Superhedging a Claim That Lets the Buyer Choose a Cashflow
Summary
The document asks how to price a contract that gives its buyer the right to select one cashflow from a finite set in a discrete-time, arbitrage-free, complete market. The proposed approach first prices each cashflow by summing the prices of its payments, then takes the largest of those values as the contract price. The accepted response explains why this can fail: in a simple one-period Arrow-Debreu economy, different cashflows can pay in different states, so holding only the hedge for the most expensive individual cashflow may not cover the buyer’s eventual choice.
The response points toward an upper Snell envelope as the relevant pricing concept for constructing a superhedge. This is a brief conceptual answer rather than a worked derivation: it does not specify the envelope recursion, provide a complete hedging strategy, or develop the result across multiple periods. Its central lesson is that comparing stand-alone prices does not by itself establish a hedge for a contract whose holder chooses among claims.
Key ideas
- A contract that lets its buyer select a cashflow requires a hedge for the selection right.
- Pricing each cashflow separately and taking the highest price may not produce a superhedge.
- Different cashflows can pay in different states, requiring coverage across those states.
- The response identifies an upper Snell envelope as a relevant pricing approach but does not derive it.
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Full text
# How to price a set of cashflows from which the buyer can choose one?
# 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?
## Answer by algebruh (score 0, accepted)
https://quant.stackexchange.com/a/67979
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.