Updated for the 2026-2027 CFA® Level I curriculum.
The one-period binomial model values a derivative by assuming the underlying asset can only move to one of two prices at the end of the period. This simple setup lets you calculate a no-arbitrage derivative value without forecasting the actual probability of each outcome. CFA Level I tests this model directly, so you need to build the price tree, find the payoffs, and apply either the replication method or the risk-neutral method to get the same answer.
Quick Answer
The one-period binomial model values a derivative by finding the price that prevents arbitrage between the derivative and a portfolio of the underlying asset and a risk-free bond. The underlying moves up to or down to . You calculate the derivative's payoff in each state, then discount using either a replicating portfolio or risk-neutral probabilities. Both methods produce the identical value.
Key Takeaways About One-Period Binomial Model for Derivative Valuation
The one-period binomial model assumes the underlying asset has exactly two possible prices at the end of one period: an up price and a down price.
The derivative's payoff is calculated separately in each state before any discounting happens.
Replication and risk-neutral valuation are two paths to the same no-arbitrage value, not competing answers.
Risk-neutral probabilities come from the up factor, down factor, and risk-free rate. They are not real-world probabilities.
The hedge ratio tells you how many shares of the underlying replicate one unit of the derivative.
A common error is using actual or subjective probabilities instead of risk-neutral probabilities when discounting expected payoff.
The model values the derivative today. It does not predict which state will actually occur.
What You Need to Know for CFA Level I
Identify the two possible ending prices of the underlying asset (up state and down state).
Calculate the derivative's payoff in each state separately.
Apply the replicating portfolio method: buy shares and borrow or lend to match the derivative's payoffs exactly.
Apply the risk-neutral valuation method: calculate risk-neutral probabilities, then discount expected payoff at the risk-free rate.
Confirm the no-arbitrage value from both methods matches.
Recognize the role of the up factor, down factor, hedge ratio, and risk-free borrowing or lending in the pricing logic.
What Is a One-Period Binomial Model?
A one-period binomial model prices a derivative by assuming the underlying asset can only take one of two values after one period. There is no continuum of outcomes and no path dependency. This narrow setup is exactly what makes the model useful for teaching valuation logic. It isolates the core idea behind option pricing: you can replicate a derivative's payoff using the underlying asset and a risk-free loan, and the derivative must trade at the cost of that replicating portfolio. If it does not, an arbitrage opportunity exists.
The model applies to any derivative with a payoff that depends on the underlying price, including calls, puts, and forwards. Level I focuses mainly on valuing a call or put using this structure.
Build the Up and Down Price Tree
Start with the current underlying price, . Over one period, the price moves to one of two values:
Up state: , where is the up factor
Down state: , where d is the down factor
where:
= current price of the underlying asset
= underlying price in the up state
= underlying price in the down state
= up factor, where
= down factor, where
The up and down factors reflect the range of price movement assumed for the period. There is no third outcome and no partial move. This binary structure is what makes the model solvable without needing an actual probability distribution.
Calculate Derivative Payoffs
Once you know the up and down prices, calculate the derivative's payoff in each state. For a European call with strike price :
Payoff if up:
Payoff if down:
For a European put, the payoff formula flips:
where:
= option strike price
= call payoff in the up state
= call payoff in the down state
= put payoff in the up state
= put payoff in the down state
Always calculate payoff separately in each state before moving to valuation. The payoff is not the same as the value. Payoff is what the derivative is worth at expiration in a specific state. Value is what the derivative is worth today, before you know which state will occur.
Replication Method
The replication method builds a portfolio of shares and a risk-free loan that produces the same payoff as the derivative in both states. If two portfolios have identical payoffs in every future state, they must have the same value today. Otherwise, arbitrage is possible.
The hedge ratio tells you how many shares to hold:
Where and are the derivative's payoffs in the up and down states, and and are the underlying's up and down prices.
Once you know , you solve for the amount to borrow so the portfolio's up-state and down-state payoffs match the derivative exactly. The derivative's value today equals the cost of this replicating portfolio:
where:
= hedge ratio, or number of underlying shares held in the replicating portfolio
= amount borrowed today at the risk-free rate; a positive represents borrowing
Risk-Neutral Valuation Method
The risk-neutral method skips building a replicating portfolio and instead calculates a probability, , that makes the expected return on the underlying equal to the risk-free rate. This is not the real-world probability of an up move. It is a mathematical tool that lets you discount expected payoff at the risk-free rate.
Where is the risk-free rate for the period, is the up factor, and is the down factor.
Once you have , the derivative's value is the expected payoff under risk-neutral probabilities, discounted at the risk-free rate:
where:
= risk-neutral probability of the up state
= risk-neutral probability of the down state
= risk-free rate for the period
If falls outside the 0 to 1 range, the no-arbitrage assumption is violated. This usually means , , and are inconsistent with each other.
Both methods rely on the same no-arbitrage logic. Replication builds the offsetting position directly. Risk-neutral valuation shortcuts the math using a probability that reflects that same no-arbitrage condition. Under consistent inputs, they produce the identical value.
Worked One-Period Binomial Example
Setup. A stock trades today at . Over one period, the price will move up to $60 or down to $45. The up factor is and the down factor is . The risk-free rate for the period is 5%. A European call option on this stock has a strike price of . Value the call today.
Step 1: Calculate payoffs.
Step 2: Risk-neutral valuation.
Step 3: Confirm with replication.
At the up state:
Check at the down state:
Both methods agree: the call is worth $4.76 today. This means an investor could replicate the call's payoff by buying 0.6667 shares of stock and borrowing $28.57 at the risk-free rate.
If the call traded above $4.76, an arbitrageur could sell the call and buy the replicating portfolio for a risk-free profit.
If it traded below $4.76, the reverse trade would work. The $4.76 value is not a forecast of the stock's future price. It is the price that removes arbitrage today.
Interpreting the Model
The one-period binomial value reflects no-arbitrage pricing, not a prediction about which state will actually happen. The risk-neutral probability is a pricing tool, not a real-world forecast of an up move.
Candidates should keep price, value, payoff, and profit distinct: the underlying's price moves to or , the derivative's payoff is calculated in each state, the derivative's value is the no-arbitrage price today, and profit depends on what an investor paid relative to the eventual payoff.
Ultimately, the binomial model does not eliminate risk. It removes arbitrage, which is a narrower and more precise claim.
Common Exam Traps
Using actual probabilities instead of risk-neutral probabilities
Candidates sometimes plug in a subjective or historical probability of an up move when the question calls for risk-neutral valuation. The risk-neutral probability is derived from , , and , not from market forecasts.
Discounting the wrong cash flow
Some candidates discount the underlying's expected future price rather than the derivative's expected payoff. Only the derivative's payoff gets discounted to find its value.
Skipping separate payoff calculations
Payoff must be calculated independently in the up state and the down state before any valuation step. Combining these too early leads to errors.
Ignoring an invalid risk-neutral probability
If π falls below 0 or above 1, the inputs violate no-arbitrage conditions. This usually signals an error in , , or rather than a valid model output.
Practice Questions
A stock currently trades at $80. Over one period, it will move up to $96 or down to $68. The risk-free rate for the period is 4%. A European put option on the stock has a strike price of $80. What is the value of the put today using risk-neutral valuation?
$3.08
$5.27
$8.42
Correct Answer: B. $5.27
Calculation steps:
The put pays only in the down state. The risk-neutral down probability is 0.4571, so the discounted value is 0.4571 × $12 / 1.04 = $5.27.
Option A: Uses the actual up-state probability instead of the risk-neutral down-state probability, understating the payoff contribution.
Option C: Fails to discount the expected payoff at the risk-free rate, overstating the value.
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FAQs About One-Period Binomial Model for Derivative Valuation
Does the one-period binomial model use real-world probabilities?
No. It uses risk-neutral probabilities derived from the up factor, down factor, and risk-free rate. These are pricing tools, not forecasts of actual market behavior.
Do replication and risk-neutral valuation give different answers?
No. Under consistent assumptions, both methods produce the same no-arbitrage derivative value. They are two routes to the same result.
What happens if the risk-neutral probability is negative or above 1?
This signals that the inputs violate the no-arbitrage condition. Check the up factor, down factor, and risk-free rate for consistency.