# Asian Option

Asian option (also known as average price option) is an option whose payoff is determined with respect to the (arithmetic or geometric) average price of the underlying asset over the term of the option.

While the payoff of a standard (American and European) option depends on the price of the underlying asset at a specific point of time i.e. the exercise date, the payoff of an Asian option depends on the average price of the underlying asset that prevailed over a period of time i.e. the term of the option.

There are two types of Asian options with respect to the method of averaging: in arithmetic Asian option, the arithmetic average of the price of the underlying is used in payoff calculations; while in geometric Asian options, geometric average is used.

Asian options have relatively low volatility due to the averaging mechanism. They are used by traders who are exposed to the underlying asset over a period of time such as consumers and suppliers of commodities, etc.

## Formula

Following are simplified payoff formulas for four different variants of Asian option:

Arithmetic Asian Call Option Payoff

= max [0, arithmetic average of underlying's price – exercise price]

Geometric Asian Call Option Payoff

= max [0, geometric average of underlying's price – exercise price]

Arithmetic Asian Put Option Payoff

= max [0, exercise price – arithmetic average of underlying's price]

Geometric Asian Put Option Payoff

= max [0, exercise price – geometric average of underlying's price]

## Examples

**Example 1:** Arithmetic Asian call option

On 1 January 20Y3, a trader purchased a 90 day arithmetic call option on AOL, Inc. The option has an exercise price of $35 and the payoff is based on arithmetic average price of the underlying stock determined after the end of each 30 day period. The stock price of the underlying asset at 30th, 60th and 90th day of the option was $30.65, $36.9 and $38.49.

Arithmetic Asian call option payoff = max [0, ($30.65+$36.9+$38.49)/3 – $35] = $0.35

Payoff of geometric Asian call option can be calculated by substituting the arithmetic average price of the underlying asset with its geometric equivalent.

**Example 2:** Geometric Asian put option

Refer to data in Example 1 above, but assume that another trader bought a geometric put option with the same exercise price i.e. $35

Geometric price of the underlying financial asset

= (30.65 × 36.9 × 38.49)^{1/3}

= $35.81

Geometric Asian put option payoff = max [0, $35 – $35.81] = 0

The put option is out of the money because the geometric average of the underlying price is higher than the exercise price.