In Section 1.2 in Shreve's Stochastic Calculus for Finance I, he introduces the Multiperiod Binomial Model. There is something about it that I don't quite understand.
He assumes that coins are tossed and depending on heads/tails we go up or down, every step. However, it is unclear to me how this exactly works:
(1) Are all coins, at different time steps, identically distributed (i.e., each having same prob. for heads), or what is the assumption about that?
(2) If they are all identically distributed, I wonder why we even have to assume this (why this assumption is necessary). Namely, it seems to me that the (real-life) probabilities do not matter, since we're going to use the risk-neutral probabilities anyway for pricing.
(3) So: Can we assume in this model that the coin tosses at different times may have different distributions?
If the real-life probabilities are irrelevant, then I find it a bit confusing that he mentions them explicitly. It seems to me that he assumes that the "up" and "down" (real-life) probabilities at different time steps are the same (and I don't understand why such assumption would be necessary).
Thanks in advance.