What is the Garman-Kohlhagen model?

Garman-Kohlhagen (1983) extends Black-Scholes to currency options by treating the foreign currency as an asset paying a continuous dividend equal to the foreign interest rate. The pricing formula uses two rates: the domestic (quote currency) rate r_d and the foreign (base currency) rate r_f.

How does FX option pricing differ from stock options?

FX options have two interest rates — one for each currency. The cost of carry is b = r_d − r_f. If r_f > r_d (e.g., AUD interest > USD interest), the forward rate is below spot, and calls are cheaper relative to puts. This 'interest rate differential' is essential in FX option pricing and cannot be ignored.

What is the quote convention for USD/IDR or EUR/USD?

In FX, the pair is written BASE/QUOTE. A quote of USD/IDR = 16,500 means 1 USD costs 16,500 IDR. The 'domestic' currency in the Garman-Kohlhagen model is the QUOTE currency (IDR here), so r_d = IDR rate. The 'foreign' currency is the BASE (USD), so r_f = USD rate. A call on USD/IDR is the right to BUY USD (pay IDR) at the strike.

Why do traders quote FX vol in Delta instead of strike?

FX option markets quote volatility surfaces by Delta (25D put, 25D call, ATM) rather than absolute strike, because Delta is constant across maturities while strikes scale with spot. A 25-delta call has similar moneyness whether EUR/USD is 1.05 or 1.20. This calculator uses absolute strikes — for delta-quoted vol, invert via the delta-to-strike formula separately.

Are FX options usually American or European?

Interbank FX options are overwhelmingly European. Retail brokers sometimes offer American FX options. This calculator prices European options exclusively — which matches ~99% of institutional FX option volume.

FX Options Calculator — Garman-Kohlhagen Currency Options

The Garman-Kohlhagen Model

Mark Garman and Steven Kohlhagen published their FX option pricing model in 1983, generalising Black-Scholes to currency options by treating the foreign currency as a continuously dividend-paying asset with yield equal to the foreign interest rate r_f. The formula:

Call = S·e^−r_f·T·N(d₁) − X·e^−r_d·T·N(d₂)
Put = X·e^−r_d·T·N(−d₂) − S·e^−r_f·T·N(−d₁)

d₁ = [ln(S/X) + (r_d − r_f + σ²/2)T] / (σ√T), d₂ = d₁ − σ√T

In the generalised BSM framework, this is simply the cost-of-carry form with b = r_d − r_f. The interest rate differential IS the cost of carry in FX.

Quote Convention — Base/Quote

Every FX pair is written BASE/QUOTE. The spot price tells you how many units of QUOTE currency equal 1 unit of BASE currency:

EUR/USD = 1.0850 → 1 EUR = 1.0850 USD. Base = EUR, Quote = USD.
USD/JPY = 150.00 → 1 USD = 150 JPY. Base = USD, Quote = JPY.
USD/IDR = 16,500 → 1 USD = 16,500 IDR. Base = USD, Quote = IDR.

In Garman-Kohlhagen, "domestic" always means the quote currency, and "foreign" always means the base currency. So for EUR/USD: r_d = USD rate (e.g., Fed funds), r_f = EUR rate (e.g., ECB depo). Getting this inverted is the #1 mistake in FX options.

What Does "Call on EUR/USD" Mean?

A call on the base currency (EUR in EUR/USD) is the right to BUY the base and PAY the quote at strike. If EUR/USD is 1.08 spot and you own a 1.10 call, you profit if EUR/USD rises above 1.10 at expiry.

A put on the base is the right to SELL the base and RECEIVE the quote at strike. Critically:

A call on EUR/USD is exactly equivalent to a put on USD/EUR (with inverted strike). This is why interbank traders always agree which currency is base first.

Interest Rate Parity & the Forward

The forward rate is pinned by no-arbitrage to the interest rate differential:

F = S · e^{(r_d − r_f)·T}

If the base currency has a higher rate than the quote (e.g., AUD vs USD historically), then r_f > r_d, so F < S — the base currency trades at a forward discount. Long-dated AUD/USD calls are therefore cheaper relative to puts, all else equal. This is the rate differential effect on option pricing — ignoring it is a classic pricing error.

Greeks in FX Context

Delta — traded in FX as "% of base notional". A 25-delta call on €10M EUR/USD has a spot delta of 25% × 10M = €2.5M equivalent.

Gamma — similarly normalised. High gamma means the delta hedge needs frequent re-balancing.

Vega — biggest risk in FX options given vol surfaces are actively traded. Typical G10 1-month ATM vega is 0.03-0.05 per 1% vol move on a notional of 1.

Theta — time decay is faster in FX than equity because FX vols are usually lower → shorter effective duration.

Rho_d and Rho_f — unlike equity, FX options have TWO rate sensitivities. Rho_d = dPrice/dr_d (quote rate). Rho_f = dPrice/dr_f (base rate). For a call, Rho_d is positive and Rho_f is negative.

Worked Example — EUR/USD 1-month ATM Call

EUR/USD = 1.0850, 1-month ATM call (X = 1.0850), USD rate 5.25%, EUR rate 3.50%, vol 7%.

Premium ≈ 0.0085 USD per 1 EUR (0.78% of spot)
Delta ≈ 0.54 (slightly above 0.5 because forward is above spot)
Vega ≈ 0.0012 per 1% vol — a vol move from 7% → 8% adds ~0.00012 USD/EUR
Theta ≈ −0.00015 per day — premium bleeds 15 pips/day

For €10M notional, premium ≈ $85,000, vega ≈ $12,000 per vol point, daily theta ≈ −$1,500.

Emerging Market FX Options (IDR, TRY, ZAR)

EM FX options have higher vols (10–30%+) and larger rate differentials, which skew forwards dramatically:

USD/IDR: IDR rates ~6%, USD rates ~5% → modest forward premium; vol 6–12%
USD/TRY: TRY rates 40%+, USD rates 5% → massive forward premium → TRY puts expensive
USD/ZAR: ZAR rates ~8%, USD rates 5% → moderate premium; vol 14–18%

Always verify rates against current interbank curves before using this model in live trading.

Model Limitations

Constant vol — real FX surfaces have smile and skew. Use market-quoted vol for the specific delta/strike you're pricing.
No counterparty / NDF risk — for restricted currencies (CNY, BRL, INR), NDF quirks matter.
Constant rates — for long-dated FX options, stochastic rates matter; use HJM or LMM.
Continuous hedging — FX markets trade 24/5 but still have gaps at weekly open.

For FX exposure at portfolio level, see VaRisk Kancil's multi-currency VaR engine.

Related Tools

Black-Scholes (Stocks) — options on equities and indices
Bond YTM — underlying rate curves for FX discount factors
Compound Interest — discount factor intuition

FX Options Calculator

FX Option Inputs

Implied Volatility

3D Greek Surface