# An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price will at some point dip below 40 and then subsequently rise above 60 before time $N$. How do you prove that $S(k)$ cannot be a martingale?