I want to change my country to South Korea and I can't do it now. I don't have visa, master card or amex. I want to change it because I downloaded a game there (coz the game was available on Korea only) but now I can't. There's no 'none' option :(

You are being asked for payment details because you are trying to change countries, it's not appearing because you do/don't owe anything, asking for payment details is a way of checking that you are able to use a country's store (e.g. that you have a billing address in it) : Why can’t I select None when I edit my Apple ID payment information? - Apple Support

When you change the country or region of an existing Apple ID, you must provide a payment method and update your billing options. If you want to remove your payment method after you change the country or region, you can change your payment information to None.

Are you currently in South Korea ? If you are then try contacting Korean Support and see if they can remove the requirement to enter payment details : http://www.apple.com/kr/support/itunes/contact/