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:





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?

  • $\begingroup$ Suppose we ignore the third asset. Then we can solve the system and get $q_1 = q_2 =\frac{1}{2}$ and $q_3=0$. Thus even then we do not have an EMM. Can you find the arbitrage opportunity in this case? $\endgroup$ – Borun Chowdhury Mar 27 '16 at 9:03
  • $\begingroup$ You don't mention the existence of a risk-free asset. Are we to understand that there is no such asset? If there is one, the answer is trivial. If there isn't any, then what does the value of 1 represent exactly? $\endgroup$ – ocstl Mar 27 '16 at 15:42
  • 1
    $\begingroup$ Risk free asset is the one with zer interest rate. $\endgroup$ – Borun Chowdhury Mar 28 '16 at 16:05

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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.