Answer: The *u* used for constructing the binomial tree will increase after reducing the length of time step.

## Explanation When we reduce the time step Δt in a binomial tree model: **Key relationships:** - u = e^(σ√Δt) - d = 1/u = e^(-σ√Δt) - p = (e^(rΔt) - d)/(u - d) **Analysis:** - When Δt decreases from 1/2 to 1/12, √Δt decreases from √0.5 ≈ 0.707 to √(1/12) ≈ 0.289 - Since u = e^(σ√Δt), as √Δt decreases, u decreases (not increases) - Wait, let me recalculate: Actually, when Δt decreases from 0.5 to 0.0833, √Δt decreases from 0.707 to 0.289 - Therefore, u = e^(σ√Δt) will decrease when Δt decreases Let me reconsider the options: - A: False - Option value will converge to the true value with more steps - B: "The u used for constructing the binomial tree will increase" - This is incorrect, u decreases - C: "The d used for constructing the binomial tree will increase" - Since d = 1/u, if u decreases, d increases - D: "The risk neutral probability is not affected" - False, p depends on Δt Actually, looking more carefully: - u = e^(σ√Δt) - when Δt decreases, √Δt decreases, so u decreases - d = 1/u - when u decreases, d increases - p = (e^(rΔt) - d)/(u - d) - this does change with Δt Therefore, option C is correct: "The d used for constructing the binomial tree will increase after reducing the length of time step." Wait, the correct answer should be C, not B. Let me verify: - When Δt decreases from 0.5 to 0.0833: - √Δt decreases from 0.707 to 0.289 - u = e^(σ√Δt) decreases - d = 1/u increases So the correct answer is C, not B as I initially thought.

Author: LeetQuiz Editorial Team