Altering the length of time step will alter the $u$ and $d$. Thus, the stock prices on the nodes will change, so as the future option values (which will be discounted back). Further, the risk-neutral probabilities and discounting procedure will be affected as well. As a result, the option value must also change.

The formula of $u$ is given by $e^{\sigma \sqrt{\Delta t}}$. Clearly, if $\Delta t$ drops, $u$ will decrease, not increase.

According to the formula, the risk-neutral probability of an upward movement is affected by $\Delta t$: $p^* = \frac{e^{r \Delta t} - d}{u - d} = \frac{e^{r \Delta t} - e^{-\sigma \sqrt{\Delta t}}}{e^{\sigma \sqrt{\Delta t}} - e^{-\sigma \sqrt{\Delta t}}}$*