Martingale constructed from a random walk

Question

I am trying to solve this problem for a while now but I am not coming to a solution. Could anyone help or give me a hint?

Let $S_n=\sum_{i=1}^{n}X_i$ be a random walk on $\mathbb{Z}$ with $S_0=0$, $\mathbb{P}(X_i=1)=\frac{2}{3}$, $\mathbb{P}(X_i=-1)=\frac{1}{3}$. For which strictly positive constant $c\neq1$ is $M_n:= c^{S_n}$ a martingale?

Answer 1

Here is how you get it. So, note that $$ E(c^{S_{n+1}}\mid \mathcal F_n) = E(c^{S_n}\cdot c^{X_{n+1}} \mid \mathcal F_n)= c^{S_n} E(c^{X_{n+1}})= c^{S_{n}} $$

Therefore, you need that $E(c^{X_{n+1}})=1$. Hence,

$$ E(c^{X_{n+1}})= c2/3+1/(3c)= 1 $$

Solve the above equation and find $c$.