Garman-Kohlhagen FX Option Model

A stock pays no interest just for existing (dividends aside), so Black-Scholes only has to discount the option's payoff at one rate, $r$. A currency is different: holding euros earns the euro interest rate, and holding dollars earns the dollar interest rate, simultaneously. An option to exchange one for the other has to account for both rates at once — the interest you're implicitly earning by holding the currency you own, and the interest you're implicitly forgoing by not holding the one you'd receive.

Two savings accounts, not one

Imagine you're comparing two savings accounts in two different currencies, and someone offers you the right to swap balances between them on a future date. What that right is worth depends on how much interest each account pays in the meantime — not just the exchange rate between them. Garman and Kohlhagen simply took Black-Scholes and gave the "dividend yield" slot a real, permanent job: the foreign interest rate, $r_f$, standing in for the interest the euro side keeps earning while you hold the option instead of the currency itself.

$$C = S\,e^{-r_f T}\,N(d_1) - K\,e^{-r_d T}\,N(d_2), \qquad d_1 = \frac{\ln(S/K) + (r_d - r_f + \tfrac12\sigma^2)T}{\sigma\sqrt{T}}$$

In plain English: $S$ is the spot exchange rate (say, dollars per euro), $r_d$ is the domestic (dollar) rate, $r_f$ is the foreign (euro) rate, and everything else is ordinary Black-Scholes with $r_f$ doing the job a dividend yield $q$ would do — discounting the "asset" leg because holding the option instead of the actual euros forgoes euro interest.

Worked example 1: pricing a EUR/USD call. $S=1.10$ ($1.10 per €1), $K=1.10$, $r_d=5\%$ (USD rate), $r_f=2\%$ (EUR rate), $\sigma=10\%$, $T=0.5$. Then $\ln(S/K)=0$, and $(r_d-r_f+\tfrac12\sigma^2)T = (0.05-0.02+0.005)\times0.5 = 0.0175$. With $\sigma\sqrt{T}=0.10\times0.7071=0.0707$, $d_1=0.0175/0.0707=0.2475$, $d_2=0.2475-0.0707=0.1768$. Using $N(0.2475)\approx0.5977$ and $N(0.1768)\approx0.5702$: $C = 1.10\,e^{-0.01}(0.5977) - 1.10\,e^{-0.025}(0.5702) = 1.10(0.99)(0.5977) - 1.10(0.9753)(0.5702) = 0.6511 - 0.6119 = 0.0392$, i.e. $0.0392 per euro of notional.

Worked example 2: checking it with put-call parity. For FX, Put-Call Parity reads $C - P = S e^{-r_f T} - K e^{-r_d T}$. Plugging in the same numbers: $S e^{-r_f T} = 1.10 \times 0.9753 = 1.0728$, and $K e^{-r_d T} = 1.10 \times 0.99 = 1.089$, so $C-P = 1.0728 - 1.089 = -0.0162$. With $C=0.0392$ from above, the matching put must be $P = 0.0392 + 0.0162 = 0.0554$, i.e. $0.0554. Notice the put is worth more than the call here even though both are struck at the money — because the dollar rate exceeds the euro rate, the market expects the euro to drift down against the dollar under the risk-neutral measure, tilting value toward the put.

What this means in practice

Every FX option desk quotes in delta and volatility rather than strike, but underneath, every one of those quotes runs through this two-rate discounting. When a central bank moves rates, FX option prices shift even if spot and volatility don't, purely through $r_d$ or $r_f$ changing — a risk that's easy to overlook if you think of an FX option as "just Black-Scholes on an exchange rate."