .18) implied volatility. USD and GBP continuously compounded risk free interest rates are .0294 and .0327. Applying formula , the option has market value USD .0524 per GBP. Based upon the notional amount of GBP 10MM, that becomes USD 0.524MM. Because the option is out-of-the-money, that value is entirely time value.
Because their prices are affected by two—one domestic and the other foreign—risk free rates, currency options have two rho sensitivities. All the Greeks—delta, gamma, vega, theta, domestic rho and foreign rho—for a call are:
where denotes the standard normal probability density function. For a put, the Greeks are: