Answer: 0.459

## Explanation In a binomial model, the risk-neutral probability of an up movement (p) is calculated using the formula: $$p = \frac{e^{r\Delta t} - d}{u - d}$$ Where: - $r$ = risk-free rate = 4.0% = 0.04 - $\Delta t$ = time step = 1 year - $u$ = up factor = $e^{\sigma\sqrt{\Delta t}}$ = $e^{0.38\sqrt{1}}$ = $e^{0.38}$ ≈ 1.462 - $d$ = down factor = $1/u$ = $1/1.462$ ≈ 0.684 First, calculate $e^{r\Delta t}$: $$e^{0.04 \times 1} = e^{0.04} ≈ 1.0408$$ Now apply the formula: $$p = \frac{1.0408 - 0.684}{1.462 - 0.684} = \frac{0.3568}{0.778} ≈ 0.459$$ Therefore, the risk-neutral probability of an up movement is approximately 0.459, which corresponds to option B. Note: The fact that this is for an American-style put option on a stock with 38% volatility doesn't affect the calculation of the risk-neutral probability, as p depends only on the risk-free rate, time step, and the up/down factors (which are determined by volatility).

Author: LeetQuiz .