Consider the following model with assets $S_1, S_2$ and three states, and suppose that $r = 10\%$ \begin{array}{|c|c|c|c|} \hline n&S_n(0)& S_n(1,\omega_1) & S_n(1,\omega_2) & S_n(1,\omega_3) \\ \hline 1&4 & 5&6 &3\\ \hline 2&11 &12 & 9&7\\ \hline \end{array}
For clarity, time 1 means one year.
Show that there is a sure-thing arbitrage.
So, what I have done is I tried to find risk-neutral probabilities and I got $\Bbb Q = (\frac{10}{11},\frac{-3}{11},\frac{4}{11})$ which has a negative probability so a risk-neutral probability measure does not exist which means we may not be able to replicate the claims. But I am not quite sure how to construct a portfolio for this question.
A friend suggested a portfolio consisting of 11 units of asset 1 and 4 units of asset 2 and then clearly there is arbitrage between the 2. I am just not quite sure about the reasoning for this.
Note, a portfolio $H$ is a sure-thing arbitrage if value $V_0(H) = 0$ and, for every ω ∈ Ω, $V_T (H, ω) > 0$
Take a long position of $11$ units of $A_1$ and a short position of $4$ units of $A_2$. That costs $0$ initially. Verify that this portfolio has positive value in each of the three scenarios.
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10\%$, whoever loaned you $A_1$ should get a $10\%$ return on the proceeds from the short sale. That comes to $4.4$ here and one can verify that, even with this cost, the position is still a perfect arbitrage. The only close one is the third scenario and in that the portfolio is worth $33-28=5>4.4$.