A special type of Asian options (average price options) in which the payoff is determined with respect to the arithmetic average price of the underlying asset over the lifespan of the option.

While the payoff of a standard American and European option is determined based on the price of the underlying asset at a specific point of time i.e., the exercise date or maturity date, the payoff of an Asian option depends on the average price of the underlying as observed over a period of time i.e., the term of the option.  The method of averaging in an arithmetic Asian option takes into consideration the arithmetic average of the price of the underlying in calculating the payoff.

The payoff of an arithmetic Asian call option is given by:
Payoff = max [0, arithmetic average of underlying price – exercise price]

The payoff of an arithmetic Asian put option is determined as:
Payoff = max [0, exercise price – arithmetic average of underlying price]

For example, on a given day, an investor bought a 60-day arithmetic call option on XYZ share. The exercise price of the call is $20 and the payoff is determined based on the arithmetic average price of the underlying share as established every time at the end of a period of 30 days. If the share price on the 30th and 60th day of the option’s term was $21 and $23, respectively, then the option’s payoff is:

Payoff = max [0, (21 + 23)/2 – 20] = $2

The payoff per contract is: 2 × 100 = $200