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.