Currency Option Pricing Calculator
Price a call option on a currency pair using the Garman-Kohlhagen model. Essential for FX traders and treasurers valuing European-style currency options with foreign interest rate adjustments.
About this calculator
The Garman-Kohlhagen model extends the Black-Scholes framework to currency options by treating the foreign interest rate as a continuous dividend yield. The call price formula is: C = S·e^(−rf·T)·N(d1) − K·e^(−r·T)·N(d2), where d1 = [ln(S/K) + (r − rf + σ²/2)·T] / (σ·√T) and d2 = d1 − σ·√T. Here S is the spot rate, K is the strike price, T is time to expiry in years, r is the domestic risk-free rate, rf is the foreign risk-free rate, σ is implied volatility, and N(·) is the cumulative standard normal distribution. The foreign rate discounts the spot price because holding foreign currency earns the foreign risk-free return. Higher volatility or longer expiry increases the option premium, while a higher foreign rate reduces it.
How to use
Suppose EUR/USD spot = 1.0800, strike = 1.0900, expiry = 90 days, domestic (USD) rate = 5%, foreign (EUR) rate = 3.5%, implied volatility = 8%. T = 90/365 ≈ 0.2466. d1 = [ln(1.08/1.09) + (0.05 − 0.035 + 0.0032)·0.2466] / (0.08·0.4966) ≈ (−0.00922 + 0.00449) / 0.03973 ≈ −0.119. d2 = −0.119 − 0.040 ≈ −0.159. N(d1) ≈ 0.453, N(d2) ≈ 0.437. C = 1.08·e^(−0.00864)·0.453 − 1.09·e^(−0.01233)·0.437 ≈ 0.4881 − 0.4712 ≈ $0.0169 per unit of base currency.
Frequently asked questions
How does the Garman-Kohlhagen model differ from the standard Black-Scholes model for currency options?
The key difference is that Garman-Kohlhagen incorporates two interest rates: a domestic rate and a foreign rate, whereas Black-Scholes uses a single risk-free rate. The foreign interest rate acts like a continuous dividend yield, reducing the effective cost of carrying the spot position. This adjustment is necessary because holding foreign currency earns the foreign risk-free return, which lowers the fair value of a call option compared to a plain equity option. Ignoring the foreign rate would systematically misprice currency options.
What does implied volatility mean in the context of currency option pricing?
Implied volatility (IV) is the market's consensus forecast of how much the currency pair will fluctuate over the option's life, expressed as an annualised standard deviation. Rather than being calculated from historical price moves, IV is backed out from observed market option premiums using the Garman-Kohlhagen formula. A higher IV means the market expects larger exchange-rate swings, which increases the option premium for both calls and puts. Traders monitor IV surfaces (plotting IV against strike and expiry) to identify cheap or expensive options relative to their own volatility forecasts.
When should a corporate treasurer use a currency call option instead of a forward contract to hedge FX exposure?
A call option is preferable when the treasurer wants protection against adverse currency moves but still wishes to benefit if the rate moves favourably — something a forward contract prevents. For example, a US importer paying in euros might buy a EUR call to cap their cost while still profiting if the euro weakens. Options are also useful when the foreign-currency cash flow is uncertain in size or timing, since an unexercised option carries no obligation. The trade-off is the upfront premium cost, which must be weighed against the value of retaining upside participation.