Binomial-market probability density ($L_n=(q/p{)}^{N_n}((1-q)/(1-p){)}^{n-N_n}$)

With independent up moves of physical probability $p$ and risk-neutral probability $q$, the Radon-Nikodym derivative on the first $n$ periods is the displayed likelihood ratio, where $N_n$ counts up moves. Its conditional one-step multiplier has physical expectation one, so $L_k$ is a positive mean-one martingale. The terminal state-price density is $L_n/B_n$ for deterministic bank-account value $B_n$.