This paper studies the pricing of American options. An upper bound of the price can be made from a martingale and an optimal martingale attains the true price. But it is not easy to find an optimal martingale, and then the improvement of the upper bound is an important problem. In this study, we propose a simple improvement method of the upper bound by stopping times. The stopping times are made from a lower bound process of the continuation value of the American option. We show that a higher lower bound process improves an upper bound more. Finally we show numerically that our method works in the Black-Scholes model.
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