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

A short note on super-hedging an arbitrary number of European options with integer-valued strategies