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).