# Example of one-period model that satisfies law of one price but is not free of arbitrage

We know that by the law of one price: in a one-period model $$(\overline{\pi},\overline{S})$$ for an arbitrage-free market model it follows that for two strategies $$\overline{\rho}$$ and $$\overline{\xi}\in \mathbb R^{d+1}$$: $$\overline{S}\cdot\overline{\xi}=\overline{S}\cdot\overline{\rho}\; \; \mathbb P-\text{a.s.}\implies \overline{\pi}\cdot\overline{\xi}=\overline{\pi}\cdot\overline{\rho}\; \; \mathbb P-\text{a.s.}$$

Question: Find a market model that is not arbitrage-free but still obeys the law of one price.

My idea:

Consider the space $$\Omega:=\{\omega_{0},\omega_{1}\}$$, with $$\mathbb P(\{\omega_{i}\})>0,\; i =0,1$$ and two assets (one risk-free, the other risky) such that $$S^{1}(\omega_{0})=0$$ and $$S^{1}(\omega_{1})=1$$. It can be proven that for $$r>0$$ as the interest rate on the risk-free asset, i.e. $$(\pi^{0},S^{0})=(1,1+r)$$ that the model is arbitrage free if and only if:

$$S^{1}(\omega_{0})=0< \pi^{1}(1+r)<1=S^{1}(\omega_{1})$$

Now consider two strategies $$\overline{\rho}=(\rho^{0},\rho^{1})$$ and $$\overline{\xi}=(\xi^{0},\xi^{1})$$ such that

$$\overline{S}\cdot\overline{\xi}=\overline{S}\cdot\overline{\rho}\; \; \mathbb P-\text{a.s.}$$ i.e.

on $$\{ \omega_{1}\}$$: $$(1+r)\xi^{0}+\xi^{1}=(1+r)\rho^{0}+\rho^{1}$$

and on $$\{ \omega_{0}\}$$: $$(1+r)\xi^{0}=(1+r)\rho^{0} \implies \xi^{0}=\rho^{0}$$

and thus $$\xi^{1}=\rho^{1}$$. Thus fundamental idea here is that we have not had to place any restrictions on the risk-free rate, $$r >0$$. Thus for a fixed $$\pi^{1}$$ we choose $$r> \max\{\frac{1}{\pi^{1}}-1,\varepsilon\}$$ for $$\varepsilon >0$$ such that $$\pi^{1}(1+r)> 1$$.

Therefore $$\overline{\xi}=\overline{\rho}$$ and thus trivially $$\overline{\pi}\cdot\overline{\xi}=\overline{\pi}\cdot\overline{\rho}$$. But the market model admits arbitrage.

I feel like I may be "cheating" by setting $$S^{1}(\omega_{0})=0$$ since aren't are asset prices assumed positive (does this mean strictly positive?). Is this indeed cheating? Are there other examples where the prices are strictly positive?

Thanks for the help!