Two No-Arbitrage Definitions and Their Equivalence

Summary

The document compares two mathematical formulations of no arbitrage in a market modeled by a multidimensional price process. In the first, a self-financing strategy is an arbitrage when it starts with nonpositive value, ends with nonnegative value almost surely, and has a strictly positive terminal payoff with positive probability. The second defines a set of attainable terminal values and the bounded claims that can be dominated by those values; no arbitrage means this set contains no nonzero nonnegative claim.

It asks whether the definitions are equivalent but provides no proof or answer. The formulations involve different conditions on strategies and terminal claims, so equivalence depends on the precise assumptions about admissible strategies, self-financing, and the relevant function spaces. Those assumptions are not fully laid out in the document. It therefore serves as a theoretical question about arbitrage definitions rather than a completed argument or a practical trading method.

Key ideas

• One definition identifies arbitrage through initial and terminal portfolio values.
• The other rules out nonzero nonnegative bounded claims that can be super-replicated.
• Both formulations express a no-arbitrage condition using terminal payoffs.
• The document asks about equivalence but does not establish it.
• Comparing the definitions requires care about admissible strategies and mathematical assumptions.

Tags

# Different definitions of arbitrage


# Different definitions of arbitrage












Consider the following setup: Let $S=\left(S_1,\ldots,S_n\right)$ be a $n$-dimensional price process and denote by $V$ its value process defined by $V_t=\phi_t\dot\ S_t$ for $t=0,\ldots,T$. In "Stochastic Finance" by Föllmer and Schweizer, we have the following definition for arbitrage in chapter 5:

> Definition: A self-financing trading strategy $\phi$ is called an arbitrage opportunity if its value process V satisfies $V_0\leq 0$ a.s., $V_T\geq 0$ a.s. and $P(V_T>0)>0$. If there is no arbitrage opportunity, then the financial market is said to satisfy NA.

In the book "The mathematics of arbitrage" by Schachermayer and Delbaen, we have the following definition of arbitrage in chapter 2:

> Definition: The set $K=\{(\phi\dot\ S)_T|\phi\in\mathcal{H}\}$, where $\mathcal{H}$ is the set of trading strategies. Define $C=\{g\in L^\infty(\Omega,\mathcal{F},P)|\exists f\in K\ f\geq g \}$ as the set of contingent claims super-replicable at time T. A financial market satisfy the (NA) (no arbitrage), if $$C\cap L^\infty_{\geq 0}=\{0\}$$

Is it possible to prove the equivalence of these definitions?

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