Cox--Ross--Rubinstein model

In the Cox--Ross--Rubinstein model, a stock changes at each discrete time by an up factor $U$ or a down factor $D$, while the money-market account changes by a deterministic factor $R$. After discounting by the money-market account, the factors are $u=U/R$ and $d=D/R$. The model is arbitrage-free exactly when $D<R<U$, equivalently $d<1<u$.

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