Georgia Institute of Technology College of Sciences

Based on a sequence of discretized American option price processes under the multinomial model proposed by Maller, Solomon and Szimayer (2006), the sequence converges to the counterpart under the original L\'{e}vy process in distribution for almost all time. We prove a weak convergence in this case for American put options for all time. By adapting Skorokhod representation theorem, a new sequence of approximating processes with the same laws with the multinomial tree model defined by Maller, Solomon and Szimayer (2006) is obtained. The new sequence of approximating processes satisfies Aldous' criterion for tightness. And, the sequence of filtrations generated by the new approximation converges to the filtration generated by the representative of L\'{e}vy process weakly. By using results of Coquet and Toldo (2007), we give a complete proof of the weak convergence for the approximation of American put option prices for all time.