Foreign exchange option#Valuation: 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.{{cite web|title=British Pound (GBP) to Euro (EUR) exchange rate history|url=http://www.exchangerates.org.uk/GBP-EUR-exchange-rate-history.html|website=www.exchangerates.org.uk|access-date=21 September 2016}}

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 $r\_d$ 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 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.{{cite web|title=Currency options pricing explained|url=http://www.derivativepricing.com/blog/bid/59633/Currency-options-pricing-explained|website=www.derivativepricing.com|access-date=21 September 2016}}

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

:$c\; =\; S\_0e^\{-r\_f\; T\}\backslash mathcal\{N\}(d\_1)\; -\; Ke^\{-r\_d\; T\}\backslash mathcal\{N\}(d\_2)$

The value of a put option has value

:_{$p\; =\; Ke^\{-r\_d\; T\}\backslash mathcal\{N\}(-d\_2)\; -\; S\_0e^\{-r\_f\; T\}\backslash mathcal\{N\}(-d\_1)$}

where :

:$d\_1\; =\; \backslash frac\{\backslash ln(S\_0/K)\; +\; (r\_d\; -\; r\_f\; +\; \backslash sigma^2/2)T\}\{\backslash sigma\backslash sqrt\{T\}\}$

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

:$S\_0$ is the current spot rate

:$K$ is the strike price

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

:$r\_d$ is domestic risk free simple interest rate

:$r\_f$ is foreign risk free simple interest rate

:$T$ is the time to maturity (calculated according to the appropriate day count convention)

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

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Category:Options (finance)

Category:Derivatives (finance)