Garman–kohlhagen

What Is Garman–Kohlhagen?

The Garman–Kohlhagen model is a mathematical framework used to determine the fair value of European options on foreign currencies. It falls under the broader category of Financial Derivatives Valuation and is a widely recognized option pricing model in the foreign exchange market. The Garman–Kohlhagen model considers factors such as the current exchange rate, the strike price of the option, the time to expiration, the volatility of the exchange rate, and, crucially, the prevailing interest rates in both the domestic and foreign currencies.

History and Origin

The Garman–Kohlhagen model was developed by Mark B. Garman and Steven W. Kohlhagen and first published in their paper "Foreign Currency Option Values" in the Journal of International Money and Finance in 1983. This mo⁹del emerged as an extension of the seminal Black-Scholes model, adapting its principles to account for the unique characteristics of currency options. While the original Black-Scholes model was designed for options on dividend-paying stocks, Garman and Kohlhagen recognized that foreign currencies could be treated similarly, where the foreign risk-free rate effectively acts as a "dividend yield" on the foreign currency. This innovation allowed for more accurate valuation of currency options, which are critical financial derivatives for international trade and investment.

Key Takeaways

• The Garman–Kohlhagen model is used for pricing European-style currency options.
• It is an extension of the Black-Scholes model, adapted for foreign exchange markets.
• A key feature is its inclusion of separate domestic and foreign interest rates.
• The model assumes that exchange rate movements follow a geometric Brownian motion.
• It helps market participants assess the fair value of call option and put option contracts on currencies.

Formula and Calculation

The Garman–Kohlhagen model provides separate formulas for pricing European call option and put option contracts on currencies. The core structure is similar to the Black-Scholes model, but with adjustments for the two distinct interest rates and treating the foreign risk-free rate as a continuous dividend yield.

For a Euro⁸pean call option (C):
$C = S e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2)$

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

Where:

• (S) = Current spot exchange rate (domestic currency per unit of foreign currency)
• (K) = Strike price
• (T) = Time to expiration (in years)
• (r_d) = Domestic risk-free rate (annualized, continuously compounded)
• (r_f) = Foreign risk-free rate (annualized, continuously compounded)
• (\sigma) = Volatility of the exchange rate
• (N(x)) = Cumulative standard normal distribution function
• (d_1) and (d_2) are calculated as:

$d_1 = \frac{\ln\left(\frac{S}{K}\right) + \left(r_d - r_f + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}$

$d_2 = d_1 - \sigma\sqrt{T}$

Interpreting the Garman–Kohlhagen Model

The output of the Garman–Kohlhagen model is a theoretical fair value for a currency option. This value represents what the option should be worth given the current market conditions and the model's assumptions. Traders and financial institutions use this value to determine if an option is undervalued or overvalued in the market. If the market price is higher than the Garman–Kohlhagen model's output, the option might be considered overvalued, potentially indicating a selling opportunity. Conversely, if the market price is lower, it could suggest an undervalued option, presenting a buying opportunity. The model also allows for the calculation of option Greeks, such as delta, which measures the sensitivity of the option price to changes in the underlying exchange rate. Understanding these sensitivities is crucial for managing currency risk.

Hypothetical Example

Consider an investor evaluating a European call option on the Euro (EUR) against the US Dollar (USD).

• Current spot exchange rate (EUR/USD): $1.0800 (S)
• Strike price: $1.0900 (K)
• Time to expiration: 0.5 years (6 months) (T)
• US (domestic) risk-free rate: 5.0% (0.05) (rd)
• Eurozone (foreign) risk-free rate: 4.0% (0.04) (rf)
• Volatility of EUR/USD exchange rate: 10% (0.10) (σ)

First, calculate (d_1) and (d_2):
$d_1 = \frac{\ln\left(\frac{1.0800}{1.0900}\right) + \left(0.05 - 0.04 + \frac{0.10^2}{2}\right)0.5}{0.10\sqrt{0.5}}$
$d_1 = \frac{\ln(0.9908) + (0.01 + 0.005)0.5}{0.10 \times 0.7071}$
$d_1 = \frac{-0.0092 + 0.0075}{0.07071} = \frac{-0.0017}{0.07071} \approx -0.0240$

$d_2 = -0.0240 - 0.10\sqrt{0.5} = -0.0240 - 0.07071 \approx -0.0947$

Next, find (N(d_1)) and (N(d_2)) from a standard normal distribution table:
(N(-0.0240) \approx 0.4904)
(N(-0.0947) \approx 0.4623)

Finally, calculate the call option price (C):
$C = 1.0800 e^{-0.04 \times 0.5} (0.4904) - 1.0900 e^{-0.05 \times 0.5} (0.4623)$
$C = 1.0800 e^{-0.02} (0.4904) - 1.0900 e^{-0.025} (0.4623)$
$C \approx 1.0800 \times 0.9802 \times 0.4904 - 1.0900 \times 0.9753 \times 0.4623$
$C \approx 0.5193 - 0.4913 = 0.0280$

Based on the Garman–Kohlhagen model, the theoretical value of this European call option is approximately $0.0280. This calculated value helps the investor decide whether to buy or sell the option at its current market price.

Practical Applications

The Garman–Kohlhagen model is extensively used by financial professionals involved in the foreign exchange market. Its primary application is in pricing currency options, which are vital tools for hedging against foreign exchange rate risk. Businesses engaged in ⁷international trade, such as exporters and importers, utilize these options to protect their profit margins from adverse currency fluctuations. For example, an importer expecting to pay a foreign supplier in a few months can buy a call option to lock in an exchange rate, mitigating the risk of the foreign currency appreciating.

Beyond corporate hedging, the model is employed by:

• Banks and Financial Institutions: For quoting prices on currency derivatives to clients and for managing their own trading books.
• Fund Managers: To evaluate the fair value of currency options within their portfolios and execute appropriate trading strategies.
• Analysts and Researchers: For academic studies, market analysis, and developing more sophisticated option pricing methodologies.
  The model's ability to incorporate differential interest rates makes it particularly suited for the global currency markets, where interest rate parity often influences exchange rate dynamics.

Limitations and Cr⁶iticisms

While widely used, the Garman–Kohlhagen model, as an extension of the Black-Scholes model, shares many of its underlying assumptions and, consequently, its limitations. One significant assumpti⁵on is that the volatility of the exchange rate remains constant over the life of the option. In reality, market volatility is dynamic and can fluctuate significantly, leading to potential inaccuracies in the model's output. Critics also point out that the model assumes continuous trading and no transaction costs or taxes, which are not reflective of real-world market conditions.

Furthermore, the model ⁴assumes that the underlying exchange rate follows a geometric Brownian motion, implying a lognormal distribution of returns. However, empirical evidence often shows that financial asset returns, including exchange rates, exhibit "fat tails" (more extreme events than a normal distribution would predict) and skewness. This discrepancy can lea³d to the model mispricing out-of-the-money options. The model's reliance on constant domestic and foreign risk-free rate is another simplification, as interest rates can change over time due to economic policies or market forces. For these reasons, financial professionals often use the Garman–Kohlhagen model as a starting point, adjusting its outputs with market observations and employing more advanced numerical methods or stochastic volatility models for more accurate pricing.

Garman–Kohlhagen vs. Black-Scholes Model

The Garman–Kohlhagen model is a direct adaptation of the Black-Scholes Model, primarily distinguished by its application to currency options rather than stock options. The fundamental difference lies in how they account for the "income" generated by the underlying asset.

The Black-Scholes Model traditionally applies to non-dividend-paying stocks. When extended to dividend-paying stocks, it incorporates a dividend yield. The Garman–Kohlhagen model treats the foreign currency's risk-free rate as an analogous continuous "dividend yield" paid by the foreign currency. This allows it to explicitly acc²ount for the differential interest rates between the domestic and foreign currencies, which is a crucial factor in foreign exchange markets. Unlike the Black-Scholes model, which typically assumes a single risk-free rate, Garman–Kohlhagen incorporates both a domestic and a foreign risk-free rate, making it more suitable for evaluating currency options.

FAQs

What type of options¹ does the Garman–Kohlhagen model price?

The Garman–Kohlhagen model is specifically designed to price European options on foreign currencies. European options can only be exercised at their expiration date.

How does the Garman–Kohlhagen model account for two different interest rates?

In the Garman–Kohlhagen model, the domestic risk-free rate is used for discounting the strike price, similar to the Black-Scholes model. The foreign risk-free rate is treated as a continuous dividend yield on the foreign currency, impacting the forward exchange rate and thus the option's value. This differentiation is critical for accurately valuing currency options.

Can the Garman–Kohlhagen model be used for American options?

No, the Garman–Kohlhagen model is a closed-form solution based on the assumptions of European-style options, meaning it assumes the option can only be exercised at expiration. American options, which can be exercised any time up to expiration, require more complex valuation methods like binomial tree models or Monte Carlo simulations to account for the early exercise premium.

What is the role of volatility in the Garman–Kohlhagen model?

Volatility represents the expected fluctuation of the underlying exchange rate over the option's life. Higher volatility generally leads to higher option premiums for both call option and put option contracts because it increases the probability of the option expiring in the money. The Garman–Kohlhagen model assumes constant volatility, which is a simplification compared to real-world market behavior.