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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.

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The Asian option is cash settled, so the bank will transfer you $0.5. Delivering the shares and doing some trades is not possible. You can't buy the spot for the average price over a period, you just pay the spot price.

Since you're into Asian options, I assume asian option valuation is useful for you to assess whether you're not paying too much.

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    $\begingroup$ So can I presume the Asian option is over-the-counter instead of traded on an exchange? $\endgroup$ – William Zhang Jan 15 '15 at 12:37
  • $\begingroup$ Although there have been examples in the past, its quite uncommon them being traded on an exchange. $\endgroup$ – QuantK Jan 16 '15 at 15:41
  • $\begingroup$ Depending on your definition, you might consider Hang Seng index options a flavour of exchange-traded Asian option where t_i are spaced five minutes apart on expiry day. $\endgroup$ – VommaNeutral Jul 23 '17 at 11:15

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