Questions about the replicating portfolio in the binomial model

I'm starting to teach myself quantitative finance and I've got several questions ( marked in bold ) regarding the replicating portfolio of a security in the binomial model. I'm following, among others, the classical book " Stochastic Calculus for Finance I: The Binomial Asset Pricing Model ". First, I'll start with some notation so that there is no confusion. As always, let $d$ , $u$ , $r$ be the down-factor, up-factor, and risk-free interest rate, respectively, verifying that $d < 1 + r < u$ . Then, if $V_{n}$ , $0 \leq n \leq N$ , is the value at time $n$ of a security that has a unique payoff $V_N$ at maturity, we know that the discounted process $\dfrac{V_n}{(1+r)^n}$ is a martingale under the risk-neutral probability measure $\widetilde{\mathbb{P}}$ , where the probability of heads is $p := \dfrac{1+r-d}{u-d}$ , and thus we can easily compute the value of each $V_n$ via $V_{n} = \widetilde{E}_{n}\Big(\frac{V_{N}}{(1+r)^{N-n}}\Big).$ If $d = d_{n}$ , $u = u_{n}$ , and $r = r_{n}$ are not constant numbers but an adapted stochastic process, the result is the same but now the discounted process is given by $\dfrac{V_n}{(1+r_{0})·\dots·(1+r_{n-1})}$ , and the risk-neutral pricing formula still holds true provided that $\widetilde{\mathbb{P}}(w_{n+1} = H|w_{1},\dots,w_{n}) := p_{n} := \dfrac{1+r_n-d_n}{u_n-d_n},$ $\widetilde{\mathbb{P}}(w_{n+1} = T|w_{1},\dots,w_{n}) = 1- p_{n}.$ To show the previous result, one usually constructs the following portfolio: suppose that $V$ is, for instance, an European call. Start with $X_{0}$ wealth, buy $\Delta_{0}$ shares of the underlying, and invest (or borrow) the remaining money at the risk-free rate $r$ . At time $1$ , sell the portfolio and reinvest the money doing the same strategy. At time $n+1$ , the value of the replicating portfolio is given by $X_{n+1} = \Delta_{n}S_{n+1} + (1+r)(X_{n}-\Delta_{n}S_{n}).$ Now that the notation is clear, my first question is the following: 1) Do we need to replicate the derivative security using the underlying security $S$ ? I'm aware of the hedging benefits of combining a derivative and its underlying, but since here the goal is to construct a replicating portfolio, could it be constructed trading with another security? What are the advantages of using the underlying over the rest of securities? The only benefit I see is that you only need to model the prices of one stock. Now suppose that we want to price zero-coupon bonds using the binomial model. Assume that the interest rates form an stochastic adapted process, in such a way that 1 dollar invested at time $n$ yields $(1+r_{n})$ at time $n+1$ . Let $B_{n,m}$ be the value at time $n$ of a zero-coupon bond that pays $1$ dollar at time $m$ . Since the risk-neutral pricing formula also applies here, we can easily conclude that $B_{n,m} = \widetilde{E}_{n}\Big(\frac{1}{(1+r_{n})·\dots·(1+r_{m-1})}\Big).$ However, here goes another question: 2) How would one construct a replicating portfolio in this case? In what securities does it make sense to trade? Finally, I see that in the book that I mention at the beginning, a portfolio process is constructed by trading in the zero-coupon bonds and the money market via the following equation: where $\Delta_{n,m}$ is the number of zero-coupon bonds of maturity $m$ held by the investor between times $n$ and $n+1$ . I understand that this portfolio process, properly discounted, is a martingale, and hence there can't be arbitrage when trading in the zero-coupon bonds and the money market. My final question is: 3) How is this formula related to the proof that $B_{n,m} = \widetilde{E}_{n}\big(\frac{1}{(1+r_{n})·\dots·(1+r_{m-1})}\big)$ ? If some of my questions are not clear enough, please let me know. Thanks a lot!