Sub-Replication Prices as Minimum Risk-Neutral Values

Summary

The document considers the lower price bound for a claim in an incomplete market, where multiple risk-neutral measures may be consistent with observed prices. It asks whether sub-replication is the counterpart to super-replication: while the super-replication cost is described as the maximum discounted risk-neutral expectation, the proposed lower bound is the minimum across those measures.

The answer gives a duality relation: if p(G) denotes the super-replication price of claim G, then the sub-replication price is −p(−G). It also points to the infimum of discounted risk-neutral expectations of G as the lower bound. This produces a price interval between lower and upper replication bounds. The document offers a concise mathematical pointer rather than a derivation, and does not state the market conditions or technical assumptions needed for the duality result.

Key ideas

• In an incomplete market, multiple risk-neutral measures can imply different valuations for the same claim.
• The sub-replication price of G can be expressed as the negative super-replication price of −G.
• The lower bound is described as the infimum of discounted risk-neutral expectations over admissible measures.
• Super- and sub-replication bounds define an interval for the claim’s price.
• The answer does not detail the assumptions or proof behind the duality.

Tags

# Why sub-replication is not studied in literature


# Why sub-replication is not studied in literature












There are numerous paper about super-hedging and super-replication in an incomplete market where the risk neutral measures are not unique. The most fundamental result is that the super-replication cost equals the maximum of risk neutral expectation of the option. I wonder why no one is studying the dual, sub-replication, and see if it equals the minimum of the risk neutral expectation? Thus we will have an interval for the option price.

## Answer by siou0107 (score 1)

https://quant.stackexchange.com/a/50929

If you denote by $p\left(G\right)$ the super replication of a claim $G$, the sub replication price is simply $-p\left(-G\right)$.

I think I saw that it is the infimum across risk-neutral measures of the (discounted) risk-neutral expectations of $G$ in Bouchard & Chassagneux Fundamentals and Advanced Techniques in Derivatives Hedging. Quite mathematical book.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.