Skip to content
All library documents

Compactness of Nonnegative Payoff Portfolios in an Arbitrage-Free Market

Article Quant Q&A · Author: StefanH
Read the original

Summary

The document poses a mathematical finance question about a market with one risk-free asset and several risky assets. It defines arbitrage in terms of a portfolio with nonpositive initial cost and a nonnegative payoff that is sometimes positive, and defines non-redundancy as the absence of a nonzero portfolio with an identically zero payoff. The question asks whether portfolios with a fixed positive price and nonnegative payoffs form a compact set under these assumptions.

The text suggests proving compactness by showing the set is closed and bounded, but it does not provide a proof. It also asks whether non-redundancy alone implies absence of arbitrage; this is posed as a conjecture rather than established. The discussion is therefore a theorem-proving prompt, not a completed result. Any proof would depend on the stated market assumptions and definitions, which the document does not further qualify.

Key ideas

  • The market model includes a risk-free asset alongside risky assets and assigns a price vector to portfolios.
  • Non-redundancy rules out nonzero portfolios whose payoff is almost surely zero.
  • The question concerns compactness of portfolios with a fixed positive price and nonnegative payoff.
  • In finite-dimensional Euclidean space, compactness can be shown through closedness and boundedness.
  • The document does not provide a proof or establish whether non-redundancy implies absence of arbitrage.

Tags

StatisticsPortfolio constructionRisk management
Full text
# Show that in an arbitrage-free and non-redundant market a certain set is compact


# Show that in an arbitrage-free and non-redundant market a certain set is compact












Some notation: We consider a financial market with $d+1$ assets, the $0$-th asset is considered the risk-free asset, the others are the risky ones. The vector $\overline \pi \in \mathbb R^{d+1}$ denotes the pricing vectors, and the random vector $\overline S = (S^0,S^1,\ldots, S^d)$ denotes the vector of the random variables corresponding to the assets. Further we have $\pi^0 = 1, S^0 \equiv 1 + r$, as these are risk-free.

Here a portfolio $\overline \xi \in \mathbb R^{d+1}$ is called an arbitrage opportunity if $\overline \xi \cdot \overline \pi \le 0$ but $\overline \xi \cdot \overline S \ge 0$ P-a.s. and $P(\overline \xi \cdot \overline S > 0) > 0$. And the market model is called non-redundant if $$ \overline \xi \cdot \overline S = 0 ~\mbox{P-a.s.} \Rightarrow \overline \xi = 0 $$ i.e. non of the assets $S^i$ could be synthezied by investing in the others.

> Show that in a non-redundant and arbitrage-free market model the set $$ \{ \overline \xi \in \mathbb R^{d+1} : | \overline \pi \cdot \overline \xi = w \mbox{ and } \overline \xi \cdot \overline S \ge 0 ~ \mbox{P-a.s.} \} $$ is compact for any $w > 0$.

First, as this is a subset of the euclidean space $\mathbb R^{d+1}$, compactness is equivalent with bounded and closed. But already showing boundedness gives me a hard time, I do not see where to start?

Also, I guess a non-redundant market model also implies that it is arbitrage-free, so the presupposition of just arbitrage-free'ness should be enough.

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.