  This paper concerns optimal stopping problems driven by the running maximum
of a spectrally negative L\'{e}vy process $X$. More precisely, we are
interested in modifications of the Shepp-Shiryaev optimal stopping problem
[Avram, Kyprianou and Pistorius Ann. Appl. Probab. 14 (2004) 215-238; Shepp and
Shiryaev Ann. Appl. Probab. 3 (1993) 631-640; Shepp and Shiryaev Theory Probab.
Appl. 39 (1993) 103-119]. First, we consider a capped version of the
Shepp-Shiryaev optimal stopping problem and provide the solution explicitly in
terms of scale functions. In particular, the optimal stopping boundary is
characterised by an ordinary differential equation involving scale functions
and changes according to the path variation of $X$. Secondly, in the spirit of
[Shepp, Shiryaev and Sulem Advances in Finance and Stochastics (2002) 271-284
Springer], we consider a modification of the capped version of the
Shepp-Shiryaev optimal stopping problem in the sense that the decision to stop
has to be made before the process $X$ falls below a given level.
