Martingale under risk neutral probability

Question

I have a question to prove martingale under risk-neutral measure:

Question

Consider a discrete time process $S$, which at time $n \in \mathbb{N}$ has value $S_n = S_{0}\prod_{j=1}^{n}Z_j$, $S_0>0$ and $Z_j$ a sequence of independent and identically distributed random variables to either 1.15 or 0.85. Assume a zero risk free rate of rate.

Show that under the risk neutral martingale measure $\mathbb{\hat{P}}$ the process $S_n$ is a martingale.

My thoughts

1. I do understand that under the risk neutral measure $\mathbb{\hat{P}}$, which is that $e^{-rt}C_t = \mathbb{\hat{E}}[e^{-rt}C_T|\mathbb{F}_t]$.
2. I understand that 1 - period $\mathbb{\hat{P}}$ is that:
   • $\mathbb{\hat{P}}_u = \frac{1-0.85}{1.15-0.85}=0.5$, and so is $\mathbb{\hat{P}}_d$ = 0.5.
3. I think I seek to prove the martingale property of $S_n$,but I am not sure how is that related to the $\mathbb{\hat{P}}$ if I just want to prove that:
   • $\mathbb{E}|S_n|< \infty$
   • $\mathbb{E}[S_{n+1}|\mathbb{F}_t]=S_n$

Answer 1

You already have it. Risk neutral measure is one where tosses are still independent but each individual toss has probability of 0.5 up or down.

Say you're at step n, with $S_{n}$ known.

$E(S_{n+1}|F_{n})=S_{n}*(0.5*1.15+0.5*0.85)=S_{n}$

and unconditional expectation is also finite using tower law as it equals $S0$.