Is it possible to buy or sell Bitcoin options? Like put and call options for a stock, but for the future price of Bitcoin?
It is possible, there various sites that provide option (or option like) trading. Each have their pros and cons. Of the ones I know of theses seemed the best. However YMMV and caveat emptor.

At time of writing here are some simple ones:


btc oracle &lt;-- Looks like a very elegant (bitcoin) solution. Just send some bitcoins to the addresses and if (in 15min, 3hr, 1day, 1week respectively later) you are in the money you are sent back X a Multiplier. 
btc Levels &lt;-- Similar to btc oracle in concept. Except you have to login to place your (high low positions) bets. Plus you can only place positions for up to the next 24 hr. 


However neither of those let you buy/sell an option that is in the money(or out of), you have to wait until the option expires. And neither of them let you write options. 

Some others:


icbit.se &lt;-- Futures contract  
mpex.co &lt;--  Has a large buy in.
coinbr.com &lt;-- They act as a broker to mpex.co (So has a much lower buy in)
1broker.com &lt;-- CFD  for btc/USD
campbx.com &lt;-- Options coming soon. (so is xmas though :P )