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

Question

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?

Answer 1

Ok, this falls apart rather quickly. Consider a simple one period arrow-debreu economy. If i have two different states and their respective arrow-debreu assets then I can only super hedge the above deal if I buy both assets, not only the more expensive one.

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