# How to price a set of cashflows from which the buyer can choose one?

Lets consider an arbitrage free and complete Model.Let also focus the analysis on the discrete time setting.Assume you have a finite set of random Cashflows $$\mathcal{A}$$. That means all elements of $$\mathcal{A}$$ are adapted to the filtration of the market. Note that those are not european derivatives. Now if I sell someone such a set with the condition that the buyer can only choose one of those. How would I price that? My idea was that if we take a cashflow $$\mathcal{A}\ni A = (A_{t_1},\ldots, A_{t_n})$$ we can replicate it by replicating the payoffs for the respective times. Since we are in a complete arbitrage-free model, for all $$A_{t_i}$$ there is a selffinancing replicating strategy with initiial costs $$p_i$$ which is the arbitrage free price. Which also equals to $$\mathbb{E}_{\mathbb{Q}}[\frac{A_{t_i}}{B_{t_i}}]$$ where $$B$$ is the numeraire. Then the arbitragefree price $$p_A$$ of the whole cashflow would be the sum of the single payoffs. That is $$p_A = \sum_i p_{i}$$. I then thought that the folowing price for the whole deal would be arbitragefree

$$p_{\mathcal{A}}= \max_{A\in \mathcal{A}}p_A$$

However I am not sure how to set up a (super)-hedging strategy with this money. Is it possible? if yes/no, why?

Thus we need some kind of upper snell envelope for $$\mathcal{A}$$ to price it.