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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.

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  • $\begingroup$ I do not have the book you mention so I will comment on the Cox, Ross, & Rubinstein method (CRR): The "real-life" ($p$) probabilities could vary at each time step and each node. In fact they are not affecting the price at all. Instead you calculate the risk-neutral probabilities ($q$) using the (all constant) values up- and down-movementes ($u$ and $d$) together with the volatility $\sigma$, the risk-free rate $r$ and the size of the time steps $\Delta t$. Hence, the answers your questions are: (1) No. (2) Not necessary for pricing of the option. (3) Yes. $\endgroup$
    – Landscape
    Aug 3 at 12:33

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