Given $\Omega=\{\omega_1,...,\omega_4\}$ and a probability measure $\mathbb{P}$ on $(\Omega, \mathcal{P}(\Omega))$ where $\mathbb{P}(\{\omega_i\})>0$ for all $i$. Let, furthermore, $r\geq 0$, $S_0=5$ and

$S_1(\omega_1)=S_1(\omega_2)=8$, $S_1(\omega_3)=S_1(\omega_4)=4$,

$S_2(\omega_1)=9$, $S_2(\omega_2)=S_2(\omega_3)=6$, $S_2(\omega_4)=3$.

For which interest rates $r$ is the model arbitrage-free?

I looked at the one-period arbitrage possibilities and got $r\in [0,1/3)$. I'd appreciate it if someone could double-check it since my textbook gives no solutions.

  • 1
    $\begingroup$ If it is a 2 period problem, aren't there going to be 2 interest rates, i.e. $r$ changes over time? Don't we need to look at them separately? $\endgroup$
    – nbbo2
    Sep 11 at 6:53
  • $\begingroup$ Maybe you could show us your analysis for the 1st period and we can check whether it makes sense or not. $\endgroup$
    – nbbo2
    Sep 11 at 7:02
  • 2
    $\begingroup$ @nbbo2 I think the situation is even worse than that -- might be a question of finding the set of rates that work at time 0, time 1 if the stock goes up, and time 1 if the stock goes down $\endgroup$
    – Rylan
    Sep 11 at 10:43
  • $\begingroup$ $S_t^0=(1+r)^t$., so $1,1+r,(1+r)^2$. $\endgroup$
    – Analysis
    Sep 11 at 11:03
  • $\begingroup$ Where should I post my work? $\endgroup$
    – Analysis
    Sep 11 at 11:05


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