Risk neutrality and risk-neutral probabilities

In short

• The risk-neutral probability of an up move is $\pi = \dfrac{1 + r - R^d}{R^u - R^d}$; the down move gets $1 - \pi$.
• Option value: $c_0 = \dfrac{\pi c_1^u + (1-\pi)c_1^d}{(1+r)^T}$, and the same formula with p for a put.
• π depends only on $R^u$, $R^d$ and r, so the same π values every option on the same underlying over the same period.
• π is not a forecast. It is the probability that would make the underlying's expected return equal to the risk-free rate.
• Risk-neutral pricing: the value does not depend on actual probabilities, on the underlying's expected return or on investors' risk aversion.