# Binomial pricing model: When the Cox-Ross-Rubinstein assumption is not arbitrage-free

I understand that in an arbitrage-free Binomial model, we assume that $S_{t+1} = S_t \cdot u$ in the event of an up-jump and $S_{t+1} = S_t \cdot d$ in the event of a down-jump. We call $u$ and $d$ the growth factors. A neccessary condition for no-arbitrage is that $d < e^{r \delta t} < u$, where $r$ is the risk-free lending rate and $\delta t$ is the time elapsed during each step. A common assumption, due to Cox, Ross and Rubinstein, is to let $u = 1/d = e^{\sigma \sqrt{\delta t}}$.

It seems that this assumption is inconsistent with no-arbitrage in the case where $e^{r \delta t} \geq e^{\sigma \sqrt{\delta t}}$ or, equivalently, when $r \geq \sigma / \sqrt{\delta t}$. What is the typical recourse in this situation?

if you let $\delta t$ be small enough, this won't happen. So the solution is to take more steps.