C is correct. The formula for $d$ in the binomial model is $d = e^{-\sigma \sqrt{\Delta t}}$. When $\Delta t$ decreases, the term $\sqrt{\Delta t}$ becomes smaller, making the exponent less negative, which causes $d$ to increase. This is mathematically clear from the exponential function behavior.

A is incorrect because while altering $\Delta t$ does affect the binomial tree construction, it doesn't mean the option value must change - the binomial model should converge to the same theoretical value regardless of step size.

B is incorrect because $u = e^{\sigma \sqrt{\Delta t}}$ decreases when $\Delta t$ decreases, not increases.

D is incorrect because while it's true that $p^*$ is affected by $\Delta t$, this doesn't contradict the fact that $d$ increases when $\Delta t$ decreases.