Currency Options & the Garman-Kohlhagen Model

Published: June 25, 2026
Ryan O'Connell, CFA, FRM
Article by Ryan O'Connell, CFA, FRM
Table of Contents

    Currency options give investors the right to buy or sell one currency for another at a predetermined exchange rate. Unlike equity options, pricing currency options requires accounting for two interest rates — the domestic and foreign risk-free rates. The Garman-Kohlhagen model extends Black-Scholes-Merton to handle this two-rate environment, making it the standard framework for valuing European-style currency options.

    What Are Currency Options?

    A currency option gives the holder the right, but not the obligation, to exchange one currency for another at a specified exchange rate (the strike price) on or before a specified date.

    Key Concept

    A call option on a currency gives the right to buy the foreign currency. A put option gives the right to sell the foreign currency. The exchange rate is quoted as domestic currency per unit of foreign currency (e.g., USD per EUR).

    Currency options trade primarily in the over-the-counter (OTC) market, where banks and institutions customize strike prices, expiration dates, and notional amounts to match specific hedging needs. Exchange-traded currency options exist on venues like Nasdaq PHLX and CME (as options on currency futures), but the OTC market handles the vast majority of volume.

    Common uses include:

    • Hedging FX exposure — A U.S. company expecting to receive euros can buy EUR puts to protect against euro depreciation
    • Speculation — Taking directional views on currency movements with limited downside
    • Range forwards — Combining calls and puts to create zero-cost hedging structures

    The Garman-Kohlhagen Formula for Currency Option Pricing

    The Garman-Kohlhagen (GK) model, published in 1983, adapts Black-Scholes-Merton for currency options. The key insight: a foreign currency can be treated like a stock that pays a continuous dividend yield equal to the foreign risk-free interest rate.

    Garman-Kohlhagen Call Option
    c = S0 × e-rfT × N(d1) − K × e-rT × N(d2)
    Value of a European call option on foreign currency
    Garman-Kohlhagen Put Option
    p = K × e-rT × N(-d2) − S0 × e-rfT × N(-d1)
    Value of a European put option on foreign currency
    d1 and d2 Calculations
    d1 = [ln(S0/K) + (r − rf + σ²/2) × T] / (σ × √T)
    d2 = d1 − σ × √T
    Intermediate terms used in the GK formulas

    Where:

    • S0 — Spot exchange rate (domestic currency per unit of foreign currency)
    • K — Strike price (in the same quote convention as S0)
    • r — Domestic risk-free interest rate (continuously compounded)
    • rf — Foreign risk-free interest rate (continuously compounded)
    • σ — Volatility of the exchange rate
    • T — Time to expiration (in years)
    • N(·) — Cumulative standard normal distribution function
    Pro Tip

    The GK model assumes European exercise (exercise only at expiration), continuous compounding of interest rates, and constant volatility. If you have rates quoted with annual compounding, convert them: rcontinuous = ln(1 + rannual).

    Understanding Quote Conventions

    For EUR/USD quoted as “1.0800 USD per EUR,” the domestic currency is USD and the foreign currency is EUR. If you invert the quote to EUR per USD, you must also swap which rate is domestic and which is foreign. Always ensure S0, K, r, and rf are consistent with your quote convention.

    Connection to Interest Rate Parity

    The forward exchange rate is directly linked to the interest rate differential:

    Forward Exchange Rate
    F0 = S0 × e(r − rf)T
    Covered interest rate parity relationship

    This explains why the foreign rate enters the GK model: the forward rate already incorporates the interest rate differential, and option pricing must be consistent with no-arbitrage forward pricing.

    What Affects Currency Option Prices

    Understanding how each input affects option values helps traders and hedgers anticipate price movements:

    Input Call Value Put Value Intuition
    Spot rate (S0) ↑ ↑ Increases ↓ Decreases Higher spot makes calls more valuable, puts less valuable
    Domestic rate (r) ↑ ↑ Increases ↓ Decreases Higher domestic rate reduces PV of strike (benefits calls)
    Foreign rate (rf) ↑ ↓ Decreases ↑ Increases Higher foreign rate reduces PV of receiving foreign currency
    Volatility (σ) ↑ ↑ Increases ↑ Increases More uncertainty benefits option holders (limited downside)
    Time to expiration (T) ↑ ↑ Increases* ↑ Increases* More time = more potential movement (*usually)
    Strike price (K) ↑ ↓ Decreases ↑ Increases Higher strike = call further OTM, put further ITM
    Pro Tip

    The foreign interest rate has the opposite effect of the domestic rate. Think of it this way: holding foreign currency earns the foreign rate, so a higher foreign rate makes the foreign currency more attractive to hold — reducing call values (you’re giving up that yield) and increasing put values.

    Currency Option Example

    Let’s price a 3-month European call option on EUR/USD using the Garman-Kohlhagen model.

    EUR/USD Call Option Calculation

    Given:

    • Spot rate (S0): 1.0800 USD per EUR
    • Strike price (K): 1.1000 USD per EUR
    • Time to expiration (T): 0.25 years (3 months)
    • USD risk-free rate (r): 5.0% per annum (domestic)
    • EUR risk-free rate (rf): 3.5% per annum (foreign)
    • Volatility (σ): 8% per annum

    Step 1: Calculate d1

    d1 = [ln(1.0800/1.1000) + (0.05 − 0.035 + 0.08²/2) × 0.25] / (0.08 × √0.25)

    d1 = [ln(0.9818) + (0.015 + 0.0032) × 0.25] / (0.08 × 0.5)

    d1 = [-0.0184 + 0.0046] / 0.04 = -0.345

    Step 2: Calculate d2

    d2 = -0.345 − 0.08 × 0.5 = -0.385

    Step 3: Look up cumulative normal values

    N(d1) = N(-0.345) = 0.3651

    N(d2) = N(-0.385) = 0.3502

    Step 4: Apply the GK call formula

    c = 1.0800 × e-0.035×0.25 × 0.3651 − 1.1000 × e-0.05×0.25 × 0.3502

    c = 1.0800 × 0.9913 × 0.3651 − 1.1000 × 0.9876 × 0.3502

    c = 0.3909 − 0.3804 = $0.0105 per EUR

    Result: The call option is worth approximately 1.05 cents per euro. For a notional of €1,000,000, the premium would be $10,500.

    This call is out-of-the-money (spot of 1.08 is below the strike of 1.10), which explains the relatively modest premium. The option gives the holder the right to buy euros at 1.10 USD when the current rate is 1.08 USD.

    Currency Options vs Equity Options

    While both are priced using similar frameworks, currency and equity options differ in important ways:

    Currency Options

    • Underlying is an exchange rate
    • Two interest rates: domestic (r) and foreign (rf)
    • Foreign rate acts like a continuous dividend yield
    • Priced using Garman-Kohlhagen model
    • Directly linked to interest rate parity
    • Premium in domestic currency per unit of foreign

    Equity Options

    • Underlying is a stock price
    • One interest rate plus dividend yield (q)
    • Dividend yield reduces call value, increases put
    • Priced using Black-Scholes-Merton
    • No explicit interest rate parity relationship
    • Premium in currency per share

    The mathematical connection is straightforward: in the BSM model for stocks paying continuous dividends, setting the dividend yield q equal to the foreign risk-free rate rf produces the Garman-Kohlhagen formulas.

    Put-Call Parity for Currency Options

    The put-call parity relationship for currency options is modified to account for both interest rates:

    Currency Option Put-Call Parity
    c + K × e-rT = p + S0 × e-rfT
    Relationship between call and put prices on the same currency

    This differs from standard put-call parity for equities because the spot price is discounted at the foreign rate (reflecting the foreign interest earned by holding the currency) rather than appearing undiscounted.

    Practical uses:

    • Arbitrage detection — If the relationship is violated, arbitrage profits are possible
    • Pricing puts from calls — Given a call price, solve for the corresponding put price
    • Creating synthetic positions — Replicate a call using a put, spot currency, and borrowing/lending

    How to Price Currency Options

    Follow these steps to price a European currency option using the Garman-Kohlhagen model:

    1. Determine the spot rate (S0) and strike (K) — Ensure both are in the same quote convention (e.g., USD per EUR)
    2. Obtain domestic and foreign risk-free rates — Use rates matching the option’s maturity; convert to continuous compounding if needed
    3. Estimate volatility — Use implied volatility from traded options or historical volatility of the exchange rate
    4. Calculate d1 and d2 — Apply the formulas above
    5. Apply N(d1) and N(d2) — Use the cumulative normal distribution function
    6. Compute the option price — Substitute into the GK call or put formula
    Try the Currency Option Pricing Calculator

    Common Mistakes

    Currency option pricing introduces several pitfalls that don’t exist with equity options:

    Watch Out

    Inverting the quote without swapping rates — If you flip from USD/EUR to EUR/USD, you must also swap which rate is domestic and which is foreign. Failing to do this produces incorrect prices.

    1. Confusing domestic vs. foreign rates
    Domestic means the numeraire currency in your quote — the currency in which the option premium is expressed. For EUR/USD (USD per EUR), domestic is USD. Don’t think of “domestic” as “your home country.”

    2. Ignoring interest rate parity
    The forward exchange rate is determined by the interest rate differential. If your option price implies a forward rate inconsistent with interest rate parity, something is wrong.

    3. Using wrong volatility
    Historical volatility and implied volatility can differ significantly. For pricing, implied volatility from actively traded options is preferred when available.

    4. Forgetting compounding conventions
    The GK model assumes continuous compounding. If you have annually compounded rates, convert them: rcc = ln(1 + rannual).

    Limitations of the Garman-Kohlhagen Model

    While widely used, the GK model has important limitations practitioners should understand:

    Key Limitation

    The model assumes constant volatility, but FX markets exhibit a pronounced volatility smile — out-of-the-money options trade at different implied volatilities than at-the-money options. Traders adjust for this using volatility surfaces.

    Other limitations include:

    • Jump risk — Exchange rates can jump significantly on central bank announcements, elections, or geopolitical events. The GK model assumes continuous price paths.
    • Interest rate changes — The model assumes constant rates, but rates can change during the option’s life, especially for longer-dated options.
    • European exercise only — The closed-form GK formula applies only to European options. American currency options require numerical methods.
    • Transaction costs — Real-world bid-ask spreads, especially for exotic strikes or maturities, are not captured.

    For more on how volatility varies across strikes, see our article on volatility smile and skew.

    Frequently Asked Questions

    A forward contract locks in an exchange rate with an obligation to transact at maturity. A currency option provides the right but not the obligation to exchange currencies, offering downside protection while preserving upside potential. Forwards cost nothing upfront (aside from margin requirements), while options require paying a premium. Options are preferred when you want protection but also want to benefit if rates move favorably.

    The domestic rate corresponds to the currency in which the option premium is expressed — the numeraire currency. For EUR/USD quoted as “USD per EUR,” the premium is in USD, so the USD rate is domestic and the EUR rate is foreign. If you’re working with the inverse quote (EUR per USD), swap them: EUR becomes domestic, USD becomes foreign. The key is consistency: S0, K, and the premium must all be in the same currency units.

    You can use the dividend-yield version of Black-Scholes-Merton by setting the continuous dividend yield (q) equal to the foreign risk-free rate (rf). This produces the Garman-Kohlhagen formulas exactly. The plain no-dividend Black-Scholes model is not sufficient because it doesn’t account for the interest earned on foreign currency. Conceptually, holding foreign currency earns the foreign risk-free rate, which acts like a continuous dividend yield.

    The OTC market allows customization of strike prices, expiration dates, and notional amounts to precisely match corporate hedging needs. A company with a €47.3 million receivable due on March 15 can get an option with exactly those terms. Exchange-traded contracts have standardized terms that may not fit specific exposures. Additionally, OTC markets can handle very large institutional transactions efficiently, with major banks providing liquidity. Exchange-traded currency options exist (Nasdaq PHLX, CME FX options on futures) but serve different use cases.
    Explore the Options Greeks Course

    Disclaimer

    This article is for educational and informational purposes only and does not constitute investment advice. Currency option pricing involves assumptions that may not hold in real markets. Option values cited are illustrative and based on the Garman-Kohlhagen model with stated inputs. Always conduct your own analysis and consult qualified professionals before trading currency options.