Garman-Kohlhagen FX Option Pricing Lab

Simulate Mark Garman and Steven Kohlhagen's (1983) currency option pricing framework. Model dual sovereign interest rate curves, Covered Interest Parity (CIP) forward points, spot vs. forward delta hedging, and dual rho sensitivities.

FX Market Presets:

Currency Pair Parameters

Dom/For
Dom/For
Years
%

Dual Sovereign Yield Curves

4.75%
Risk-free interest rate of the quote / pricing currency (e.g. USD).
3.50%
Risk-free interest rate earned on base currency deposits (e.g. EUR).
Rate Differential (rᵢ - rᶜ): +1.25%
CIP Forward Rate (F₀): 1.0918
FX Call Option Price
0.0253
2.33% of Spot • Δᵣᵠᵛ: 0.531
FX Put Option Price
0.0186
1.71% of Spot • Δᵣᵠᵛ: -0.451
Dual Rho Sensitivity (ρᵢ vs ρᶜ)
+0.252 / -0.283
Domestic: +0.252 • Foreign: -0.283
Forward Delta & Vega (ν)
0.541 • 0.298
Forward Points: +68.0 pips

Quantitative Visualizer

Call Price Put Price CIP Forward (F₀)

Garman-Kohlhagen (1983) Closed-Form Currency Option Valuation

C = S₀ e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2),   P = K e^{-r_d T} N(-d_2) - S₀ e^{-r_f T} N(-d_1)

Where d_1 = [ ln(S₀/K) + (r_d - r_f + 0.5σ²)T ] / (σ√T) and d_2 = d_1 - σ√T. Covered interest parity dictates F₀ = S₀ e^{(r_d - r_f)T}.

Sensitivity Grid 1: Domestic Rate (rᵢ) vs. Foreign Rate (rᶜ)

Evaluates ATM Call Option Price across interest rate differentials, demonstrating the carry trade forward premium.

Sensitivity Grid 2: FX Volatility (σ) vs. Expiry (T)

Evaluates ATM Straddle Premium (Call + Put), the primary quote metric in interbank currency option markets.

Institutional Mechanics of Foreign Currency Options

1. Covered Interest Parity (CIP) & Forward Carry

In FX trading, exchange rates cannot be analyzed in isolation from interest rates. Holding foreign currency provides a continuous dividend yield equal to the foreign deposit rate r_f. Meanwhile, borrowing domestic currency incurs financing costs at rate r_d:

  • Forward Premium (r_d > r_f): When the domestic interest rate exceeds foreign yields, the forward rate trades above spot (F_0 > S_0), inflating call options and depressing put options.
  • Forward Discount (r_d < r_f): Conversely, when foreign rates are higher (such as holding USD vs. JPY), the foreign currency forward trades at a discount, shifting moneyness toward puts.
  • Forward Points: Quoted in pips as (F_0 - S_0) × 10,000, governing the intrinsic bias of currency option positions.

2. Spot Delta vs. Forward Delta Hedging Conventions

Institutional FX desks differentiate strictly between Spot Delta and Forward Delta:

  • Spot Delta (Δ_spot = e^{-r_f T} N(d_1)): The exact amount of spot foreign currency needed to create a delta-neutral hedge today. Because foreign balances accrue interest at r_f, fewer units of spot currency are needed today to cover terminal obligations.
  • Forward Delta (Δ_fwd = N(d_1)): The hedge ratio against an FX forward contract maturing on the same date. FX vanilla options are universally quoted by forward delta (e.g., 25Δ risk reversals and 10Δ butterflies).
  • Dual Rho Dynamics: Hedging FX options requires two interest rate hedges—one in the domestic money market and another in the foreign bond market.

Garman-Kohlhagen Mastery Quiz

Frequently Asked Questions

Published in 1983 by Mark B. Garman and Steven W. Kohlhagen, the Garman-Kohlhagen model is the market-standard extension of the Black-Scholes-Merton framework for currency options. In foreign exchange, holding foreign currency earns the foreign risk-free interest rate r_f, while borrowing or funding occurs at the domestic risk-free rate r_d. Garman-Kohlhagen incorporates both sovereign yield curves via Covered Interest Rate Parity, correctly discounting the spot exchange rate by exp(-r_f * T) and the strike by exp(-r_d * T).

Spot Delta measures the option's sensitivity to changes in the immediate spot exchange rate (Delta_spot = exp(-r_f * T) * N(d1)), representing the cash amount of foreign currency required to hedge the position today. Forward Delta measures sensitivity to changes in the forward exchange rate (Delta_fwd = N(d1)). In interbank FX option markets, contracts are conventionally quoted and hedged in forward delta (such as 25-delta calls and 25-delta puts) because hedging is executed using FX forwards rather than spot transactions.

Unlike equity options with a single interest rate sensitivity, FX options depend on two sovereign yield curves. Domestic Rho (dC/dr_d) measures the option value shift when domestic rates move, which increases call value by lowering the present value of the strike paid. Foreign Rho (dC/dr_f) measures the sensitivity to foreign rates, which decreases call value because higher foreign interest rates create a larger carry cost or dividend drag on holding the foreign currency.

Under no-arbitrage conditions, Covered Interest Parity dictates that the forward exchange rate equals F_0 = S_0 * exp((r_d - r_f) * T). If the domestic interest rate is higher than the foreign rate (r_d > r_f), the forward exchange rate trades at a premium to spot (F_0 > S_0), reflecting forward points. If the foreign rate is higher (r_f > r_d), the foreign currency trades at a forward discount, directly altering the moneyness and pricing of FX calls and puts.

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