  This paper is intended to provide a unique valuation formula for the American
put option in a random time horizon; more precisely, such option restricts any
stopping rules to be made before the last exit time of the price of the
underlying assets at any fixed level. By exploiting the theory of enlargement
of filtrations associated with random times, this pricing problem can be
transformed into an equivalent optimal stopping problem with a semi-continuous,
time-dependent gain function whose partial derivative is singular at certain
point. The difficulties in establishing the monotonicity of the optimal
stopping boundary and the regularity of the value function lie essentially in
these somewhat unpleasant features of the gain function. However, it turns out
that a successful analysis of the continuation and stopping sets with proper
assumption can help us overcome these difficulties and obtain the important
properties that possessed by the free-boundary, which eventually leads us to
the desired free-boundary problem. After this, the nonlinear integral equations
that characterise the free-boundary and the value function can be derived. The
solution to these equations are examined in details.
