Kamis, 06 Agustus 2009

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 rd is the risk-free interest rate to expiry of the domestic currency and rf 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"). Then the domestic currency value of a call option into the foreign currency is

$c = S_0\exp(-r_f T)\N(d_1) - K\exp(-r_d T)\N(d_2)$

The value of a put option has value

$p = K\exp(-r_d T)\N(-d_2) - S_0\exp(-r_f T)\N(-d_1)$

where :

$d_1 = \frac{\ln(S_0/K) + (r_d - r_f + \sigma^2/2)T}{\sigma\sqrt{T}}$
$d_2 = d_1 - \sigma\sqrt{T}$
S0 is the current spot rate
K is the strike price
N is the cumulative normal distribution function
rd is domestic risk free simple interest rate
rf is foreign risk free simple interest rate
T is the time to maturity (calculated according to the appropriate day count convention)
and σ is the volatility of the FX rate.