Exact and instant. Analytic Greeks. The benchmark every numerical method is judged against.

European exercise only. Assumes constant volatility and rates and a lognormal underlying.

Vanilla European options; implied volatility; sanity-checking numerical methods.

Simple to explain and implement. Handles American exercise naturally. The tree recombines around the spot.

Converges slowly, order 1/N, and the price oscillates as the strike moves between nodes.

Teaching the mechanics of backward induction and early exercise.

Equal probabilities keep the arithmetic simple. Never produces negative probabilities.

Same slow, oscillating convergence as Cox-Ross-Rubinstein. The tree drifts, so the spot is not a node after step zero.

Illustrating that many trees converge to the same limit.

Matches the first three moments of the lognormal distribution at every step.

No faster than Cox-Ross-Rubinstein in practice; still oscillates.

Comparing moment-matching against the classic scheme.

Places a node on the strike, so convergence is smooth and of order 1/N². Accurate to a few basis points with a few dozen steps.

Needs an odd step count and the closed-form d1 and d2, so it is tied to the lognormal model.

Fast, accurate American pricing; the practical choice among trees.

Three branches give a better fit per step than two. Equivalent to an explicit finite-difference scheme. Theta comes straight from the tree.

More nodes per step than a binomial tree, so each step costs more. Still order 1/N convergence.

Bridging trees and finite-difference methods.

Solves the pricing PDE on a grid, giving values for every spot and time at once. Handles American exercise and barriers.

Grid design matters: explicit schemes can be unstable, and the boundary conditions must be chosen with care.

American options, barriers, and any single-factor model with a known PDE.

Handles path-dependent payoffs and many underlyings. Error is measurable through the standard error.

Slow convergence, order 1/√N. Early exercise needs regression techniques. Greeks are noisy.

Exotic and multi-asset options.

Almost as fast as the closed form. Gives the early-exercise boundary explicitly.

An approximation: error grows for long maturities and very high volatility. American only.

Quick American pricing when a few cents of accuracy is enough, such as scanning many strikes.

Fully closed form, no root finding at all, so it is the fastest American method.

Assumes a flat exercise boundary, so it is biased low. The 2002 two-step version is more accurate.

Real-time American quotes and as a control variate for Monte Carlo.