As the name suggests, a Binomial model is a “two-number” model. The two numbers referred to in the name of the model are simply the two probable values (and only two values) taken by a random variable of interest. The toss of a coin is an example of a real life event that can take on only two values – head or tail. Being the only two values that the random variable can assume, the probability of either value occuring in the model is certain with probability $1$ (A tossed coin will certainly end up with either a head or tail). In practice, if it is possible to establish the two values of the random variable at a particular point in time $t$, the model allows us to use the certainty of the binomial random variable to determine its value at a future time $t+1$ or historical time $t-1$. We can indeed model a whole series of times equidistant (futuristically or historically) from a known time $t$.
Clearly as we are talking about values of a random variable at particular times, this is a time discrete model. At each time step there are at least two values. Except at the theoretical beginning of the time reference (or origin). As a result we can represent this in a not too dissimilar way to the familiar “$XY$” cartesian coordinate system except that neither the $X$ or $Y$ axis is a time axis. The time axis (not always drawn) is the 45degree line through the origin. If the 45 degree line is rotated clockwise to the horizontal position, the $XY$ axes now appear diagonal and that is how we see the lattice structure of the binomial represented in mosts texts. In figure 3 below, the dotted red line is a time reference for the $2\Delta t$ time steps from the $t = 0$ time origin.
This model proposed by Cox, Ross and Rubenstein in the late 1970s is a finite model with similar properties to geometric Brownian motion. The model is specified by two parameters $\alpha > 0$ and $p \in [0,1]$. We define $$S_k = S_{k-1} \cdot e^ {\alpha X_k} \qquad K>0$$
independent random variables $X_k$ satisfy
\begin{align}
\mathbb{P}(X_k = 1 ) &= p\\
\mathbb{P}(X_k = -1 ) &= 1-p = q
\end{align}
Binomial lattice trees are used to represent the relationship.
$\require{\tikz}$
substituting values for the nodes A, B, C . . .
the directions may also be drawn at right angles to each other and the entire lattice rotated to make the directions either vertical or horizontal. The dotted line at -45 degrees then connects related events.
