# The relationship between no-arbitrage and the law of one price

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.

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}
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 $$V_1(\psi)=-10V_0(\psi)$$.