Risk-neutral probability p
p is the synthetic weight that makes the discounted stock a martingale: under it the stock is expected to grow at the risk-free rate, so any derivative prices as a discounted expectation. Move the up move u, down move d, and rate r and watch p respond. It is a pricing device, not a real-world probability.
p = (er − d) / (u − d)
p = (1.020 − 0.900) / (1.200 − 0.900) = 0.401
p = (1.020 − 0.900) / (1.200 − 0.900) = 0.401
p (up) 0.4011 − p (down) 0.599
Martingale check: p·u + (1−p)·d = 1.020 = er = 1.020 ✓
Under p the expected one-step return is exactly the risk-free rate.
Under p the expected one-step return is exactly the risk-free rate.
Risk-neutral p0.401
1 − p0.599
Growth e^r1.020
Up factor u1.20
Down factor d0.90
Rate r / step2.0%
Assumption: one step with Δt = 1 and a continuously compounded rate, so per-step growth is er. The no-arbitrage band d < er< u must hold for p ∈ (0, 1).