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Arbitrage, Optimal Portfolios, and Investor Budget Sets

Article Quant Q&A · Author: Moeo
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Summary

The document presents an informal argument linking arbitrage to the existence of an investor’s optimal consumption and trading choice. It says that if an optimal strategy exists and utility is strictly increasing, adding an affordable arbitrage that delivers extra consumption would improve utility while preserving the budget constraint. That would contradict optimality, so an optimum rules out such an arbitrage under the stated setup.

It also sketches the converse: absence of arbitrage is said to imply a state-price vector, which supports asset prices and a convex budget set; with concave utility, an optimum is then claimed to exist. This second direction is only outlined and omits conditions needed for existence, such as assumptions on the feasible set, preferences, and market structure. The question mentions an associative ray, but the answer does not explain that concept or directly connect it to the proof in question.

Key ideas

  • A strictly improving arbitrage is incompatible with an existing optimum under strictly increasing utility.
  • The argument assumes the arbitrage can be added without violating the investor’s budget constraint.
  • The answer sketches a link from no arbitrage to state prices and a convex budget set.
  • The claimed existence result needs additional assumptions that the short proof does not state.

Tags

ArbitragePortfolio constructionRisk management
Full text
# Please explain this proof for me: (arbitrage and bounded set)


# Please explain this proof for me: (arbitrage and bounded set)












Consider this problem and subsequent proposition:

Part of the proof of this proposition is given here:

Could somebody please explain to me why the existence of the "associative ray" (which I have never heard of before) means there's an arbitrage? I've highlighted that part of the proof in red.

## Answer by phdstudent (score 1)

https://quant.stackexchange.com/a/39412

Let me proof that theorem in a different way which might help:

Theorem: There is a solution to the agents' optimization problem iff there are no arbitrage opportunities.

i) Proving the first if: Existence of optimal strategy implies no arbitrage

- Suppose that $\theta^\star$ os an optimal trading strategy for the agent with optimal consumption allocation of $c^\star$. Now assume there is an arbitrage opportunity $\theta^{arb}$ - i.e. without the investment of any initial endowment it yields a consumption bundle $c^{arb} > 0 $. This implies that $\theta^\star + \theta^{arb}$ yields $c^\star + c^{arb} > c^{\star}$. Since $U$ is stricty increasing and $\theta^\star + \theta^{arb}$ satisfies the budget constraint it implies that the initial pair $(\theta^\star, c^\star$) is a solution to the investor's problem.

(ii) Proving the second if: nor arbitrage implies a solution to the consumer problem:

- If there are no arbitrage opportunities then a state price vector must exist. As a result asset prices exist (and are unique) and therefore the investors budget set convex. Since the utility is concave we have a solution to the optimization problem.

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.