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CFA Level I Exam · Valuing a Derivative Using a One-Period Binomial Model

One-Period Binomial Model: Basics and Assumptions

Updated 7 October 2026 · Fact-checked

The one-period binomial model assumes an underlying's price moves over a single step to only one of two values: up (S × u) or down (S × d). With a risk-free rate and no arbitrage, you can price a derivative from those two outcomes. Up factor u = 1 + up return; d = 1 + down return.

Understand Binomial Model Basics and Assumptions

A derivative gets its value from an underlying asset, such as a share. To value it, you need a view of where the underlying could be later. The binomial model makes this simple. It looks at one time step and allows only two outcomes.

Today the underlying price is S₀. At the end of the step it becomes either S₁⁺ = S₀ × u (the up move) or S₁⁻ = S₀ × d (the down move). The factors u and d are gross returns. If the price rises 25%, u = 1.25. If it falls 20%, d = 0.80. The tree is drawn as one node splitting into two.

The second input is the risk-free rate r for the same period. You can invest or borrow at r. Together with the two prices, r lets you build a portfolio of the underlying and risk-free bonds that copies the derivative's payoff. This is the idea behind later pricing steps.

The model rests on a few assumptions. The price has only two possible outcomes in the period. The risk-free rate is known and constant, and you can lend and borrow at it. Markets are frictionless, with no transaction costs or taxes. And there is no arbitrage: two assets with the same payoffs must have the same price. For the tree to be consistent with no arbitrage, d < 1 + r < u must hold.

This topic is the base. Hedge ratios, risk-neutral probabilities and option values come next. Here you must read the tree correctly and know what each input means.

Key formulas to remember

Up and down prices
S₁⁺ = S₀ × u ; S₁⁻ = S₀ × d
u and d are gross factors (1 + return), not percentage moves.
Factors from returns
u = 1 + up return ; d = 1 + down return
A 15% fall gives d = 0.85, not 0.15.
No-arbitrage condition on factors
d < 1 + r < u
If 1 + r is outside this range, one asset dominates the other and arbitrage exists.
Risk-free growth over the period
Bond value at end = B₀ × (1 + r)
Use the rate for the length of the single step, not an annual rate unless the step is one year.
Call payoffs at expiry
c₁⁺ = max(0, S₁⁺ − X) ; c₁⁻ = max(0, S₁⁻ − X)
Put payoffs: max(0, X − S₁). Shown here so you can read the tree outputs.

How to solve Binomial Model Basics and Assumptions questions

For any question on tree set-up, assumptions or inputs, work through the same checks.

  1. 1Identify the step length and make sure r is for that same period.
  2. 2Write down S₀, then convert any stated returns into factors u and d.
  3. 3Compute the two end prices S₀ × u and S₀ × d.
  4. 4Check d < 1 + r < u. If it fails, the model allows arbitrage.
  5. 5If a derivative is given, compute its payoff at each node using max(0, ...).
  6. 6For assumption questions, test each statement against: two outcomes, constant known r, borrowing and lending at r, no frictions, no arbitrage.
  7. 7Check units and rounding, then pick the option that matches your value.

Quickest way: Factor and bounds check

When to use it: Use when the question gives returns or prices and asks for a node value or whether the tree is valid.

  1. Turn percentages into factors at once: add 1 to each return.
  2. Multiply S₀ by each factor to get both nodes.
  3. Compare 1 + r with d and u in one glance.
  4. With three options, estimate the value first; usually only one option fits the right magnitude and the others come from using u − 1 or d − 1 by mistake.

Common mistakes in Binomial Model Basics and Assumptions

  • Using the percentage move instead of the factor, e.g. d = 0.20 for a 20% fall.

    The question states the move as a percentage, and it is easy to copy it directly.

    Fix: Always write d = 1 − fall and u = 1 + rise before multiplying.

  • Using an annual risk-free rate for a step shorter than a year.

    The rate is quoted per year and the time step is ignored.

    Fix: Match r to the step length. For a one-year step use r as given; for a different period, adjust it as the question states.

  • Forgetting the condition d < 1 + r < u.

    Students focus on calculating prices and skip validity checks.

    Fix: Compare 1 + r with both factors before any pricing. If it fails, the model implies arbitrage.

  • Believing the model says the price will truly go only up or down by those amounts.

    The two-outcome rule is read as a forecast.

    Fix: It is a simplifying assumption for pricing, not a prediction. Real prices have many outcomes.

  • Treating the call payoff at the down node as negative.

    S₁⁻ − X is negative when the option is out of the money.

    Fix: Apply max(0, ...). An option holder never loses more than the premium from exercise; payoff is zero.

Worked examples

Example 1

A share trades at €80. Over one period it can rise 25% or fall 10%. The one-period risk-free rate is 5%. Find u, d, both end prices and state whether the tree is consistent with no arbitrage.

Show the solution
  1. u = 1 + 0.25 = 1.25 and d = 1 − 0.10 = 0.90.
  2. Up price = 80 × 1.25 = €100.
  3. Down price = 80 × 0.90 = €72.
  4. 1 + r = 1.05.
  5. Check d < 1 + r < u: 0.90 < 1.05 < 1.25 is true.

Answer: u = 1.25, d = 0.90, prices €100 and €72; the tree is consistent with no arbitrage.

Example 2

A stock is at $50. In one period it will be $60 or $45. The risk-free rate for the period is 4%. A call has a strike of $52. What are u, d and the call payoffs at the two nodes?

Show the solution
  1. u = 60 ÷ 50 = 1.20 and d = 45 ÷ 50 = 0.90.
  2. 1 + r = 1.04, and 0.90 < 1.04 < 1.20, so the tree is valid.
  3. Up payoff = max(0, 60 − 52) = $8.
  4. Down payoff = max(0, 45 − 52) = max(0, −7) = $0.

Answer: u = 1.20, d = 0.90; call payoff is $8 in the up state and $0 in the down state.

Exam tips

  • Read whether the question gives returns or prices; convert to factors before anything else.
  • Check that r is for the same period as the tree step.
  • For assumption questions, a statement that the price can take many values or that borrowing costs more than r contradicts the model.
  • With three options and no penalty, a quick u and d calculation usually removes the options built from the percentage moves.

Practice questions from Valuing a Derivative Using a One-Period Binomial Model

Binomial Model Basics and Assumptions in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Binomial Model Basics and Assumptions: frequently asked questions

What is a one-period binomial model?

It is a model where an underlying's price moves over one time step to just one of two values, up or down. You use those two values, plus the risk-free rate, to value a derivative. It is the building block for multi-period trees.

How do I calculate the up and down factors?

Add the percentage rise to 1 for u, and subtract the percentage fall from 1 for d. For a 12% rise, u = 1.12. For an 8% fall, d = 0.92. You can also divide each end price by the starting price.

What are the assumptions of the binomial option pricing model?

Two possible outcomes in the period, a known constant risk-free rate at which you can borrow and lend, frictionless markets, and no arbitrage. For consistency, d < 1 + r < u must hold.

Why must 1 + r lie between d and u?

If 1 + r ≥ u, the bond never does worse than the underlying, so you short the underlying and lend at r for a sure profit. If 1 + r ≤ d, the underlying never does worse than the bond, so you borrow at r and buy the underlying. Either case gives arbitrage.