Building martingales from mean-zero, unit-variance increments

Let $\{Y_n\}$ be a sequence of independent random variables with mean zero and variance 1. Let $\mathbf F_n=\sigma(Y_1,\dots,Y_n)$. Consider the statements: (I) $X_n=Y_1+Y_2+\dots+Y_n$ is a $\mathbf F_n$-martingale. (II) $X_n=\prod_{k=1}^nY_k$ is a $\mathbf F_n$-martingale. (III) $X_n=(\sum_{k=1}^nY_k)^2-n$ is a $\mathbf F_n$-martingale.