  For a L\'evy process $X$ on a finite time interval consider the probability
that it exceeds some fixed threshold $x>0$ while staying below $x$ at the
points of a regular grid. We establish exact asymptotic behavior of this
probability as the number of grid points tends to infinity. We assume that $X$
has a zooming-in limit, which necessarily is $1/\alpha$-self-similar L\'evy
process with $\alpha\in(0,2]$, and restrict to $\alpha>1$. Moreover, the
moments of the difference of the supremum and the maximum over the grid points
are analyzed and their asymptotic behavior is derived. It is also shown that
the zooming-in assumption implies certain regularity properties of the ladder
process, and the decay rate of the left tail of the supremum distribution is
determined.
