  In this article, we consider a risky asset $X$ for which evolution follows a
model proposed by D.G. Hobson and L.C.G. Rogers\cite{HR98}. We assume that the
volatility of $X$ depends on the ratio of the present value and the
exponentially weighted average of the past value. Using the Markovian modelling
of the enlarged two-dimensional process, we show that, for the American put
option with $X$ as the underlying asset, the continuation region and the
stopped region are separated a striking curve . This striking curve lies
between the two striking curves from the basic BSM model, yet is {\em not}
monotone.
