Does it make a difference at which stock exchange I buy a special security(e.g. whether I buy at Frankfurt or New York)?
Yes, I have in mind that there are different currencies, but I mean just things regarding the security alone.
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In a simple statement, no doesn't matter. Checked on my trade portal, everything lines up. Same ISIN, same price(after factoring in FX conversions, if you were thinking about arbitrage those days are long gone).
But a unusual phenomenon I have observed is, if you aren't allowed to buy/sell a stock in one market and try to do that in a different market for the same stock you will still not be allowed to do it. Tried it on French stocks as my current provider doesn't allow me to deal in French stocks.
Different exchanges sometimes offer different order types, and of course have different trading fees.
But once a trade is finished, it should not matter where it was executed.
Also important to keep in mind is the difference in liquidity. The stock could be very liquid in 1 exchange but not in another. When times get bad, liquidity could dry up 1 one exchange, which results in a trading discount.
Really arbitrage means that, currency risk aside, it shouldn't matter which exchange you buy on in price terms alone. Arbitrage will always make sure that the prices are equivalent otherwise high frequency traders can make free money off the difference. In practical terms liquidity and brokerage costs usually make trading on the "home" exchange more worthwhile as any limit orders etc will be filled at a better price as you will more easily find a counterparty to your trade. Obviously that will only be an issue where your quantity is significant enough to move the market on a given exchange. The volume needed to move a market is dependent upon the liquidity of the particular stock.
Yes, it does matter very much.
There's a thing called fungible instruments. These are the instruments where it doesn't matter. E.g. most options I ever dealt with in the US are fungible no matter which US exchange you trade them on.
A fungible instrument is an instrument where you buy 1 lot on exchange A, then sell 1 lot on exchange B, and as a result you have 0 lots.
With another instrument, you can buy 1 lot on exchange A, sell 1 lot on exchange B, and even tough they are the exact same thing, you now own 1, and owe 1 - they don't cancel each other out.
Other than that, in different countries there are obviously different laws and regulations.
For a small at home trader who just gambles for fun and isn't interested in negative positions (sell what you don't have), there isn't much of a difference . I think that's what the other answers are saying.