Something mathematical!

By some sort of coincidence, if we define $q=\frac{s_{0}e^{rT}-s_{down}}{s_{up}-s_{down}}$, then we can rewrite p as

$$\large {p=\exp(-rT)((1-q)f(1)+qf(2))}$$ (8)

Seemingly q here acts as some sort of `probability' (even though q has no relationship with the transition probability m!), so that if we average the potential payout of the derivative according to this `probability', we will get the (time discounted) price of the derivative! (At least in this format, the price p of the derivative reduces to the formula in section ).