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

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.