From a Nonnegative Payoff to an Admissible Arbitrage

Summary

This post asks about a step in a proof from mathematical finance: whether a self-financing strategy with zero initial value, a nonnegative terminal value, and positive expected terminal value implies the existence of an arbitrage. The point of confusion is a statement that, if the strategy is not admissible, there must be a time and event on which its value is negative and remains nonpositive at all later times. The author questions how that condition can coexist with a nonnegative terminal payoff and wonders whether the inequality may be a typo.

The document defines admissibility through nonnegative portfolio value over time and notes that a strategy satisfying this condition would itself qualify as an arbitrage under the stated lemma. It does not provide the resolution or the proof’s construction of a new admissible strategy. Its value is therefore as a focused question about the role of pathwise constraints, stopping or continuation arguments, and self-financing in arbitrage proofs, rather than as a complete derivation. The issue is framed in a continuous-trading mathematical setting.

Key ideas

• The lemma connects a zero-cost self-financing strategy with a nonnegative terminal payoff and positive expected value to arbitrage existence.
• Admissibility is defined as keeping portfolio value nonnegative over time.
• The question concerns how to handle a strategy that violates admissibility at intermediate times.
• The author highlights a potential tension between later nonpositive values and a nonnegative terminal payoff.
• The post raises the proof issue but does not supply its resolution.

Tags

# Question about proving the existence of an arbitrage opportunity


# Question about proving the existence of an arbitrage opportunity












I am having a hard time understanding the reasoning behind a statement in the proof of the following lemma from page 14 (228) of the paper "Martingales and stochastic integrals in the theory of continuous trading" by Harrison and Pliska in 1981.

Lemma: If there exists a self-financing strategy $\phi$ (not necessarily admissible) with $V_{0}(\phi)=0, V_{T}(\phi)\geq 0$ and $\mathbb{E}[V_{T}(\phi)]>0 $, then there exists an arbitrage opportunity.

Here the admissible stands for $V(\phi)\geq 0$, $\phi,S\in\mathbb{R}^n $ and $V_t(\phi)=\phi_t\cdot S_t$ which is a scalar product.

Proof: If $V(\phi)\geq 0$, then $\phi$ is admissible and hence is an arbitrage opportunity itself, and we are done. Otherwise there must exist $t<T,A\in\mathcal{F_t}$ such that $\phi_t\cdot S_t=a<0$ on $A$ and $\phi_u\cdot S_u\leq 0$ on $A$ for all $u>t$.

Here $\phi_t\in\mathcal{F_{t-1}}$ and $ S_t \in \mathcal{F_{t}}$

What I don't understand is why all values after a given time $t$ have to be non-positive.

My intuition tells me that I should have some which are non-positive, but not necessarily consecutively. And also the statement contradicts $ V_{T}(\phi)\geq 0$ in my opinion.

With this, I want to create a new strategy which is both self financing and admissible. The problem with taking $t$ non-consecutive is that the new strategy doesn't satisfy either the admissibility condition or the self-financing condition.

I also suspect that $\phi_u\cdot S_u\leq 0$ is a typo and it should be $\phi_u\cdot S_u\geq 0$.

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