CFA Level I Exam · Pricing and Valuation of Options

Binomial Option Pricing Model for CFA Level I

Updated 7 October 2026 · Fact-checked

The binomial option pricing model values an option by building a tree where the underlying price moves up or down each period. You compute the option payoff at expiry, then discount the expected payoff using risk-neutral probabilities at the risk-free rate, working backward to today. For American options, you also compare with exercise value at each node.

Understand Binomial Option Pricing Model

An option's value depends on where the underlying price ends up. The binomial model keeps this simple: in each period the price can only go up by a factor u or down by a factor d. Starting from today's price S0, you get S0 × u and S0 × d after one period. After two periods you get S0u², S0ud and S0d².

The key idea is replication. You can build a portfolio of the underlying and a risk-free bond that has exactly the same payoffs as the option in both the up and down states. If two things have the same payoffs in every state, they must have the same price today, or there is arbitrage. The number of units of the underlying in this portfolio is the hedge ratio.

There is a shortcut. Replication gives the same answer as pretending investors are risk neutral. You find a risk-neutral probability of an up move, π, that makes the underlying earn exactly the risk-free rate. You then take the expected option payoff using π and discount it at the risk-free rate. These are not real-world probabilities. They are a pricing tool.

For a two-period tree, you work backward. Find option values at the two nodes after one period, using the final payoffs. Then discount those values one more period to get today's value. A European option can only be exercised at expiry, so you just discount. An American option can be exercised early, so at each node you take the larger of the continuation value and the immediate exercise value.

Key formulas to remember

Up and down factors: u = S_up ÷ S0 ; d = S_down ÷ S0 ; d = 1 ÷ u (when given as a recombining tree); The d = 1 ÷ u link holds only when the question builds the tree that way. Use the numbers given.
Risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d); r is the risk-free rate per period. The down probability is 1 − π. For a stock with a dividend yield, the numerator changes, so use the formula the question gives.
One-period option value: c0 = [π × c_up + (1 − π) × c_down] ÷ (1 + r); Works for calls and puts. Use the payoffs of the option you are pricing (for a put, replace c with p).
Hedge ratio: h = (V_up − V_down) ÷ (S_up − S_down), where V is the option value (c for a call, p for a put); Number of units of the underlying held against one short option to remove risk. For a call, h is between 0 and 1. For a put, h is negative.
Replicating portfolio value: c0 = h × S0 + B, where B = (c_down − h × S_down) ÷ (1 + r) and h = (c_up − c_down) ÷ (S_up − S_down); B is the bond position. It is negative when you borrow. It gives the same price as the risk-neutral method. For a put, use p in place of c.
Terminal payoffs: Call = max(0, S − X) ; Put = max(0, X − S); X is the exercise price. Compute at the final nodes first.
American option node value: Value = max(continuation value, exercise value); Apply at every node before expiry. For European options, use continuation value only.

How to solve Binomial Option Pricing Model questions

Use this order for any binomial question. It works for calls, puts, one or two periods, European or American.

1Write down S0, X, u, d, r per period and the number of periods. Check that r is per period.
2Build the tree of underlying prices at each node. Multiply by u or d.
3Compute π = (1 + r − d) ÷ (u − d) and 1 − π.
4Compute the option payoff at every final node: max(0, S − X) for a call, max(0, X − S) for a put.
5Work backward one step at a time. At each node: value = [π × up value + (1 − π) × down value] ÷ (1 + r).
6For an American option, at each earlier node compare that value with the exercise value and keep the larger one.
7If asked for the hedge ratio, use (option up − option down) ÷ (stock up − stock down) at that node.
8Check the answer is sensible: a call value should be positive and not above S0, and a put value should be positive and not above X (for a European put, not above the present value of X).

Quickest way: Risk-neutral roll-back on the calculator

When to use it: Use when the question gives u, d and r and asks only for the option value. Skip the replicating portfolio unless the hedge ratio is requested.

Compute π once and store it in memory (on the BA II Plus: type the value, press STO 1; recall with RCL 1).
Write payoffs next to each final node, and mark zero payoffs quickly.
If only some final nodes have a non-zero payoff, you can skip the others. Each such node contributes (number of paths to it) × π^k × (1 − π)^(n − k) × payoff ÷ (1 + r)^n, where k is the number of up moves and n the number of periods. Add the contributions. Use this only for a European option.
Discount each step by dividing by (1 + r). Do not discount twice in one step.
Eliminate options: for a non-dividend-paying stock, a European call must be at least max(0, S0 − X ÷ (1 + r)^n) and cannot exceed S0.

Common mistakes in Binomial Option Pricing Model

Using a real-world probability of an up move instead of the risk-neutral π.
The question may mention a chance of the stock rising, and it feels natural to use it.
Fix: Always compute π from u, d and r. Ignore real-world probabilities when pricing.
Using an annual risk-free rate for each period of a multi-period tree.
Students copy the rate from the stem without checking the length of a period.
Fix: Confirm r is per period. If the stem gives an annual rate and a half-year step, convert it as the question instructs.
Forgetting to compare with exercise value for an American option.
The roll-back method is the same as for European options, so the extra check is easy to skip.
Fix: At each node before expiry, write both continuation and exercise values and take the larger. This matters most for puts.
Discounting the final payoff for two periods in one step and then again at the middle nodes.
Mixing shortcut and step-by-step methods.
Fix: Pick one method. Either discount one period at each node, or use the full-path formula once.
Swapping the hedge ratio signs or the numerator for a put.
The formula is memorised without thinking about direction.
Fix: Use (option up − option down) ÷ (stock up − stock down). A put's value falls as the stock rises, so the ratio is negative.
Using the call payoff formula for a put.
Rushing through the final nodes.
Fix: Write max(0, S − X) or max(0, X − S) beside the tree before you start.

Worked examples

Example 1

A stock trades at $50. In one period it will go to $60 or $40. The risk-free rate is 5% per period. Find the value of a one-period European call with exercise price $50 and the hedge ratio.

Show the solution

u = 60 ÷ 50 = 1.2 and d = 40 ÷ 50 = 0.8.
π = (1 + 0.05 − 0.8) ÷ (1.2 − 0.8) = 0.25 ÷ 0.40 = 0.625. So 1 − π = 0.375.
Call payoffs: up = max(0, 60 − 50) = 10; down = max(0, 40 − 50) = 0.
Value = (0.625 × 10 + 0.375 × 0) ÷ 1.05 = 6.25 ÷ 1.05 = 5.952.
Hedge ratio = (10 − 0) ÷ (60 − 40) = 0.5.

Answer: The call is worth about $5.95 and the hedge ratio is 0.5 shares per call.

Example 2

A stock is at $100. Each period it rises 10% or falls by 10%. The risk-free rate is 5% per period. Value a two-period American put with exercise price $100, assuming the put is exercised early only if exercise value exceeds continuation value.

Show the solution

u = 1.10, d = 0.90. π = (1.05 − 0.90) ÷ (1.10 − 0.90) = 0.15 ÷ 0.20 = 0.75; 1 − π = 0.25.
Stock tree: period 1: 110 and 90. Period 2: 121, 99 and 81.
Put payoffs at expiry: at 121 = 0; at 99 = 1; at 81 = 19.
Node up (S = 110): continuation = (0.75 × 0 + 0.25 × 1) ÷ 1.05 = 0.2381. Exercise = max(0, 100 − 110) = 0. Value = 0.2381.
Node down (S = 90): continuation = (0.75 × 1 + 0.25 × 19) ÷ 1.05 = (0.75 + 4.75) ÷ 1.05 = 5.2381. Exercise = 100 − 90 = 10. Exercise is larger, so value = 10.
Today: continuation = (0.75 × 0.2381 + 0.25 × 10) ÷ 1.05 = (0.1786 + 2.5) ÷ 1.05 = 2.5510. Exercise today = 0. Value = 2.551.

Answer: The American put is worth about $2.55. It is optimal to exercise early at the down node where the stock is $90.

Exam tips

Questions are standalone three-option MCQs, so do the one-period roll-back and check which option is closest. Options are listed smallest to largest, which helps you spot a sign or discounting slip.
Compute π first and check that it lies between 0 and 1. If not, you have made an arithmetic error.
For American options, the early-exercise check is the likely trap. Look for a deep in-the-money put at a down node.
A hedge ratio question needs only the stock and option values at the two next nodes. Do not build the full tree.
Keep four decimals through the roll-back and round only at the end, as options may be close in value.

Practice questions from Pricing and Valuation of Options

Binomial Option Pricing Model in other exams

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

Binomial Option Pricing Model: frequently asked questions

What is the risk-neutral probability in the binomial model?

It is the probability of an up move that makes the underlying earn exactly the risk-free rate on average. It is π = (1 + r − d) ÷ (u − d). It is a pricing tool and not a forecast of the real chance of a rise.

How do I find the hedge ratio in a one-period binomial model?

Divide the difference in option values by the difference in stock prices: (option up − option down) ÷ (stock up − stock down). This is the number of shares held against each short option to remove risk.

How does an American option differ in a two-period tree?

You still roll back from expiry. At each earlier node, though, you compare the discounted expected value with the value of exercising right away and keep the larger. European options skip this comparison.

Can I price an option using replication instead of risk-neutral probabilities?

Yes. Both give the same value. Replication builds a portfolio of shares and a risk-free bond that matches the option payoffs. Risk-neutral pricing is usually faster in the exam.