Separating Hyperplanes and the No-Arbitrage Condition

Summary

The document concerns a proof in a single-period mathematical finance setting: how a separating hyperplane theorem supports the connection between no-arbitrage and the existence of a risk-neutral probability measure. The question references normalized prices and a stated no-arbitrage equivalence, but does not reproduce the book’s full setup or specify the exact version of the theorem being applied.

The responses offer only a brief geometric hint. One suggests choosing a vector orthogonal to the separating plane and oriented so that its inner product with each relevant vector is positive. This points toward how separation can yield a strictly positive pricing direction, a key ingredient in the finite-dimensional argument. However, the document does not show the sets being separated, establish the required conditions, or derive the probability measure. It is therefore useful as a pointer to the proof’s geometric idea, but insufficient as a standalone explanation; readers need the cited textbook context or a fuller treatment of the theorem and its assumptions.

Key ideas

• The question links a single-period no-arbitrage condition to the existence of a risk-neutral probability measure.
• A separating hyperplane provides a geometric route to a positive pricing direction.
• The response suggests using a vector orthogonal to the separating plane with the appropriate orientation.
• The document omits the full proof, its assumptions, and the construction of the probability measure.

Tags

# How does this follow from the separating hyperplane theorem?


# How does this follow from the separating hyperplane theorem?












This is from Pliskas book in mathematical finance. I do not know what was best to write the question so I included the pages from the book. He has not written what form of the separating hyperplane theorem he uses, and why this follows. If someone understands it, could you please explain it? I have outlined in red the sentence I do not understand.

They refer to 1.3 which is: $V_t^*=V_t^*/B_t$, $t=0,1$

and 1.16: There are no arbitrage oppurtunities if and only if there exists a risk neutral probability measre on Q.

He also uses a single period model.

## Answer by emcor (score 2)

https://quant.stackexchange.com/a/14492

In our lecture, we were told to omit the proof because it was too difficult. Maybe it will help you though if you can read it here:

## Answer by Lucas Morin (score 0)

https://quant.stackexchange.com/a/14490

After applying the theorem you can take an $Y$ orthogonal to the separation plane, pointing in the right direction, you will easily have the property $X.Y>0$ for all $X$.

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