What does and does not move a binomial option value

In short

• A wider spread between $R^u$ and $R^d$ (more volatility) raises both call and put values.
• A higher risk-free rate raises π and lowers the discount factor: the call value rises, the put value falls.
• The actual up-probability q, the underlying's expected return and investors' risk aversion do not affect the value.
• Two investors with opposite views on direction, but the same $R^u$, $R^d$ and r, must agree on the price.
• Binomial call and put values satisfy put-call parity: $S_0 + p_0 = c_0 + X(1+r)^{-T}$.