Show that $z^{Sn}$ gives a martingale and for which z

Question

I'm stuck with this problem

Consider a game where you bet on 1 each time. With probability $p= 1/2$ you win $1$ and with probability $1-p$ you lose $1$. Let $S_n$ be your fortune after n games. Find for what values of z, such that the process $z^{Sn}$ is a martingale.

So here I know that I can say that $S_{n+1} = S_n + Y_{n+1}$ where $Y_i$ are i.i.d and $Y_i = 1$ with probability $p$ and $-1$ with probability $q = 1-p$

Logically, in order to be a martingale I need to find the value of z such that:

So I find that:

Using independence and $F_n$-measurability.

I know from the book that:

But how do they found that result using the information about $Y_i$ ?

I know how to compute a normal expected value but I must admit that this exponent is really blocking me. Any help will be welcomingly accepted !

Answer 1

We have $$ \mathbb{E}\left[z^{Y_{n+1}}\right] = P(Y=1)\cdot z^{(Y=1)} + P(Y=-1)\cdot z^{(Y=-1)} = pz + (1-p)z^{-1} = pz + q/z $$ To solve the full problem follow the comment by Curious above.