How Market Completeness Connects Unique Pricing Measures and Replication

Summary

The document asks whether an equivalent martingale measure guarantees a replicating portfolio for every derivative, then distinguishes that question from the case of a complete market with a unique equivalent martingale measure. It notes that a trinomial discrete-time model can be an incomplete-market example, while familiar binomial and Black–Scholes settings are cited as cases where replication is available.

The included answer points to the second fundamental theorem of asset pricing: under broad conditions, completeness is linked to uniqueness of the equivalent martingale measure. Its intuition is that an unhedgeable derivative can admit multiple arbitrage-free prices, which corresponds to flexibility in the choice of pricing measure. The post offers only this high-level explanation and an external reference, not a proof or the theorem’s precise assumptions. Conclusions therefore depend on the chosen market framework and admissible claims.

Key ideas

• An equivalent martingale measure alone does not guarantee that every derivative can be replicated.
• Market completeness is associated with uniqueness of the equivalent martingale measure under suitable conditions.
• An unreplicable claim can have multiple arbitrage-free prices, suggesting multiple pricing measures.
• The document gives intuition for the second fundamental theorem but omits its assumptions and proof.

Tags

# Under an EMM, does there necessarily exist a replicating portfolio?


# Under an EMM, does there necessarily exist a replicating portfolio?












In general, under an EMM, does there necessarily exist a replicating portfolio for every derivative?

I believe the answer to this is false. A simple example is a discrete time, trinomial model.

However:

In a complete market, i.e. the EMM is unique, does there necessarily exist a replicating portfolio for (at least all European) derivatives?

In the only 2 models I know - being Black-scholes and Binomial model, this is true, but is this true in general?

## Answer by quasi (score 0, accepted)

https://quant.stackexchange.com/a/9887

This is known to be true in very wide generality. In the mathematical finance literature, it is often called the second fundamental theorem of asset pricing. The proof echoes, in some respects the martingale representation theorem. Intuitively, if there are some derivatives that cannot be hedged perfectly, there is some flexibility in arbitrage-free ways of pricing these derivatives, and this leads to the existence of multiple EMM's.

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