Many finance books introduce the pricing on geometric asian call/put options underlying black-scholes model, since its price has its explicit formula. I am not sure, if geometric asian option is indeed traded in the real market, or otherwise, it is merely an academic interests. If it is traded, which stock (ex. AAPL) has its geometric asian option?
Asian options are based on the average price of something during a period.
The average price of a stock is not very interesting, so Asian options on stock are not traded.
The average price of oil (or other commodity) during a season is important because it determines the cost of heating or transport when you burn fuel constantly during the season. Similarly the average price of a foreign currency is important if you are constantly making small purchases of foreign currency during the year. For this reason it may be interesting to have Asian options that can be used in these situations. So Asian options tend to be on this kind of continuously purchased commodity, not on stocks like AAPL.
They are not traded, even Over-The-Counter (OTC).
Asian options with arithmetic averaging are traded. The geometric Asian may be used to derive a closed-form approximation for the arithmetic variety, or as a control variate in a Monte-Carlo simulation to significantly reduce the variance of the estimate.
Arithmetic Asian options are interesting, not only for commodities, but also for stocks, as it decreases the effective volatility, and thus makes the derivative cheaper and less risky. In Asia, because the markets are more volatile, it is common for example to settle on a 5 days averaging period before the maturity date.