foreign exchange option

In finance, a foreign exchange option (commonly shortened to just fx option) is a derivative security where the owner has the right but not the obligation to exchange money denominated in one currency into another currency at pre-agreed exchange rate on a specified date. For example a USD/GBP FX option might be specified by a contract allowing the purchaser to exchange 1,000,000 into $2,000,000 on December 31st. In this case the pre-agreed exchange rate, or strike price, is 2USD/GBP or 0.5GBP/USD and the notional is 1,000,000. This type of contract may be called either a dollar call or a sterling put depending on the market convention. If the dollar is stronger than 0.5GBP/USD come December 31st (say at 0.55GBP/USD) then the option will be exercised, making a profit of (2 - 1/0.55)*1,000,000 = $181,818 or 100,000.

Valuing FX options : The Garman-Kohlhagen model

As in the Black-Scholes model for stock options and the Black model for certain interest rate options, the value of a european option on an FX rate is typically calculated by assuming that the rate follows a log-normal process. In 1983 Garman and Kohlhagen extended the Black-Scholes model to cope with the presence of two interest rates (one for each currency). Suppose that r_d is the risk-free interest rate to expiry of the domestic currency and r_f is the foreign currency risk-free interest rate (where domestic currency is the currency in which we obtain the value of the option; the formula also requires that FX rates - both strike and current spot be quoted in terms of "units of foreign currency per unit of domestic currency"). Then the value of a call option into the foreign currency has value

exp(-r_f T) S N(d_1) - K exp(-r_d T) N(d_2)

where

S is the current spot rate

K is the strike rate

N is the cumulative normal distribution function

d_1 = \frac{log(S/K) + (r_d - r_f + \sigma^2/2)T}{\sigma^2T}

d_2 = d_1 - \sigma\sqrt{T}

and

\sigma

is the volatility of the FX rate.

Foreign Exchange Option

Valuing FX options : The Garman-Kohlhagen model

See also