1
$\begingroup$

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.

$\endgroup$
6
  • 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
    Commented Sep 11, 2023 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
    Commented Sep 11, 2023 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
    Commented Sep 11, 2023 at 10:43
  • $\begingroup$ $S_t^0=(1+r)^t$., so $1,1+r,(1+r)^2$. $\endgroup$
    – Analysis
    Commented Sep 11, 2023 at 11:03
  • $\begingroup$ Where should I post my work? $\endgroup$
    – Analysis
    Commented Sep 11, 2023 at 11:05

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.