Summary

The binomial tree models the underlying as moving up by u or down by d each time step Δt, and values options by rolling expected risk-neutral payoffs backward through the tree. Its decisive advantage over Black–Scholes–Merton is early exercise: at each interior node the model takes the greater of continuation value and immediate exercise payoff, which is exactly what American options require¹. As the number of steps grows, the tree price converges to the BSM price for European options¹.

Mechanics

One step builds a riskless portfolio (Δ shares long, one option short); requiring it to earn r yields the risk-neutral valuation formula:

• f = e^(−rΔt) [ p·fu + (1−p)·fd ], with p = (a − d)/(u − d), a = e^(rΔt)

Multistep trees apply the same formula node by node. Worked example: S₀ = 20, u = 1.1, d = 0.9, K = 21, r = 12%, 3 months → call worth 0.633; the real-world up-probability (≈0.70 if μ = 16%) never enters the calculation¹.

Cox–Ross–Rubinstein parameters match volatility: u = e^(σ√Δt), d = 1/u. Example: S₀ = 50, K = 52, r = 5%, σ = 30%, two 1-year steps → American put 7.43; at 500 steps 7.47; the European twin prices at 6.76, matching BSM¹.

Early Exercise (American Options)

Feature	European	American
Node value	continuation only	max(continuation, exercise payoff)
Where early exercise shows up	never	typically deep-ITM puts; calls just before ex-dividend dates
Closed-form price	BSM exists	none — tree or other numerical method required

In the worked put example, early exercise is optimal at the lower middle node (payoff 12 vs. continuation 9.46), raising the value from 4.19 to 5.09¹. This node-by-node check is the entire reason trees exist: they operationalize early-exercise theory, which BSM cannot.

Extensions change only the growth factor a: dividend-yield stock or index a = e^((r−q)Δt); currency a = e^((r−rf)Δt); futures a = 1 (zero drift in the risk-neutral world)¹.

Convergence and Accuracy

• 30–50 steps give reasonable accuracy in practice; the tree converges to BSM as Δt → 0¹.
• Control variate: value the American option fA and its European twin fE on the same tree; since the European tree error is known (fBS − fE), correct with fA + (fBS − fE). Worked example improves a 5-step estimate from 4.49 to 4.25 vs. the accurate 4.278¹.
• Greeks from the tree: delta from the first step, gamma from a central difference of the two deltas at time 2Δt, theta by differencing across Δt; vega and rho by rebuilding with perturbed σ or r¹.
• Dividends: known yields keep the tree recombining; known dollar dividends break recombination — model S* = S − PV(dividends) instead and add the PV back at each node¹.

Why Traders Should Care

The tree is the conceptual bridge between model price and market behavior: American premium over European is visible as the early-exercise value at nodes, and put–call parity divergences in listed chains trace to exactly this optionality. For pricing work in this system, trees (or py_vollib-class engines) are the fallback wherever early exercise or discrete dividends matter — see the EV Contract tooling note.

Links

• Black–Scholes–Merton Model — the limiting model
• Delta and Gamma — tree-derived hedge ratios
• Theta, Vega, Rho

Footnotes

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1. Hull, Options, Futures, and Other Derivatives — binomial-trees topic: one/two-step trees, CRR parameters, early exercise, dividends, greeks from trees, control variates. ↩↩↩↩↩↩↩↩↩↩