We consider the Russian option introduced by Shepp and Shiryayev (Ann. Appl. Probab. 3 (1993) 631, Theory Probab. Appl. 39 (1995) 103) but with finite expiry and show that its space-time value function characterizes the unique solution to a free boundary problem. Further, using a method of randomization (or Canadization) due to Carr (Rev. Financ. Stud. 11 (1998) 597) we produce a numerical algorithm for solving the aforementioned free boundary problem.