Answer: The risk neutral probability of an upward movement is not affected by reducing the length of time step.

## Explanation In the binomial option pricing model: - **u (up factor)** and **d (down factor)** are calculated as: - u = e^(σ√Δt) - d = e^(-σ√Δt) = 1/u - **Risk-neutral probability (p)** is calculated as: - p = (e^(rΔt) - d) / (u - d) When we reduce Δt from 1/2 to 1/12: 1. **u decreases** (not increases) because u = e^(σ√Δt) and √Δt decreases 2. **d increases** (not decreases) because d = e^(-σ√Δt) and √Δt decreases 3. **Option value changes** - with smaller time steps, the binomial tree becomes more granular and the option price converges to the Black-Scholes value 4. **Risk-neutral probability (p) remains approximately the same** - this is the correct answer because: - p = (e^(rΔt) - d) / (u - d) - As Δt becomes smaller, both numerator and denominator change proportionally - For small Δt, p ≈ (1 + rΔt - (1 - σ√Δt + σ²Δt/2)) / ((1 + σ√Δt + σ²Δt/2) - (1 - σ√Δt + σ²Δt/2)) - This simplifies to p ≈ (rΔt + σ√Δt) / (2σ√Δt) = 1/2 + (r - σ²/2)√Δt/(2σ) - The change in p is very small and often negligible for practical purposes The key insight is that while u and d change significantly with Δt, the risk-neutral probability p is relatively stable and not meaningfully affected by changes in time step size.

Author: LeetQuiz .