CoachA trinomial tree moves the log-price up, down or not at all at each step, by (for instance ), with probabilities chosen to match the mean and variance of the log-return; it is an explicit finite-difference scheme with a stability condition built into its geometry, and its extra middle branch lets the node grid be placed more freely than a binomial tree’s.
Examples
Example 22.8 (A tenth of a cent)
The one-year American put (spot and strike 100, rate 5%, volatility 20%) is worth 6.0904, from a Richardson-extrapolated grid (1 600 and 800 nodes and steps). The Cox–Ross–Rubinstein tree’s error alternates in sign with the parity of the step count. Its two neighbours are both within 0.1 cent from 1 500 steps, 1 125 750 node updates. The trinomial tree converges without the zigzag but at first order: its error is 0.11 cent at 1 600 steps and 0.054 at 3 200. The grid, with a node on the strike, payoff smoothing and Rannacher start-up, errs by 1.04 cent with 50 nodes and steps, 0.30 with 100, 0.091 with 200 and 0.029 with 400: second order. It is within 0.1 cent at 200, 40 600 node updates. That is 28 times less work than the tree (Figure 22.4). Without smoothing and start-up the grid errs by 0.125 cent at 200.