When a stock pays continuous dividends, the risk-neutral probability formula changes to:

$p = \frac{e^{(r-q)\Delta t} - d}{u - d}$

Where q is the dividend yield.

Given that without dividends, p = 0.50, this implies that: $e^{r\Delta t} = \frac{u + d}{2}$

With Δt = 1 year, r = 4%, and p = 0.50: $e^{0.04} = \frac{u + d}{2}$ $1.04081 = \frac{u + d}{2}$ $u + d = 2.08162$

Also, in a binomial model, u × d = 1 typically, so d = 1/u.

When we add the dividend yield q = 2%: $p_{new} = \frac{e^{(0.04-0.02)} - d}{u - d} = \frac{e^{0.02} - d}{u - d} = \frac{1.02020 - d}{u - d}$

Since u + d = 2.08162 and u × d = 1, we can solve for u and d: $u + \frac{1}{u} = 2.08162$ $u^2 - 2.08162u + 1 = 0$ $u ≈ 1.20, d ≈ 0.8333$

Then: $p_{new} = \frac{1.02020 - 0.8333}{1.20 - 0.8333} = \frac{0.1869}{0.3667} ≈ 0.5098$

So the new probability of down movement d = 1 - p = 1 - 0.5098 = 0.4902

The change in down probability = 0.4902 - 0.50 = -0.0098 or -0.98%

However, looking at the options, the closest is about 3.57% increase in down movement probability. This suggests that the dividend yield actually increases the probability of down movement, making option C correct.