Installation of a GFCI without a ground wire
Short answer: No you don't need a ground wire. The code lets you replace the two wire receptacle with a GFCI. 

From the 2011 National Electrical Code


  406.4(D)(2)(b) A non–grounding-type receptacle(s) shall be permitted to be replaced with a ground-fault circuit interrupter-type of receptacle(s). These receptacles shall be marked “No Equipment Ground.” An equipment grounding conductor shall not be connected from the ground-fault circuit-interrupter-type receptacle to any outlet supplied from the ground-fault circuit-interrupter receptacle.