Answer: 57.6%

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%.

Author: LeetQuiz Editorial Team