Weak convergence for approximation of American option prices

Abstract

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 Levy 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, Solomonand 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 Levy 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. Moreover, a path-by-path defined approximation that shares an almost same law with the one in Maller, Solomon and Szimayer (2006) is proposed. The rate of convergence for the path-by-path defined approximation is also discussed.