Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 30K / b / Solution

If every increment $\xi_n=M_n-M_{n-1}$ of a discrete martingale takes values in $\{-1,1\}$, then
$$
\mathbb P(\xi_n=1\mid\mathcal F_{n-1})
=\mathbb P(\xi_n=-1\mid\mathcal F_{n-1})=\frac12.
$$
Iterated conditioning shows that the increments are independent symmetric signs, so $M_n-M_0$ is a <simple symmetric random walk>.

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 Incoming links: Symmetric signs forced by the martingale property