A call option involves purchasing the right (but not the obligation) to purchase later an amount of stock at a fixed price. Fundamentally, this makes the transaction different from futures which are [discouraged|forbidden] because of the general prohibition against selling one obligation for another obligation (e.g. a debt for a debt).
Unlike a futures contract, a call option does not involve any debt on the buyer's part; he is under no obligation to buy anything. The only debt involved is on the seller, as she has to agree to the sale if the buyer chooses to go through with it.
Fundamentally, this is analogous to providing a down payment toward a future purchase (with the down payment treated as part of the final price if the sale goes through, forfeited if it does not). There is some disagreement whether such transactions are halal, but the Hanbali position is that it's permissible, citing a report that Umar himself partook in such a transaction.
A number of Islamic financial institutions have thus permitted the use of such contracts (termed 'arbun), however their use is still controversial. Even if considered permissible, there are caveats which should be considered:
- The contract should be fixed at the time the option is contracted, not only in price and number of shares but also in length of time, to prevent uncertainty.
- The stock involved must exist at the time the option is contracted until the time it is redeemed/expired, to prevent selling what one does not possess.
- This should not be done for currencies (e.g. FX options); I don't know if buying the right to buy future money for future money with current money is haram per se, but it's a whole extra level of complexity that I really don't feel like getting into with this answer.
Also note that this is only regarding call options. Put options, which is to say purchasing the right to sell instead of buy stock, are a different beast altogether; there are shari'ah-compliant vehicles which can be (and have been) used to emulate put options to a point, but they not quite as close a fit as 'arbun is to call options. I would argue that puts should be just as permissible as calls since they're effectively the same thing in reverse, but that's just a personal opinion and not backed by any actual evidence.
Also note that this permissibility is entirely in the context of a market that meets Islamic ideals. Of particular concern, if in the non-Islamic options market the contracted stock does not exist at the beginning of and during the period of the contract (i.e. the stocks were short-sold) then the whole transaction basically becomes pure gambling. As always, you should investigate any potential investment thoroughly before actually investing.