DP
Binomial Treesintermediate

Two-step Binomial Tree

A two-step binomial tree divides the option's life into two equal periods of length Δt\Delta t, generating three terminal stock prices S0u2S_0 u^2, S0udS_0 u d, and S0d2S_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.

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.

Formulas

European closed form (two steps)
f0=e−2rΔt[p2fuu+2p(1−p)fud+(1−p)2fdd]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
fu=e−rΔt[pfuu+(1−p)fud]f_u = e^{-r\Delta t}\bigl[p f_{uu} + (1-p) f_{ud}\bigr]
Intermediate down-node
fd=e−rΔt[pfud+(1−p)fdd]f_d = e^{-r\Delta t}\bigl[p f_{ud} + (1-p) f_{dd}\bigr]

Worked examples

Scenario

European put on a stock at S0=S_0 = A$100, strike K=K = A$100, u=1.1u = 1.1, d=0.9d = 0.9, Δt=0.5\Delta t = 0.5, r=0.05r = 0.05.

Solution

Terminal stock prices are A$121, A$99, A$81 with payoffs fuu=0f_{uu} = 0, fud=1f_{ud} = 1, fdd=19f_{dd} = 19. Risk-neutral p=(e0.025−0.9)/0.2=0.6282p = (e^{0.025} - 0.9)/0.2 = 0.6282. Intermediate up-node f_u = e^{-0.025}(0.6282 \times 0 + 0.3718 \times 1) = A\0.36$. Intermediate down-node f_d = e^{-0.025}(0.6282 \times 1 + 0.3718 \times 19) = A\7.50$. Root f_0 = e^{-0.025}(0.6282 \times 0.36 + 0.3718 \times 7.50) = A\2.94$.

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⁡(intrinsic,fd)=max⁡(19,7.50)=19\max(\text{intrinsic}, f_d) = \max(19, 7.50) = 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)O(1/N) where NN is the number of steps, with oscillation as NN varies. Practitioners use 50 to 200 steps for typical equity options (Hull 2022, §21).

Revision bullets

  • •Three terminal prices S0u2S_0 u^2, S0udS_0 ud, S0d2S_0 d^2
  • •Backward induction from payoffs to root
  • •Intermediate nodes expose early-exercise decisions
  • •Convergence to Black–Scholes at rate O(1/N)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 ExerciseDiv. Trees

In learning paths

Pricing Models PathExam Revision Path

Sources

  1. 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.
  2. 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.
  3. 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
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