Enter Values

Call = buy foreign currency; Put = sell foreign currency
e.g., 1.10 = 1 EUR costs 1.10 USD
Exercise exchange rate
%
Continuously compounded annual rate
%
Continuously compounded annual rate
%
Annualized exchange rate volatility
Time until option expiration
Amount of foreign currency
Garman-Kohlhagen Formula
C = Se-rfTN(d1) - Ke-rdTN(d2)
S = Spot rate | K = Strike | rd = Domestic rate | rf = Foreign rate
Ryan O'Connell, CFA
Calculator by Ryan O'Connell, CFA

Option Premium

Call Premium (per unit) --
Total Premium --
Premium % of Spot --
Forward Rate (F) --
d1 --
d2 --
N(d1) --
N(d2) --

Option Greeks

Delta per unit foreign --
Gamma per unit foreign --
Theta per day --
Vega per 1% vol --

Formula Breakdown

C = Se-rfTN(d1) - Ke-rdTN(d2)
Garman-Kohlhagen pricing with dual interest rates

Moneyness Interpretation

Moneyness Condition (Call; reversed for Put) Description
Deep ITM F >> K High intrinsic value, delta near 1
ITM F > K Positive intrinsic value
ATM F ≈ K Maximum time value, delta near 0.5
OTM F < K No intrinsic value, all time value
Deep OTM F << K Low probability of expiring ITM

Moneyness is evaluated against the forward rate F = Se(rd-rf)T, not the spot rate.

Model Assumptions
  • Continuous compounding for both domestic and foreign rates
  • Lognormally distributed exchange rates
  • European exercise only (no early exercise)
  • No transaction costs or bid-ask spreads
  • Exchange rate quoted as domestic per unit foreign (e.g., USD/EUR = 1.10)

For educational purposes. Not financial advice. Market conventions simplified.

Understanding Currency Option Pricing

What is the Garman-Kohlhagen Model?

The Garman-Kohlhagen model (1983) extends the Black-Scholes framework to price European options on foreign currencies. While BSM uses a single risk-free rate, the GK model incorporates two interest rates — one for each currency — reflecting the fact that holding foreign currency earns the foreign risk-free rate.

Garman-Kohlhagen Equations (Hull Eqs. 17.11–17.12)
Call: C = Se-rfTN(d1) - Ke-rdTN(d2)
Put: P = Ke-rdTN(-d2) - Se-rfTN(-d1)
Where d1 = [ln(S/K) + (rd - rf + σ²/2)T] / (σ√T)

How Does It Differ from Black-Scholes?

Black-Scholes (Equities)

One risk-free rate (r)
Dividend yield (q) is optional. The underlying is a stock that may pay discrete or continuous dividends.

Garman-Kohlhagen (FX)

Two rates: rd (domestic) and rf (foreign)
The foreign rate replaces the dividend yield. Holding foreign currency earns rf continuously.

Interest Rate Parity and FX Options

The forward exchange rate is determined by interest rate parity: F = S × e(rd - rf)T. When domestic rates exceed foreign rates, the forward rate is above spot (domestic currency expected to depreciate). This rate differential is the key driver that distinguishes FX option pricing from equity option pricing.

Important: The Garman-Kohlhagen model applies to European options only. American-style currency options require numerical methods (binomial trees or finite differences) due to the possibility of early exercise.

When to Use This Calculator

Use the Currency Option Calculator when pricing FX options — it accounts for both domestic and foreign interest rates. A standard Black-Scholes calculator with a single risk-free rate would not correctly price currency options because it ignores the foreign interest rate that accrues on the underlying currency position.

Frequently Asked Questions

The Garman-Kohlhagen model extends Black-Scholes to price European currency options. It replaces the dividend yield with the foreign risk-free rate, accounting for the interest rate differential between two currencies. Published in 1983, it remains the standard model for FX option valuation and is widely used by banks and institutional traders for pricing over-the-counter currency options.

Currency options use two interest rates (domestic and foreign) instead of one. The foreign interest rate acts like a continuous dividend yield on the underlying currency. This dual-rate structure reflects interest rate parity — the forward exchange rate is driven by the rate differential between two countries. In contrast, equity options use a single risk-free rate with an optional discrete or continuous dividend yield.

Interest rate parity determines the forward exchange rate: F = S × e(rd - rf) × T. Higher domestic rates relative to foreign rates push the forward rate above spot, meaning the domestic currency is expected to depreciate. This forward rate is embedded in Garman-Kohlhagen option pricing through the rate differential, affecting both the option premium and the Greeks.

Delta measures sensitivity to spot rate changes and is adjusted by e-rfT to account for the foreign rate discount. Gamma measures the rate of delta change. Theta captures time decay per day (typically negative for long options, meaning the option loses value as expiry approaches). Vega shows premium sensitivity to a 1% change in FX volatility. All Greeks in the Garman-Kohlhagen model are modified from standard BSM to incorporate the foreign rate discount factor.

The key drivers are: (1) FX volatility — higher volatility increases both call and put premiums; (2) the interest rate differential between domestic and foreign rates, which affects forward rates and option values; (3) time to expiry — longer maturities increase premium through both time value and rate differential accumulation; and (4) moneyness, meaning how far the spot rate is from the strike price relative to the forward rate.

A call on EUR/USD gives the right to buy euros at the strike rate in USD. A put gives the right to sell euros at the strike rate. Due to the symmetry of currency pairs, a call on EUR/USD is equivalent to a put on USD/EUR — buying euros is the same as selling dollars. In the Garman-Kohlhagen model, both are priced using the same dual-rate framework with continuous compounding.
Disclaimer

This calculator is for educational purposes only and assumes European options with continuously compounded rates. Actual FX option pricing involves additional factors like volatility smiles, market microstructure, and counterparty risk. This tool should not be used for trading decisions.

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