Risk-neutral probability in the Cox--Ross--Rubinstein model

The unique equivalent martingale measure in the one-stock Cox--Ross--Rubinstein model assigns each up move probabilityand each down move probability $1-q$. Under this measure the discounted stock is a martingale.

For a discounted claim $H$, the arbitrage-free value process is $V_t={\mathbb{E}}_Q[H\mid {\mathcal{F}}_t]$. A terminal payoff $h(S_T)$ has value

For a payoff $h(S_T,M_T)$ with $M_t={\max }_{s\le t}{S}_s$, the pair $(S_t,M_t)$ is a sufficient Markov state for valuation. If an independent restarted stock path has initial value $y$, terminal value $z$, and running maximum $w$, then the continued payoff is