FX Option Pricer — Garman-Kohlhagen
Garman-Kohlhagen is the market-standard pricing model for European vanilla FX options — the two-rate extension of Black-Scholes. For a pair quoted as BASE/QUOTE (e.g. EUR/USD = 1.085), the spot S is in quote/base units, the premium is paid in the quote currency, rd is the quote (domestic) rate and rf the base (foreign) rate. The tool below computes the premium, the forward outright and all the Greeks live.
The model
The maturity T is measured from the spot date (T+2) to the maturity date — the standard FX convention — and the model returns the premium as the present value as of the spot date.
d₁ = [ ln(S/K) + (rd − rf + σ²/2)·T ] / (σ·√T)
d₂ = d₁ − σ·√T
Call = S·e^(−rf·T)·N(d₁) − K·e^(−rd·T)·N(d₂)
Put = K·e^(−rd·T)·N(−d₂) − S·e^(−rf·T)·N(−d₁)
Forward outright: F = S·e^((rd − rf)·T)N(·) is the standard normal cumulative distribution (Abramowitz-Stegun 26.2.17, error ~7.5·10⁻⁸). Put-call parity C − P = S·e^(−rf·T) − K·e^(−rd·T) holds exactly in the model.
The Greeks
- Delta =
e^(−rf·T)·N(d₁)(call) — sensitivity to spot; scaled by the foreign discount factor. - Gamma =
e^(−rf·T)·n(d₁) / (S·σ·√T)— identical for call and put. - Vega =
S·e^(−rf·T)·n(d₁)·√T— per 1.00 (i.e. 100%) change in vol; divide by 100 for a 1% vol move. - Theta — calendar-time decay (∂Premium/∂t), annualized. Typically negative for long positions.
- Rho (rd / rf) — sensitivity to two separate rates. In the EUR-call example, if the rd (USD) rate rises the forward goes up and the call gains value: rho_domestic positive, rho_foreign negative.
Conventions
| Parameter | Choice |
|---|---|
| Spot | T+2, Modified Following roll (weekends + TARGET/Fed holiday calendar) |
| Day count | ACT/365 |
| Rates | Continuously compounded |
| Premium payment | Spot date (standard) or maturity — if maturity is selected, the standard premium is carried forward by e^(rd·T) |
For theoretical depth on the subject, see Cross-Currency Option Hedging and Replication: EUR/TRY as a Worked Example — why an option on a cross rate is a covariance instrument and why static replication with vanilla legs is impossible.
For educational and indicative purposes only; not investment advice.