  The usual theory of asset pricing in finance assumes that the financial
strategies, i.e. the quantity of risky assets to invest, are real-valued so
that they are not integer-valued in general, see the Black and Scholes model
for instance. This is clearly contrary to what it is possible to do in the real
world. Surprisingly, it seems that there is no many contributions in that
direction in the literature, except for a finite number of states. In this
paper, for arbitrary {\Omega}, we show that, in discrete-time, it is possible
to evaluate the minimal super-hedging price when we restrict ourselves to
integer-valued strategies. To do so, we only consider terminal claims that are
continuous piecewise affine functions of the underlying asset. We formulate a
dynamic programming principle that can be directly implemented on an historical
data and which also provides the optimal integer-valued strategy. The problem
with general payoffs remains open but should be solved with the same approach.
