Two-step Binomial Tree

Q: A two-step recombining binomial tree produces how many distinct terminal stock prices?

3. Because ud = du, the two middle paths land on the same node, giving terminal prices S_0 u^2, S_0 u d, and S_0 d^2. With n steps a recombining tree has n+1 terminal nodes, much fewer than the 2ⁿ total paths (Hull 2022, §13.4).

A two-step binomial tree divides the option's life into two equal periods of length $\Delta t$ , generating three terminal stock prices $S_0 u^2$ , $S_0 u d$ , and $S_0 d^2$ . Pricing proceeds by backward induction. Find the option value at the two intermediate nodes, then discount one more step back to today. Adding steps refines the approximation and, in the Cox–Ross–Rubinstein parameterisation, the tree converges to Black–Scholes as the number of steps grows.

Try it yourself

Binomial option pricing tree

Price each node forward (S₀·uᵏ·dⁿ⁻ᵏ), then fold the payoffs back at the risk-neutral probability p. The asset price sits in each circle; the option value is beneath it in the accent colour.

Steps

Type

Circles show the asset price; the figure beneath, in accent, is the option value. Edges are up-moves (u) and down-moves (d).

Call value today (root)

10.48

Risk-neutral p0.40

1 − p0.60

Discount e⁻ʳ0.98

Spot S₀100.00

Strike K100.00

Up factor u1.20

Down factor d0.90

Rate r / step2.0%

Assumption: r is a continuously compounded per-step rate with one step per period (Δt = 1), so the per-step growth is e^r and the discount factor is e^−r. p = (e^r − d) / (u − d); each node = e^−r·[p·V_up + (1−p)·V_down].

Why it matters

One step is a toy. Two steps already give you three possible outcomes and an intermediate snapshot of how the option's value evolves with the stock. The middle layer is where you can see the early-exercise decision for American options and where you start to see how delta changes with the stock price. The same idea scaled to 50 or 500 steps is what production pricing engines run.

Before you read on — recall

A two-step recombining binomial tree produces how many distinct terminal stock prices?

Formulas

European closed form (two steps)

f_0 = e^{-2r\Delta t}\bigl[p^2 f_{uu} + 2p(1-p) f_{ud} + (1-p)^2 f_{dd}\bigr]

Intermediate up-node

f_u = e^{-r\Delta t}\bigl[p f_{uu} + (1-p) f_{ud}\bigr]

Intermediate down-node

f_d = e^{-r\Delta t}\bigl[p f_{ud} + (1-p) f_{dd}\bigr]

Worked examples

Scenario

European put on a stock at $S_0 =$ A$100, strike $K =$ A$100, $u = 1.1$ , $d = 0.9$ , $\Delta t = 0.5$ , $r = 0.05$ .

Solution

Terminal stock prices are A$121, A$99, A$81 with payoffs $f_{uu} = 0$ , $f_{ud} = 1$ , $f_{dd} = 19$ . Risk-neutral $p = (e^{0.025} - 0.9)/0.2 = 0.6266$ . Intermediate up-node $f_u = e^{-0.025}(0.6266 \times 0 + 0.3734 \times 1) =$ A$0.36. Intermediate down-node $f_d = e^{-0.025}(0.6266 \times 1 + 0.3734 \times 19) =$ A$7.53. Root $f_0 = e^{-0.025}(0.6266 \times 0.36 + 0.3734 \times 7.53) =$ A$2.97.

Common mistakes

✗Two-step European and American puts always have the same value. They do not. At the intermediate down-node the American put is worth $\max(\text{intrinsic}, f_d) = \max(19, 7.53) = 19$ , giving a larger root value than the European version. Early exercise is exactly where the two diverge.
✗More steps always improve accuracy linearly. CRR tree pricing error decays at rate $O(1/N)$ where $N$ is the number of steps, with oscillation as $N$ varies. Practitioners use 50 to 200 steps for typical equity options (Hull 2022, §21).

Revision bullets

•Three terminal prices $S_0 u^2$ , $S_0 ud$ , $S_0 d^2$
•Backward induction from payoffs to root
•Intermediate nodes expose early-exercise decisions
•Convergence to Black–Scholes at rate $O(1/N)$
•Workhorse model in Hull (2022) §13.4

Quick check

A two-step recombining binomial tree produces how many distinct terminal stock prices?

Connected topics

Backward Ind.Early Exercise Div. Trees Exam Hotspots

More in Binomial Trees

1-Step Tree Risk-Neutral p Delta Δ Delta Hedge

In learning paths

Pricing Models Path Exam Revision Path

Sources

Hull (2022), §13.4
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Worked two-step put and call examples, including the intermediate-node calculation that exposes early-exercise differences.
Cox, Ross & Rubinstein (1979)
Cox, John C., Stephen A. Ross and Mark Rubinstein. "Option Pricing: A Simplified Approach." Journal of Financial Economics 7(3), 1979, pp. 229–263.
Establishes the multi-step binomial recursion and the parameter choice $u = e^{\sigma\sqrt{\Delta t}}$ , $d = 1/u$ that recovers Black–Scholes in the limit.
ASX Equity Options
Australian Securities Exchange. Equity options product overview. ASX, accessed 2026.
Local reference for listed single-stock options on ASX whose American style and discrete dividend events make tree-based pricing the standard tool.

How to cite this page

Dr. Phil's Quant Lab. (2026). Two-step Binomial Tree. Derivatives Atlas. https://phucnguyenvan.com/concept/binomial-two-step