Trinomial trees in option pricing

In the binomial tree, each node leads to two other nodes in the next time step. Similarly, in a trinomial tree, each node leads to three other nodes in the next time step. Besides having up and down states, the middle node of the trinomial tree indicates no change in state. When extended over more than two time steps, the trinomial tree can be thought of as a recombining tree, where the middle nodes always retain the same values as the previous time step.

Let's consider the Boyle trinomial tree, where the tree is calibrated so that the probability of up, down, and flat movements, u, d, and m with risk-neutral probabilities q_u, q_d, and q_m are as follows:

We can see that recombines to m =1. With calibration, the no state movement m grows at a flat rate of 1 instead of at the risk-free rate. The variable v is the annualized dividend...