Valuing FX options: The Garman-Kohlhagen model, Examples

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 an european option on a FX rate is typically calculated by assuming that the rate follows a log-normal process.

Examples

Suppose a United Kingdom manufacturing firm is expecting to be paid US$100,000 for a piece of engineering equipment to be delivered in 90 days. If the GBP strengthen against the US$ over the next 90 days the UK firm will lose money, as it will receive less GBP when the US$100,000 is converted into GBP. However, if the GBP weaken against the US$,then the UK firm will gain additional money. In this case, to protect the GBP value that the firm will receive in 90 day's time, the UK firm can purchase a GBP call/ USD put option (the right to sell part or all of their expected income for pounds sterling at a given rate near today's rate) to mitigate their risk of exchange rate fluctuation over the 90 days. Conversely another party may wish to have the reverse option for a similar reason. A market maker will buy and sell these options with the aim of making a profit while not incurring too much risk.

The advantage of using an option instead is that it gives unlimited profit potential to the buyer at a limited cost (this cost is known as option premium).

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\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\exp(-r_f T)\N(-d_1)

where :

d_1 = \frac{\ln(S/K) + (r_d - r_f + \sigma^2/2)T}{\sigma\sqrt{T}}
d_2 = d_1 - \sigma\sqrt{T}

S is the current spot rate
K is the strike rate
N is the cumulative normal distribution function
rd is domestic risk free rate
rf is foreign risk free rate
and σ is the volatility of the FX rate.