Binomial Tree
Value a European or American option by working back through a recombining binomial tree, with moves set from a volatility or given directly, and see how the value settles as the steps increase.
Learning focus: at each node the option is worth its discounted risk-neutral expected value, or, if American, its exercise value when that is higher.
Early exercise
Each node shows the underlying's price (S) and the option's value (V). Work right to left: end nodes hold the payoff, and each earlier node discounts the probability-weighted values of the two nodes after it. Highlighted nodes are where exercising immediately is worth more than holding.
CRR trees from 1 to 60 steps at a volatility of 20.27%, the volatility the given factors imply over one 1.0000-year step, σ = ln(u ÷ d) ÷ (2√Δt). The European value zigzags toward Black–Scholes–Merton as the strike falls between or on nodes, and the gap shrinks as steps are added. The American line sits above it by the early-exercise premium, which has no closed form.