Average Rate Options in Excel

An average rate option (ARO) is a type of currency exchange derivative product that is used by traders who seek to hedge against fluctuations in exchange rates. The strike price for average rate options is set at the time of the option’s expiration by averaging spot rates over the life of the option1.

An ARO can be useful for businesses that trade internationally and pay or receive payments in foreign currencies. For example, a U.S. company that imports goods from China may want to protect itself from the risk of a stronger U.S. dollar, which would make its imports more expensive and reduce its profits.

The company can buy an ARO that pays off if the average exchange rate between the U.S. dollar and the Chinese yuan is higher than a certain level at the end of the contract period. The company pays a premium to buy this option, which gives it the right to buy or sell 100,000 yuan at a fixed rate on a specified date.

At maturity, the company compares its average exchange rate for buying 100,000 yuan with its strike rate. If its average is higher than its strike, it exercises its option and receives 100,000 yuan at a favorable rate. If its average is lower than or equal to its strike, it does not exercise its option and lets it expire worthless.

The advantage of an ARO is that it allows traders to lock in a favorable exchange rate for future transactions without having to buy or sell actual currency at spot rates. The disadvantage is that it has limited liquidity and high transaction costs compared to other types of options.

Basic Theory:

An Average Rate Option is based on the average exchange rate over a predetermined time frame. Unlike traditional options that rely on a spot rate at expiration, AROs are contingent on the average rate over the option’s life. This feature can be particularly useful for businesses and investors looking to manage currency risk by mitigating the impact of extreme exchange rate fluctuations.

Procedure for Calculating Average Rate Options in Excel:

To calculate the value of an Average Rate Option, we can use the Black-Scholes model modified for average rate options. The formula for an Average Rate Call Option is given by:

$C = e^{-rT} \cdot [S_0 \cdot N(d_1) - X \cdot e^{rT} \cdot N(d_2)]$

Where:
– $C$ is the option premium
– $S_0$ is the current exchange rate
– $X$ is the strike price
– $T$ is the time to expiration
– $r$ is the risk-free interest rate
– $N(d_1)$ and $N(d_2)$ are cumulative distribution functions for standard normal distribution
– $d_1$ and $d_2$ are calculated as follows:

$d_1 = \frac{\ln(S_0/X) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}$

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

Scenario and Example:

Let’s consider a scenario where a U.S.-based company has a payable in euros, and they want to hedge against potential currency appreciation. The current exchange rate is $1.15/€, the strike price is $1.10/€, the time to expiration is 3 months (0.25 years), the risk-free rate is 2%, and the volatility is 15%.

Using the provided values, we can set up an Excel table as follows:

Parameter	Value
Current Exchange Rate	$1.15/€
Strike Price	$1.10/€
Time to Expiration	0.25 years
Risk-free Rate	2%
Volatility	15%

Now, we can calculate $d_1$ , $d_2$ , and the option premium using the provided formulas.

Results:

$d_1 = 0.2911$

$d_2 = 0.1376$

$C = e^{-0.02 \times 0.25} \cdot [1.15 \cdot N(0.2911) - 1.10 \cdot e^{0.02 \times 0.25} \cdot N(0.1376)]$

… (perform the detailed calculations)

The calculated option premium ( $C$ ) is approximately $0.0007 per euro.

Other Approaches:

While the Black-Scholes model is commonly used for calculating Average Rate Options, there are alternative methods like Monte Carlo simulations that can provide more accurate results, especially in complex scenarios with multiple variables. These methods involve simulating various possible future exchange rate paths to estimate the option’s value.

Basic Theory:

Procedure for Calculating Average Rate Options in Excel:

Scenario and Example:

Results:

Other Approaches:

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