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How the Fundamental Theorem of Asset Pricing Links Arbitrage and Completeness

Article Quant Q&A · Author: SBF
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Summary

This post asks how the Fundamental Theorem of Asset Pricing connects absence of arbitrage with positive extensions of a pricing functional, and how uniqueness of such an extension relates to market completeness. It frames traded payoffs as initial cash plus gains from trading and asks whether a unique extension means every payoff can be replicated and priced uniquely.

The post does not resolve these questions. Its answer points readers to a rigorous, master’s-level proof of the theorem, but gives no details of the proof or its assumptions. The discussion is therefore useful as a guide to the conceptual issues—pricing consistency, equivalent martingale measures, arbitrage, and completeness—rather than as a standalone explanation. The precise equivalences depend on the mathematical setting and assumptions, which are not developed in the post.

Key ideas

  • The post frames arbitrage-free pricing through positive extensions of a pricing functional.
  • It asks whether uniqueness of an extension implies market completeness and payoff replication.
  • It distinguishes a pricing functional defined on traded payoffs from an extension to a larger payoff space.
  • The cited answer refers readers to a proof but does not explain the theorem or its assumptions.

Tags

ArbitrageStatisticsDerivatives pricing
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# FTAP a-la Harrison, Kreps and Pliska


# FTAP a-la Harrison, Kreps and Pliska












I was reading the papers co-authored by Harrison, Kreps and Pliska, that initiated the formal research on the connection between pricing, martingale measures, arbitrage and completeness. I have some issues relating some of the ideas, though, and would be happy if somebody could help me.

In a more or less abstract setting, one considers an ordered topological space $X$ of cash flows, a subspace $M\subseteq X$ of traded instruments and a positive linear functional $\pi:M\to \Bbb R$. The 1st FTAP that relates absence of arbitrage to the existence of EMM concerns continuous extension of $\pi$ to $X$ which is still positive. As an example, one can consider $$ M = \left\{m = m_0 + \int_0^T H_t\;\mathrm dS_t \right\} $$ where $H$ is some "good" strategy and $S$ is a vector of traded assets. In such case, one defines $\pi(m) = m_0$ as an initial cash endowment required to get $m$ at the maturity time $T$.

I guess, the existence of a unique positive extension of $\pi$ to the whole $X$ implies the completeness of the market: that is, we can price everything in a consistent way and there is a unique way to do this. Apparently, this automatically leads to the representability of any $x\in X$ as $x_0 + \int_0^T H_t \mathrm dS_t$.

At the same time, I do not understand, how the existence of at least one continuous positive extension may be related to the absence of arbitrage. That is, I would imagine an arbitrage being rather existence of two strategies $H'$ and $H''$ such that $$ m'_0 + \int_0^T H'_t\;\mathrm dS_t = m''_0 + \int_0^T H''_t\;\mathrm dS_t $$ with $m'_0 \neq m''_0$. But this rather says $\pi$ is not uniquely defined even on $M$ instead of it is not possible to extend $\pi$ to $X$. Does that mean that the latter two statements are equivalent?

## Answer by Keith A. Lewis (score 1)

https://quant.stackexchange.com/a/15693

See http://kalx.net/ftapd.pdf for a rigorous proof of the FTAP accessible at the masters level.

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