# Martingale under risk neutral probability

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:
• $$\mathbb{E}|S_n|< \infty$$
• $$\mathbb{E}[S_{n+1}|\mathbb{F}_t]=S_n$$

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 $$S0$$.