An Asian option also called average value option is a name given to a special type of option contract. The average underlying price over some pre-set period of time determines the payoff for Asian options. This is quite different from the case of the usual European option and American option, where the price of the underlying instrument at maturity determines the payoff of the option contract.
One advantage that is offered by the Asian options is that the risk of market manipulation of the underlying instrument at maturity is reduced by them.
Payout of Asian Call Options with Arithmetic Average
Below I have described the payout of some Asian call options.
For the continuous case we have the payout 
where
- time to maturity is denoted by T ,
- the price is denoted by S and,
- the strike price is denoted by K.
