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.