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

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.

• 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? Sep 11 at 6:53
• Maybe you could show us your analysis for the 1st period and we can check whether it makes sense or not. Sep 11 at 7:02
• @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 Sep 11 at 10:43
• $S_t^0=(1+r)^t$., so $1,1+r,(1+r)^2$. Sep 11 at 11:03
• Where should I post my work? Sep 11 at 11:05