Garman–kohlhagen model: Meaning, Criticisms & Real-World Uses

Garman–Kohlhagen Model

The Garman–Kohlhagen model is a financial model used to determine the theoretical fair value of currency options. This model falls under the broad category of financial derivatives and serves as an extension of the well-known Black-Scholes model, adapted specifically for the unique characteristics of the foreign exchange market. It accounts for the existence of two distinct interest rates: one for the domestic currency and another for the foreign currency, which is crucial for accurate option pricing in international markets.

History and Origin

Developed by Mark B. Garman and Steven W. Kohlhagen, the model was first published in their 1983 paper, "Foreign Currency Option Values," in the Journal of International Money and Finance. The¹³ir work extended the foundational principles of the Black-Scholes model, which was originally formulated for pricing stock options, to the realm of foreign currency options. The key innovation of the Garman–Kohlhagen model was its ability to incorporate two different interest rates, recognizing that holding foreign currency implies earning or paying interest at a rate different from the domestic currency. This adaptation made it a pivotal tool for traders and institutions managing foreign exchange exposure.

Key Takeaways

The Garman–Kohlhagen model is used to price European-style currency options.
It is an extension of the Black-Scholes model, specifically tailored for the foreign exchange market.
The model accounts for both domestic and foreign interest rates, alongside volatility and time to maturity.
It assumes no transaction costs, continuous trading, and European-style exercise.
The Garman–Kohlhagen model is widely applied in risk management and trading strategies involving foreign exchange derivatives.

Formula and Calculation

The Garman–Kohlhagen model calculates the theoretical price of European-style call option ($C$) and put option ($P$) on a foreign currency. The formulas are as follows:

For a European call option:
$C = S e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2)$

For a European put option:
$P = K e^{-r_d T} N(-d_2) - S e^{-r_f T} N(-d_1)$

Where:

(C) = Price of the European call option
(P) = Price of the European put option
(S) = Current spot exchange rate (domestic currency per unit of foreign currency)
(K) = Strike price (exchange rate at which the option can be exercised)
(r_d) = Domestic risk-free interest rate (annualized, continuously compounded)
(r_f) = Foreign risk-free interest rate (annualized, continuously compounded)
(T) = Time to maturity (in years)
(\sigma) = Volatility of the exchange rate
(N(x)) = Cumulative standard normal distribution function of (x)

And (d_1) and (d_2) are calculated as:
$d_1 = \frac{\ln(S/K) + (r_d - r_f + \sigma^2/2)T}{\sigma\sqrt{T}}$
$d_2 = d_1 - \sigma\sqrt{T}$ ,

This stru¹²c¹¹ture mirrors the Black-Scholes model, with the foreign interest rate effectively replacing the dividend yield in stock option pricing.

Interp¹⁰reting the Garman–Kohlhagen Model

The Garman–Kohlhagen model provides a theoretical fair value for currency options, which market participants can use to assess whether an option is overvalued or undervalued. A calculated option price significantly above its market price might suggest it is overvalued, and vice versa. The model's outputs, particularly the "Greeks" (such as Delta, Gamma, Vega, Theta, and Rho), help traders understand the sensitivity of the option's price to changes in underlying parameters like the spot exchange rate, volatility, and interest rates. For instance, a high Delta indicates that the option's price is highly responsive to movements in the underlying currency pair.

Hypothetical Example

Suppose a U.S.-based importer needs to pay €1,000,000 in three months and is concerned about the EUR/USD exchange rate strengthening (meaning the euro becomes more expensive). They consider buying a European call option with a strike price of 1.1000 (USD per EUR).

Let's assume the following:

Current spot exchange rate (S): 1.0900 USD/EUR
Strike price (K): 1.1000 USD/EUR
Time to maturity (T): 0.25 years (3 months)
U.S. risk-free interest rate (r_d): 5.0% (0.05)
Eurozone risk-free interest rate (r_f): 3.0% (0.03)
Volatility ((\sigma)): 12% (0.12)

First, calculate (d_1) and (d_2):
$d_1 = \frac{\ln(1.0900/1.1000) + (0.05 - 0.03 + 0.12^2/2) \times 0.25}{0.12\sqrt{0.25}}$
$d_1 = \frac{\ln(0.9909) + (0.02 + 0.0072) \times 0.25}{0.12 \times 0.5}$
$d_1 = \frac{-0.0091 + (0.0272) \times 0.25}{0.06}$
$d_1 = \frac{-0.0091 + 0.0068}{0.06} = \frac{-0.0023}{0.06} \approx -0.0383$

$d_2 = d_1 - \sigma\sqrt{T} = -0.0383 - 0.12 \times 0.5 = -0.0383 - 0.06 = -0.0983$

Next, find (N(d_1)) and (N(d_2)) from a standard normal distribution table or calculator:

(N(d_1) = N(-0.0383) \approx 0.4847)
(N(d_2) = N(-0.0983) \approx 0.4609)

Finally, calculate the call option price:
$C = 1.0900 e^{-0.03 \times 0.25} N(-0.0383) - 1.1000 e^{-0.05 \times 0.25} N(-0.0983)$
$C = 1.0900 e^{-0.0075} (0.4847) - 1.1000 e^{-0.0125} (0.4609)$
$C = 1.0900 (0.9925) (0.4847) - 1.1000 (0.9876) (0.4609)$
$C \approx 0.5230 - 0.5011 \approx 0.0219$

The theoretical fair value of this call option is approximately $0.0219 USD per EUR. This means the importer would expect to pay roughly $0.0219 per euro for the right to buy euros at 1.1000 USD/EUR in three months.

Practical Applications

The Garman–Kohlhagen model is predominantly used in the foreign exchange market by financial institutions, multinational corporations, and traders for various purposes:

Risk Management: Companies engaged in international trade use the model to price currency options for hedging against adverse exchange rate movements. This helps mitigate the impact of currency fluctuations on their revenues and costs.
Trading and [⁹Arbitrage](https://diversification.com/term/arbitrage) Detection: Traders rely on the Garman–Kohlhagen model to evaluate whether the market price of a currency option deviates significantly from its theoretical value. Such deviations can present opportunities for profitable trades or signal potential arbitrage situations.
Portfolio Valuation: Fund managers and financial analysts use the model to value portfolios that include foreign exchange derivatives, ensuring accurate accounting and reporting of asset values.
Market Transparency: The model contributes to increased transparency in the over-the-counter (OTC) foreign exchange derivatives markets, aiding central banks and market participants in monitoring global financial activities. The Bank for International Settlements (BIS) conducts a Triennial Central Bank Survey that provides comprehensive data on these markets, highlighting their significant scale and activity.,

Limitations and⁸ ⁷Criticisms

Despite its widespread use, the Garman–Kohlhagen model, like its predecessor the Black-Scholes model, operates under several simplifying assumptions that can limit its accuracy in real-world scenarios. These limitations include:

Constant Volatility and Interest Rates: The model assumes that the volatility of the exchange rate and the domestic and foreign interest rates remain constant over the option's life. In reality, these parameters fluctuate constantly, often unpredictably.,
Efficient and Fr⁶i⁵ctionless Markets: It assumes markets are perfectly efficient, with no transaction costs, taxes, or restrictions on short selling. Such ideal conditions rarely exist in practice.
European-Style [O⁴ptions](https://diversification.com/term/option-pricing): The model is strictly for European options, which can only be exercised at expiration. It does not account for American-style options, which can be exercised any time up to expiration, thus potentially underpricing them.
Lognormal Distrib³ution: The model assumes that exchange rate movements follow a geometric Brownian motion, implying that terminal exchange rates are lognormally distributed. While often a reasonable approximation, real-world exchange rate distributions can exhibit "fat tails" (more extreme events than a normal distribution would predict).

Academics continue to explore more complex models to address these limitations, incorporating factors such as stochastic volatility or jump diffusion processes to better capture real market dynamics.,

Garman–Kohlhagen ²M¹odel vs. Black-Scholes Model

The Garman–Kohlhagen model is essentially a specialized application of the Black-Scholes model. The core difference lies in their application and the way they handle interest rates. The original Black-Scholes model was developed for pricing equity options on non-dividend-paying stocks, assuming a single risk-free rate for discounting.

The Garman–Kohlhagen model adapts this framework for currency options by explicitly incorporating two distinct risk-free interest rates: one for the domestic currency and one for the foreign currency. In the context of the formula, the foreign interest rate effectively acts like a continuous dividend yield in the Black-Scholes framework, reflecting the cost or benefit of holding the foreign currency. This modification makes the Garman–Kohlhagen model more appropriate for valuing options where the underlying asset itself is a currency and thus subject to different borrowing and lending rates.

FAQs

What type of options does the Garman–Kohlhagen model price?
The Garman–Kohlhagen model is specifically designed to price European options on foreign currencies. This means the option can only be exercised on its expiration date.

How does the Garman–Kohlhagen model account for two interest rates?
The model integrates both the domestic risk-free interest rate (for the currency in which the option is priced) and the foreign risk-free interest rate (for the underlying foreign currency). This is crucial because holding foreign currency can earn or incur interest, influencing the option's value.

Why is volatility a critical input in the Garman–Kohlhagen model?
Volatility is a measure of how much the exchange rate is expected to fluctuate. Higher volatility implies a greater chance of significant price movements, which can increase the probability of an option pricing ending "in-the-money," thus generally leading to higher option premiums for both call option and put option contracts.

Garman–kohlhagen model