Garman kohlhagen option pricing model

What Is Garman-Kohlhagen Option Pricing Model?

The Garman-Kohlhagen option pricing model is a widely recognized financial model used to determine the fair value of European options on foreign currencies. It falls under the broader category of options pricing within financial derivatives. The Garman-Kohlhagen model extends the principles of the renowned Black-Scholes model by specifically adapting it to account for the unique characteristics of currency markets, primarily the presence of two distinct interest rates: one for the domestic currency and one for the foreign currency³², ³³. This distinction is crucial for accurately valuing options where the underlying asset is an exchange rate.

History and Origin

The Garman-Kohlhagen model was developed by Mark B. Garman and Steven W. Kohlhagen and first published in 1983 in the Journal of International Money and Finance³⁰, ³¹. Their work built upon the foundational Black-Scholes model, which was initially designed for pricing stock options and gained prominence after its publication in 1973²⁹.

Before the Garman-Kohlhagen model, the nascent foreign exchange options market, which saw its first over-the-counter (OTC) foreign exchange option traded in 1982, needed a robust theoretical framework for pricing²⁸. Garman and Kohlhagen addressed this need by extending the Black-Scholes framework. They recognized that unlike stock options, where a single risk-free rate is typically considered, currency options involve two interest rates—one for the domestic currency and another for the foreign currency—which directly impact the cost of holding each currency. Th²⁶, ²⁷eir model incorporated these dual interest rates, providing a more accurate mechanism for pricing foreign exchange options and serving as a pivotal tool for financial institutions engaged in the burgeoning foreign exchange markets.

#²⁵# Key Takeaways

• The Garman-Kohlhagen model is used to price European-style call option and put option on foreign currencies.
• It is an extension of the Black-Scholes model, specifically adapted for the unique aspects of the foreign exchange market.
• A key feature of the Garman-Kohlhagen model is its inclusion of both domestic and foreign interest rates, which is essential for currency option valuation.
• The model assumes that currency exchange rates follow a lognormal distribution and that markets are efficient with no arbitrage opportunities.
• ²⁴ It is widely used for risk management and trading strategies involving currency derivatives.

#²³# Formula and Calculation

The Garman-Kohlhagen model provides closed-form solutions for pricing European-style call and put options on currencies. The formulas are extensions of the Black-Scholes formula, incorporating the foreign risk-free interest rate as a "dividend yield" on the foreign currency.

F²²or a European call option ($C$) and a European put option ($P$), the formulas are:

Where:

• ( S_0 ): Current spot rate (price of one unit of foreign currency in terms of domestic currency)
• ( K ): Strike price (exercise price)
• ( T ): Time to expiration in years
• ( r_d ): Domestic risk-free interest rates (continuously compounded)
• ( r_f ): Foreign risk-free interest rate (continuously compounded)
• ( \sigma ): Volatility of the exchange rate
• ( N(x) ): Cumulative standard normal distribution function
• ( d_1 ) and ( d_2 ) are calculated as:

Interpreting the Garman-Kohlhagen Option Pricing Model

The output of the Garman-Kohlhagen model, a specific price for a currency call option or put option, represents the theoretical fair value of that option given the specified inputs. Traders and financial institutions interpret this value to make informed decisions about buying or selling currency options.

If the market price of an option is significantly higher than the Garman-Kohlhagen model's calculated value, it might suggest the option is overvalued, potentially indicating a selling opportunity. Conversely, if the market price is lower, the option may be undervalued, suggesting a buying opportunity. The model also helps to understand how changes in input variables—such as the spot rate, strike price, volatility, or interest rates in either currency—affect the option's theoretical price. For instance, an increase in the domestic interest rate or a decrease in the foreign interest rate generally increases the value of a call option and decreases the value of a put option, reflecting the carry benefits or costs of holding the respective currencies.

Hy²¹pothetical Example

Consider a U.S.-based investor who wants to buy a European call option on the Euro, allowing them to buy EUR at a specific exchange rate.

Scenario:

• Current Spot Rate ((S_0)): 1.10 USD/EUR (meaning 1 Euro costs 1.10 US Dollars)
• Strike Price ((K)): 1.12 USD/EUR
• Time to Expiration ((T)): 0.5 years (6 months)
• Domestic (USD) Risk-Free Rate ((r_d)): 5% (0.05)
• Foreign (EUR) Risk-Free Rate ((r_f)): 2% (0.02)
• Volatility ((\sigma)): 10% (0.10)

Step-by-step Calculation using the Garman-Kohlhagen Model:

1. Calculate (d_1):

   $$d_1 = \frac{\ln\left(\frac{1.10}{1.12}\right) + \left(0.05 - 0.02 + \frac{0.10^2}{2}\right)0.5}{0.10\sqrt{0.5}}$$ $$d_1 = \frac{\ln(0.9821) + (0.03 + 0.005)0.5}{0.10 \times 0.7071}$$ $$d_1 = \frac{-0.0180 + (0.035)0.5}{0.07071}$$ $$d_1 = \frac{-0.0180 + 0.0175}{0.07071} = \frac{-0.0005}{0.07071} \approx -0.0071$$

2. Calculate (d_2):

   $$d_2 = d_1 - \sigma\sqrt{T}$$ $$d_2 = -0.0071 - (0.10 \times 0.7071)$$ $$d_2 = -0.0071 - 0.07071 \approx -0.0778$$

3. Find (N(d_1)) and (N(d_2)) using a standard normal distribution table or calculator:

   • (N(d_1) = N(-0.0071) \approx 0.4972)
   • (N(d_2) = N(-0.0778) \approx 0.4690)

4. Calculate the Call Option Price ((C)):

   $$C = 1.10 e^{-(0.02)(0.5)} (0.4972) - 1.12 e^{-(0.05)(0.5)} (0.4690)$$ $$C = 1.10 e^{-0.01} (0.4972) - 1.12 e^{-0.025} (0.4690)$$ $$C = 1.10 (0.99005) (0.4972) - 1.12 (0.97531) (0.4690)$$ $$C \approx 0.5429 - 0.5135 \approx 0.0294$$

The theoretical fair value of this European call option on the Euro is approximately 0.0294 US Dollars. This means the investor would theoretically pay 2.94 cents per Euro for the right to buy Euros at 1.12 USD/EUR.

Practical Applications

The Garman-Kohlhagen option pricing model is a fundamental tool with several practical applications in the world of investing and finance:

• Currency Hedging: Multinational corporations and financial institutions widely use the model to price currency options, which are crucial instruments for hedging foreign exchange rate risk in international transactions. For ex²⁰ample, an importer expecting to pay a foreign currency in the future can use the model to price a call option that caps their potential payment if the foreign currency appreciates.
• Trading and Speculation: Traders leverage the Garman-Kohlhagen model to evaluate the fair value of currency options. By comparing the model's calculated price to the actual market price, traders can identify potential mispricing and make informed trading decisions, whether for speculation or arbitrage strategies.
• ¹⁹Risk Management: Financial institutions employ the model as part of their broader risk management framework to quantify and manage currency exposure from their portfolios of financial derivatives. The mo¹⁸del helps in calculating "Greeks" (sensitivities like delta, gamma, vega, theta, rho), which are essential for managing the risks associated with option positions.
• Structured Products: The model serves as a building block for pricing more complex structured products that contain embedded foreign currency options. Its analytical tractability allows for relatively straightforward valuation in certain scenarios.
• Academic Research and Education: It remains a core component of financial education and academic research in the field of derivatives, providing a robust theoretical foundation for understanding currency option dynamics. For example, studies have applied the Garman-Kohlhagen model to price foreign currency options in specific markets, such as the Kenyan foreign exchange market, to assess its applicability in different economic contexts.

Lim¹⁷itations and Criticisms

While the Garman-Kohlhagen model is a widely used and significant tool in options pricing for currencies, it shares many of the limitations inherent in its predecessor, the Black-Scholes model. These limitations stem from its underlying assumptions, which may not always hold true in real-world financial markets:

• ¹⁵, ¹⁶Constant Volatility: A major criticism is the assumption that the exchange rate volatility remains constant over the option's life. In rea¹³, ¹⁴lity, currency volatility fluctuates dynamically due to various economic, geopolitical, and market-specific factors. This discrepancy can lead to inaccuracies in pricing, especially for options with longer maturities or during periods of market stress.
• ¹²Constant Interest Rates: The model assumes that both domestic and foreign risk-free interest rates are constant and known until the option's expiration. Howeve¹¹r, interest rates are subject to change by central banks and market forces, impacting the carry cost of currencies and thus the option's value.
• No Dividends (Implied): While the Garman-Kohlhagen model accounts for two interest rates, treating the foreign interest rate akin to a continuous dividend yield, it doesn't explicitly handle discrete cash flows or sudden shifts that might occur in exchange rates.
• ¹⁰European-Style Options Only: The model is designed exclusively for European options, which can only be exercised at expiration. It cannot accurately price American options, which allow for early exercise, without adjustments or the use of more complex numerical methods.
• No Transaction Costs or Taxes: The model assumes frictionless markets, meaning no transaction costs (like brokerage fees or bid-ask spreads) or taxes. In pra⁹ctice, these costs can impact profitability and pricing, especially for active traders.
• Lognormal Distribution of Exchange Rates: The assumption that exchange rates follow a lognormal distribution and continuous price movements (geometric Brownian motion) may not fully capture extreme market events or "fat tails" observed in actual currency returns. This c⁸an lead to the underpricing of out-of-the-money options and overpricing of in-the-money options compared to market prices.

Despite these criticisms, the Garman-Kohlhagen model remains a foundational and widely used tool for its simplicity and analytical tractability, particularly as a starting point for more complex financial derivatives valuation.

Ga⁷rman-Kohlhagen Option Pricing Model vs. Black-Scholes Model

The Garman-Kohlhagen option pricing model is often confused with or seen as a direct alternative to the Black-Scholes model, but it is more accurately described as a specialized extension. Both models belong to the family of analytical pricing models for European-style options, relying on similar foundational assumptions such as efficient markets, constant volatility, and continuous trading. The key distinction lies in the underlying asset and, consequently, the treatment of carrying costs.

The original Black-Scholes model was developed to price options on non-dividend-paying stocks. It considers a single risk-free interest rate for discounting future cash flows. In contrast, the Garman-Kohlhagen model explicitly addresses foreign exchange options, where the underlying asset is a currency pair. Currencies inherently involve two interest rates—a domestic rate and a foreign rate—which represent the cost of holding one currency versus the yield earned on the other. The Garman-Kohlhagen model incorporates both these rates, effectively treating the foreign interest rate as a continuous "dividend yield" received on the foreign currency, thereby adjusting the present value of the underlying asset. This adjus⁵, ⁶tment makes the Garman-Kohlhagen model more appropriate for valuing currency options, as it captures the carry benefits or costs associated with holding the foreign currency until expiration.

FAQs

What types of options can 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, not before.

How d⁴oes the Garman-Kohlhagen model account for two currencies?

The model incorporates two distinct interest rates: one for the domestic currency (in which the option is priced) and one for the foreign currency (the underlying asset). The foreign interest rate is essentially treated as a continuous dividend yield on the foreign currency, reflecting the opportunity cost or benefit of holding that currency.

What ³are the main inputs for the Garman-Kohlhagen model?

The primary inputs for the Garman-Kohlhagen model include the current spot rate of the currency pair, the option's strike price, the time remaining until expiration, the domestic risk-free interest rate, the foreign risk-free interest rate, and the volatility of the exchange rate.

Is th²e Garman-Kohlhagen model still relevant today?

Yes, despite its limitations stemming from certain simplifying assumptions (like constant volatility), the Garman-Kohlhagen model remains highly relevant. It serves as a foundational model for understanding and valuing foreign exchange options and is widely used by financial professionals for risk management and trading strategies. More advan¹ced models often build upon its principles to address its shortcomings.