What is the arbitrage opportunity in this simple one-period market?

Question

I have a single period market, and three states, and I have 3 risky assets. I assume no interest.

So I have three states $\Omega=\{\omega_1,\omega_2,\omega_3\}$. All assets start with the value 1, and for each n the price of risky asset n, is 2 in $\omega_n$ and the other assets have value 0. That is, the price function is:

$S_1(0)=1,S_2(0)=1,S_3(0)=1$

$S_1(1,\omega_1)=2,S_1(1,\omega_2)=0,S_1(1,\omega_3)=0$

$S_2(1,\omega_1)=0,S_2(1,\omega_2)=2,S_2(1,\omega_3)=0$

$S_3(1,\omega_1)=0,S_3(1,\omega_2)=0,S_3(1,\omega_3)=2$

Now, the theory says that there is no arbitrage iff there exists a risk neutral probability measure.

From what I see there is no risk neutral probability measure, because then we would have to have:

$\begin{bmatrix}2 &0 & 0 \\0&2 & 0\\0 &0 &2 \\ 1&1&1\end{bmatrix}\begin{bmatrix}q_1\\ q_2 \\q_3\end{bmatrix}=\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$.

And this set of equations have no solutions, hence there should exist an arbitrage opportunity?

However, I can't find a strategy which guarantees money, I have tried to find it numerically, but wasn't able to find one. Do you see the strategy or have I made mistake somewhere?

Answer 1

Sell 1 unit of S1,2,3 respectively, gain 3; buy 2 units of risk-free asset, cost 2.

No matter which state appears, the future payoff/loss is 0 for sure, while you will gain 1 at the beginning.

Answer 2

I have not studied this, but intuitively...

If all of the states had equal probability of occurring, then the expectation of each asset $E[S_n] = \frac{2}{3}$.
Imagine instead the probability of $\omega_1$ was $0.5$, with the other two $0.25$.
Now $E[S_1] = 1, E[S_2]=E[S_3]=\frac{1}{3}$.

It seems to me, irrespective of the probabilities of the states, the expectation of the equal weighted market as a whole is 2/3.

From wikipedia for Risk Neutral Measure:

A risk neutral measure is a probability measure Q on the set of states such that there exists an r (the "discount rate") with
$q_i = rE_Q[D_i]$ for each $i$ where $D_i$ [...] represents the payoff of security $i$

In this case there does exist an $r$, and maybe it equals $\frac{2}{3}$.

If we set if $q_1 = q_2 = q_3 = \frac{1}{2}$ then this all appears to work.

With respect to arbitrage, well, given their -ve expectation, presumably you can short the stocks, and have a positive expectation? For example sell 1 share of each of $S_1, S_2, S_3$ and keep the proceeds ($3$) at zero interest. In the second period you can buy back (in all states) for $2$. Making free money.