In finance, a foreign exchange option (commonly shortened to just FX option) is a derivative where the owner has the right but not the obligation to exchange money denominated in one currency into another currency at a 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

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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

$c\; =\; S\; \backslash exp(-r\_f\; T)\; N(d\_1)\; -\; K\; \backslash exp(-r\_d\; T)\; N(d\_2)$

The value of a put option has value

$p\; =\; K\; \backslash exp(-r\_d\; T)\; N(-d\_2)\; -\; S\; \backslash exp(-r\_f\; T)\; N(-d\_1)$

where

S is the current spot rate

K is the strike rate

N is the cumulative normal distribution function

$d\_1\; =\; \backslash frac\{\backslash ln(S/K)\; +\; (r\_d\; -\; r\_f\; +\; \backslash sigma^2/2)T\}\{\backslash sigma\backslash sqrt\{T\}\}$

$d\_2\; =\; d\_1\; -\; \backslash sigma\backslash sqrt\{T\}$

and $\backslash sigma$ is the volatility of the FX rate.

See also

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• Put option
• Call option
• Foreign exchange market

Options | FX опцион