Binomial option pricing

I just started reading about options and I am confused because I have seen two completely different approaches for pricing options using binomial.

1. There is some portfolio of stock X and bank shares (normalized to 1). Then it considers no arbitrage and says that
   (Call option @ t) = (portfolio @ t) =>
   (Call option @ 0) = (portfolio @ 0)

and after solving equations we come at the equation independent of probabilities.

C(t) = e^rt (q * C_up + (1-q) * C_down)

1. This one considers hedging and pricing options such that parameters follow portfolio_up = portfolio_down independent of probabilities and then discounting it to present to get current option value.

Which is the correct approach?
And how are they two related and are they leading to same results.