Explanation

To calculate the stock price at the lowest node in a two-step binomial tree, we need to determine the down factor and apply it twice.

Step 1: Calculate the up and down factors

For a binomial tree with continuous dividend yield, the formulas are:

• Up factor (u): $u = e^{\sigma\sqrt{\Delta t}}$
• Down factor (d): $d = \frac{1}{u}$

Where:

• $\sigma = 22\% = 0.22$ (volatility)
• $\Delta t = 1$ year (time step)

$u = e^{0.22 \times \sqrt{1}} = e^{0.22} \approx 1.2461$ $d = \frac{1}{1.2461} \approx 0.8025$

Step 2: Calculate the lowest stock price

Starting price: $30

Lowest node occurs after two down moves: $S_{dd} = S_0 \times d \times d = 30 \times 0.8025 \times 0.8025$ $S_{dd} = 30 \times (0.8025)^2 = 30 \times 0.6440 \approx 19.32$

Therefore, the stock price at the lowest node is $19.32.

Note: The risk-free rate (3.0%) and dividend yield (1.0%) are not needed for calculating the stock prices in the binomial tree - they are used for calculating risk-neutral probabilities, but the tree structure itself depends only on volatility and time step.