trading in sweden
Not sure how much value I can add here, but will make an attempt. I'm not trading professionally, but I work within derivative sales & capital markets, based in Stockholm. I've previously done prop trading and hedge funds both in London, and here's my 2 cents worth of information.
I'm not sure what kind of options you're looking to trade or what your background is. I assume you are (1) currently trading for a bank or hf and hold a good degree, or (2) you are looking to trade your own account (i.e. looking for an arcade).
(1) The only prop shop I ever heard of (now i mean a firm where you are paid a salary and bonus to speculate with their money) is a Stockholm based company called Pan Capital Group (offices in the US and HK as well). They seem to be a small and quite successfull outfit (according to Sweden's companies house). They trade both equities and fixed income + derivatives on commodities. Apart from Pan Capital, you have around 40 hedge funds, maybe 30 based in Stockholm. Most are small outfits managing long/short equity portfolios. The best known firm is Brummer & Partners (a group of 6-7 hedge funds) who are well recognized internationally, and there are a few people there trading pretty sophisticated quant strategies that includes options and more exotic derivatives. These two firms however, are very attractive work places and I know from personal experience that it's easier to get a job at Goldmans or a good London hf, due to the fact that they're not hiring very often. The banks in Sweden do mostly market-making, but some prop trading is going on as well. For equity options I'd say Carnegie is a good place, while SEB, Handelsbanken, Danske Bank, and Nordea would be covering most products. Nordea is probably the only Nordic bank that can do as sophisticated products as the London banks, but most of their trading is done from Copenhagen, which would exclude them if you're going to Sweden. For fixed income trading, Danske Bank would be the best one. To my knowledge, the only London bank trading from Stockholm is UBS, the rest do only M&A here. I'd start looking at these firms if I had the right background.
I know for sure Handelsbanken Capital Markets is currently looking for a junior fixed income options trader.
(2) Trade your own account for a prop shop/arcade. Unfortunately I don't think there are any arcades in Sweden, never heard of any. If you want to trade equity options, there are plenty of brokers, but I'm not too sure that they could offer the competetive comissions you'd like as a professional trader. If you want to trade futures/options on indexes or fixed income I think you'd have to do this through a London based firm. I think there are plenty of arcades offering remote trading. If I had some experience from the prop shop business I would def consider setting up shop in Stockholm. There should be plentty of good pedigree available here, especially among the 90% of young men spending their days playing online poker (yes I think they would make good scalpers).
Best of luck!