Why Constrained Markets Use a Local Scalability Condition

Summary

The document asks about a condition in a mathematical definition of a constrained market: sufficiently small nonzero cash-flow positions can be scaled up to a specified size while remaining feasible. The answer says this condition is included mainly to make proofs easier, particularly the Fundamental Theorem of Asset Pricing for constrained markets. It notes that related results have been established for conic constraints, which allow feasible positions to scale without the same local formulation.

The discussion offers a brief conceptual explanation rather than a derivation or worked example. It identifies missing-market and short-sale restrictions as examples of constraints, and points to research extending the theorem to more general limits on asset holdings. The answer does not explain the exact role of the condition in the proof or compare the assumptions and conclusions of the cited results, so readers seeking a formal justification would need to consult the referenced mathematical literature.

Key ideas

• The definition requires small feasible cash-flow positions to remain feasible when scaled to a fixed size.
• The answer presents this condition as a technical assumption that simplifies asset-pricing proofs.
• The Fundamental Theorem of Asset Pricing is discussed in the setting of constrained markets.
• Related work extends the theorem to conic constraints on investor holdings.

Tags

# What's the underlying idea of definition of constrained market in Skiadas' Asset Pricing Theory?


# What's the underlying idea of definition of constrained market in Skiadas' Asset Pricing Theory?












I'm self-studying Skiadas' Asset Pricing Theory, and find the definition of constrained market on page 21 confusing(you can find it here in the sample chapter).

> Definition 1.26. A constrained market is a closed convex set of cash flows $X \subseteq \Bbb R^{1+K}$ such that $0 \in X$ and for some $\epsilon > 0$, $x \in X$ and $0 < \| x \| <\epsilon$ implies $\frac{\epsilon}{\| x \|}x \in X.$

I know this definition renders missing market and short-sale constraints as special cases, but the underlying idea of this formulation still eludes me.

## Answer by pbr142 (score 3)

https://quant.stackexchange.com/a/10495

I have asked myself the very same question when I first read the book. As far as I can tell, the "scalability" condition is only imposed for technical reasons. It simplifies the subsequent proof of the Fundemental Theorem of Asset Pricing in constrained markets.

There are several papers that have shown that the theorem is valid for conic constraints. Examples are Napp or Pham & Touzi. The main result from Napp shows what steps are necessary to obtain the theorem with general conic constraints on the amount that investors are able to hold of each asset.

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