I have no problem with the mathematical definition of an Asian option. For example, assume the strike price is $K$, the expiration date is $T$, the underlying asset has price $S(t)$, and the payoff is $$\left[\frac{1}{n}\sum_{i=1}^n S(t_i)- K\right]^+,$$ with the expiration date $T=t_n$.
Say, I bought an Asian option issued by a bank with $K=\$1$ 6 months ago and today is the expiration date. Suppose the average price in the past 6 months of the underlying asset is \$1.5. So the option is worth of \$0.5 now. My question is how the payoff is achieved, i.e. where I get this \$5 from? Does the bank give the \$0.5 directly? Or I need to do some trades on the market to get the \$0.5 as a profit.
If it's the latter case, at what price can I buy the underlying asset? Presumably it's not the strike price \$1. Because if I could buy the asset at price \$1 and if the spot price is \$2, then I could achieve \$1 profit instead of \$0.5.