How do Korean stock market stay independent from US stock market?

The simple answer is - it can't. You've heard of the global economy. Well there is also a global stock market. What affects the largest stock market (and economy) in the world, also affects other stock markets. But when the u.s. market goes up, it likely also helps the S Korean market.

I recall in the 90's, the Korean economy had a meltdown. This affected other asian economies & took down asian markets. The u.s. market also went down in sympathy with other markets.

It would be nice if political events & economic events (in other parts of the world) did not affect one's local market. But in truth, it just doesn't work that way.

it is supply and demand... when a bunch of people fear the value of their stock will go down (by some other market force) they panic and sell their stocks. Just like with anything else in the market, when it gets flooded with a lot of supply, the price drops because it is less scarce. Many people invest the wrong way. They see the market is doing well, so they decide that they want a piece. Then the market starts to decline and they panic that they will lose their money, so they sell. In actuality, you are better off to invest when the market is down because you can get more shares for your buck. Then sell when the market is high... you see?