The risk-neutral probability of an upward move in the first step is calculated using the formula for the up-movement probability in a binomial tree model, which is given by:

$p_{\text{up-movement}} = \frac{e^{(r - d) \cdot \Delta t} - d}{u - d}$

where:

• $e$ is the base of the natural logarithm,
• $r$ is the annual risk-free rate,
• $d$ is 1 plus the percentage decrease in stock price when there's a down movement,
• $u$ is 1 plus the percentage increase in stock price when there's an up movement,
• $\Delta t$ is the time period in years.

Given:

• $r = 12\%$ with continuous compounding, so $e^{0.12 \cdot \frac{3}{12}}$ represents the continuous compounding over a 6-month period,
• $d = 0.8$ (since the stock price can go down by 20%),
• $u = 1.2$ (since the stock price can go up by 20%).

Plugging these values into the formula gives:

$p_{\text{up-movement}} = \frac{e^{(0.12 - 0.8) \cdot \frac{3}{12}} - 0.8}{1.2 - 0.8} = \frac{e^{-0.64 \cdot \frac{1}{4}} - 0.8}{0.4}$

After calculating the value, we get:

$p_{\text{up-movement}} = 0.5761 \text{ or } 57.61\%$

The risk-neutral probability of the stock going down is then $1 - p_{\text{up-movement}} = 1 - 0.5761 = 0.4239 \text{ or } 42.39\%$. The correct answer is C, 57.6%.