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In finance, aforeign exchange option (commonly shortened to justFX option orcurrency option) is aderivative financial instrument that gives the right but not the obligation to exchange money denominated in onecurrency into another currency at a pre-agreedexchange rate on a specified date.^[1] SeeForeign exchange derivative.^[2]

As in theBlack–Scholes model forstock options and theBlack model for certaininterest rate options, the value of aEuropean option on an FX rate is typically calculated by assuming that the rate follows alog-normal process.^[3]

The earliest currency options pricing model was published by Biger and Hull, (Financial Management, spring 1983). The model preceded the Garman and Kolhagen's Model. 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${\displaystyle r_d}$ is therisk-free interest rate to expiry of the domestic currency and${\displaystyle r_f}$ 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 domestic currency per unit of foreign currency"). The results are also in the same units and to be meaningful need to be converted into one of the currencies.^[4]

Then the domestic currency value of a call option into the foreign currency is

${\displaystyle c=S_0{e}^{-r_{f}T}\mathcal{N}(d_1)-K{e}^{-r_{d}T}\mathcal{N}(d_2)}$

The value of a put option has value

_{${\displaystyle p=K{e}^{-r_{d}T}\mathcal{N}(-d_2)-S_0{e}^{-r_{f}T}\mathcal{N}(-d_1)}$}

where :

${\displaystyle d_1=\frac{\ln (S_0/K)+(r_d-r_f+{\sigma }^2/2)T}{\sigma \sqrt{T}}}$

${\displaystyle d_2=d_1-\sigma \sqrt{T}}$

${\displaystyle S_0}$ is the current spot rate

${\displaystyle K}$ is the strike price

${\displaystyle \mathcal{N}(x)}$ is the cumulative normal distribution function

${\displaystyle r_d}$ is domestic risk freesimple interest rate

${\displaystyle r_f}$ is foreign risk free simple interest rate

${\displaystyle T}$ is the time to maturity (calculated according to the appropriateday count convention)

and${\displaystyle \sigma }$ is thevolatility of the FX rate.

• Foreign exchange market
• Options (finance)
• Derivatives (finance)

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