The construction of a binomial tree is very helpful in understanding the possible stock price movements. A node is a boxed value from where two possible outcomes are possible. The moves are independent of each node and have the constant probabilities of up move (success) and down move (failure).
The magnitude of up move and down move in percentage terms also remain constant. Suppose u is equal to 1 plus the rate of return of an up move and d is equal to 1 plus the rate of return of a down move. If the stock starts at a value of S at time t=0, then the value of the stock at time t=1 will be either uS (up move) or dS (down move). Similarly, the value of the stock at time t=1 will be one of these three - uuS, udS, and ddS. The figure below illustrates the stock price movement as per a binomial model.
There are four values possible at the time t=3. The uuuS is possible only when we have three up moves, and that is given by the binomial probability as 3C3p3(1-p)0 i.e. p3. The uudS is possible in 3 ways, and the probability by the binomial formula is given as 3C2p2(1-p)1 or 3p2(1-p).
The final stock price distribution is a function of the initial stock price, the number of up moves, and the size of the up moves and down moves. We cannot say that the stock price itself is a binomial random variable, but it is a function of a binomial random variable as well as of u and d, and initial price.
The binomial tree also makes it possible to test a condition at every node. So, this flexibility allows it to value American call option and even more complex options than that.
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