Trinomial trees in options 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 such that the probability of up, down, and flat movements, , , and with risk-neutral probabilities , , and are as follows:

We can see that recombines with . With calibration, the no state movement grows at a flat rate of 1 instead of at the risk-free rate. The variable is the annualized dividend yield and is the annualized volatility of the underlying stock. In general, with an increased number of nodes to process, a trinomial tree...