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If no-arbitrage exists, then the law of one price holds, but the existence of the law of one price does not always imply that no-arbitrage exists." To prove this, what is an example where the law of one price holds, but no-arbitrage does not? Additionally, how is this concept applied in financial engineering? And what resources would be beneficial for a mathematical proof of this concept? Thank you.

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A simple textbook example is the following. Consider a discrete market with two assets and one time step: $B_0=-0.1$, $B_1(\omega)=1 \;\forall\omega\in\Omega$ and $S_0=-0.2$ and $S_1(\omega)=2\;\forall\omega\in\Omega$. Here, $\Omega$ is the set of all outcomes. Admittedly, a simple example with $S_t=2B_t$...

Any portfolio in this market can be characterised by $\varphi=(\varphi^B,\varphi^S)$. The value process of that portfolio is \begin{align} V_0(\varphi) &= -0.1\varphi^B - 0.2\varphi^S, \\ V_1(\varphi) &= \varphi^B+2\varphi^S=-10V_0. \end{align}

This market obviously contains arbitrages. Borrow money to buy the stock, get some money now and wait one period to be even richer.

However, you'll find no violation of LOP in this market. To see this, suppose you have two self-financing trading strategies $\varphi,\psi$ with $V_1(\varphi)=V_1(\psi)$. LOP requires that $V_0(\varphi)=V_0(\psi)$, which is indeed true because $V_1(\varphi)=-10V_0(\varphi)$ and $V_1(\psi)=-10V_0(\psi)$.

While academics distinguish Law of One Price, No Arbitrage, No Free Lunch with Vanishing Risk, No Free Lunch with Bounded Risk, and No Free Lunch properties, it probably matters little in real life. Those differences are kind of technical and the concepts are almost surely identical for all practical purposes.

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