How to price an option on future using a binomial tree?

While trying to price an option on a future using a binomial tree, I found out in a manual (Actuarial Finance by Boudreault and Renaud) that we need to calculate $F_0$ , and then apply the up and down factor on this value. I am wondering why we don't model the stock value instead and use the formula $F_t^T = S_t\exp(r(T-t))$ ? For me this seems more intuitive; it would give us the same value for an option on a stock and an option on a future with the same maturity which makes sense (same payout = same initial value).