Romarie Abrazaldo 11124997 Garman – Kohlhagen Model The Garman-Kohlahgen model is an analytic valuation model for European options on currencies in the spot foreign exchange. It uses a similar approached by Merton for European options on dividend-paying stocks. The only difference is that instead of using a stock’s dividend yield, the Garman-Kohlhagen model uses the foreign currency’s continuously compounded risk-free rate. But just like Robert Merton, Mark Garman and Steven Kohlhagen generalized the Black-Scholes model. It alleviates the restrictive assumption in the Black-Scholes model that borrowing and lending uses the same risk-free rate. In the foreign exchange market, risk free rates are different in each country and the interest rate differential between two currencies will affect the foreign exchange option. To address this, Garman and Kohlhagen included two interest rates – one for a domestic currency, and one for a foreign currency, in their model. Since an option holder does not receive any cash flow paid from the underlying instrument, this model predicts that foreign exchange options are cheaper than standard European option for a call but more expensive for a put. This was published in 1983. This model was based on a number of assumptions namely: 1. The option can only be exercised on the expiry date (European style) 2. The price distribution of terminal currency exchange rate (returns) is lognormal. 3. There are no arbitrage possibilities; the market is efficient. 4. Transactions cost, margins, and taxes are zero. 5. The risk-free interest rates, the foreign interest rates, and the exchange rate volatility of the underlying

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Romarie Abrazaldo11124997

Garman – Kohlhagen Model

The Garman-Kohlahgen model is an analytic valuation model for European options on currencies in the spot foreign exchange. It uses a similar approached by Merton for European options on dividend-paying stocks. The only difference is that instead of using a stock’s dividend yield, the Garman-Kohlhagen model uses the foreign currency’s continuously compounded risk-free rate. But just like Robert Merton, Mark Garman and Steven Kohlhagen generalized the Black-Scholes model. It alleviates the restrictive assumption in the Black-Scholes model that borrowing and lending uses the same risk-free rate. In the foreign exchange market, risk free rates are different in each country and the interest rate differential between two currencies will affect the foreign exchange option. To address this, Garman and Kohlhagen included two interest rates – one for a domestic currency, and one for a foreign currency, in their model.

Since an option holder does not receive any cash flow paid from the underlying instrument, this model predicts that foreign exchange options are cheaper than standard European option for a call but more expensive for a put. This was published in 1983.

This model was based on a number of assumptions namely:

1. The option can only be exercised on the expiry date (European style)2. The price distribution of terminal currency exchange rate (returns) is lognormal.3. There are no arbitrage possibilities; the market is efficient.4. Transactions cost, margins, and taxes are zero.5. The risk-free interest rates, the foreign interest rates, and the exchange rate volatility of

the underlying instrument are constant (known functions of time over the life of the option).

6. There are no penalties for short sales of currencies.7. The market operates continuously and the exchange rates follow a continuous Ito process.

The domestic currency value of a call option into the foreign currency is:

Source: http://investexcel.net/garman-kohlhagen-model/

The value of a put option is:

Source: http://investexcel.net/garman-kohlhagen-model/

Where:

Source: http://investexcel.net/garman-kohlhagen-model/

rd – domestic risk free simple interest raterf – foreign risk free simple interest rateS0 – current spot rateK – strike priceN – cumulative normal distribution functionT – time to maturity (calculated according to the appropriate day count conversion)σ – foreign exchange rate volatility

Because the prices of options are affected by two risk-free rates (domestic and foreign), there are also two rho sensitivities.

The options risk exposure or Greeks for the call option are:

Source: http://www.riskglossary.com/link/garman_kohlhagen_1983.htm

While the Greeks for the put option are: