Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 29K / b / Solution

Under a risk-neutral measure, the discounted stock must be a martingale. If $q$ is the probability of the factor $1+b$, this condition issoThe hypothesis $a<r<b$ ensures $0<q<1$, making this equivalent to the original probability measure. Since the $N$ returns are independent under $Q$, the number of up moves has a binomial distribution, and hence