DP
Binomial Treesintermediate

Backward Induction

Backward induction prices a derivative on a tree by starting at the terminal nodes, where the payoff is known, and rolling values back one step at a time. At each interior node the value is a risk-neutral expectation of the two child values, discounted at rr for the step length Δt\Delta t. Repeating the move from leaves to root produces today's fair price.

Why it matters

Think of the tree as a decision graph read right to left. At expiry the option's worth is unambiguous because the payoff function tells you. One step earlier, you know what the option will be worth in either of the two possible next states, so you average them with risk-neutral weights and discount. The method is the same dynamic programming logic used in chess engines and reinforcement learning. Each node's value is the discounted certainty-equivalent of its future.

Formulas

Interior node value
fi=e−rΔt[p fi,u+(1−p) fi,d]f_i = e^{-r\Delta t}\bigl[p\, f_{i,u} + (1-p)\, f_{i,d}\bigr]
European options. For American add a max⁡\max against the immediate exercise payoff (Hull 2022, §13.7).
American adjustment
fi=max⁡(intrinsici,  e−rΔt[pfi,u+(1−p)fi,d])f_i = \max\bigl(\text{intrinsic}_i,\; e^{-r\Delta t}[p f_{i,u} + (1-p) f_{i,d}]\bigr)

Worked examples

Scenario

A two-step tree with terminal payoffs A$10, A$2, and A$0 in the up-up, up-down, and down-down states. Risk-neutral p=0.6p = 0.6, r=0.05r = 0.05, Δt=0.5\Delta t = 0.5.

Solution

Step 1 up-node value = e^{-0.025}(0.6 \times 10 + 0.4 \times 2) = 0.9753 \times 6.80 = A\6.63$. Step 1 down-node value = e^{-0.025}(0.6 \times 2 + 0.4 \times 0) = 0.9753 \times 1.20 = A\1.17$. Root value = e^{-0.025}(0.6 \times 6.63 + 0.4 \times 1.17) = 0.9753 \times 4.446 = A\4.34$.

Common mistakes

  • ✗Backward induction is unique to option pricing. It is not. The same technique solves dynamic programming problems in operations research, optimal stopping in statistics, and Bellman equations in macroeconomics. The binomial tree is one application of a much older idea.
  • ✗You can equivalently price by forward simulation. For European options, yes. For American options the early-exercise decision at each node depends on the continuation value, which requires the backward sweep. Forward Monte Carlo without a continuation-value estimator misses the early-exercise premium.

Revision bullets

  • •Start at terminal payoffs, walk back to root
  • •Discount the risk-neutral expectation at each step
  • •Same dynamic programming logic as Bellman equations
  • •American extension adds a max⁡\max against intrinsic value
  • •Standard pricing engine in Hull (2022) §13

Quick check

Backward induction in a binomial tree begins from:

Connected topics

2-Step TreeEarly Exercise

In learning paths

Pricing Models Path

Sources

  1. Hull (2022), §13.7
    Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
    Sets out the backward-induction algorithm for European and American payoffs on a multi-step tree.
  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.
    Introduces backward induction on the binomial lattice as the discrete-time pricing engine that converges to Black–Scholes.
  3. Bellman (1957)
    Bellman, Richard. Dynamic Programming. Princeton University Press, 1957.
    Original statement of the principle of optimality that justifies recursive backward solution of multi-stage decision problems.
How to cite this page
Dr. Phil's Quant Lab. (2026). Backward Induction. Derivatives Atlas. https://phucnguyenvan.com/concept/backward-induction
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