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

Risk-Neutral Probabilities and Valuing Calls and Puts

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

It is the weight on the up state, π = (1 + r − d) ÷ (u − d), that makes the discounted expected underlying price equal to today's price. It is a pricing tool, not a forecast. It lets you value options without estimating investors' risk preferences.

Q: How is risk-neutral probability different from real-world probability?

Real-world probability is the actual chance of an up or down move and reflects investors' risk aversion. Risk-neutral probability is derived from u, d and the risk-free rate, so that discounting at the risk-free rate gives a no-arbitrage price. Option valuation uses the risk-neutral one.

Q: Why discount at the risk-free rate?

The risk-neutral probabilities already absorb the risk adjustment. Once you use them, expected payoffs are discounted as if investors required only the risk-free return. This gives the same value as the replicating portfolio.

Q: Do calls and puts use the same risk-neutral probability?

Yes. π depends only on u, d and r for the underlying. Only the payoffs differ between a call and a put.

Updated 7 October 2026 · Fact-checked

The risk-neutral probability of an up move is π = (1 + r − d) ÷ (u − d). Compute the option payoff in the up and down states, take the expected payoff using π and 1 − π, then discount at the risk-free rate: V = [π × Vu + (1 − π) × Vd] ÷ (1 + r). This works for European calls and puts.

Understand Risk-Neutral Probabilities and Valuing Calls and Puts

A one-period binomial model says the underlying price moves to only two values at expiry: an up price S × u or a down price S × d. The option payoff in each state is known. The question is what the option is worth today.

You could build a portfolio of the underlying and a risk-free bond that copies the option payoff in both states. By no-arbitrage, the option must cost the same as that portfolio. The risk-neutral method is a shortcut to the same answer.

The risk-neutral probability π is not the real-world chance of an up move. It is the probability that makes the underlying's expected value, discounted at the risk-free rate, equal to today's price. Think of it as a pricing weight. Real-world probabilities depend on investor risk appetite and are not needed to price the option.

Once you have π, treat the world as if investors were risk neutral. Take the expected option payoff with π and 1 − π, then discount at the risk-free rate. Use the same π for calls and puts on the same underlying, because the underlying and the rate are the same.

For a no-dividend asset, π is between 0 and 1 only if d < 1 + r < u. If that fails, there is an arbitrage opportunity in the underlying itself.

Key formulas to remember

Up and down factors: u = S↑ ÷ S0; d = S↓ ÷ S0; Up factor is above down factor. Often d = 1 ÷ u in problems, but only if the question says so.
Risk-neutral probability of up move: π = (1 + r − d) ÷ (u − d); r is the risk-free rate for the period. Assumes the underlying pays no income; with income, adjust as the question directs.
Risk-neutral probability of down move: 1 − π = (u − 1 − r) ÷ (u − d); Check that π and 1 − π add to 1.
Option value today: V0 = [π × Vu + (1 − π) × Vd] ÷ (1 + r); Discount at the risk-free rate, not a required return on the stock.
European call payoffs: cu = max(0, Su − X); cd = max(0, Sd − X); X is the exercise price.
European put payoffs: pu = max(0, X − Su); pd = max(0, X − Sd); A put pays when the price at expiry is below X.

How to solve Risk-Neutral Probabilities and Valuing Calls and Puts questions

Use this order for any one-period binomial valuation question.

1Write down S0, X, r for the period, and the up and down prices or factors u and d.
2Compute the underlying price in the up and down states: Su = S0 × u and Sd = S0 × d.
3Compute π = (1 + r − d) ÷ (u − d), then 1 − π.
4Compute the option payoffs in each state using max(0, Su − X) for a call or max(0, X − Su) for a put.
5Compute the expected payoff: π × Vu + (1 − π) × Vd.
6Discount one period at the risk-free rate: divide by (1 + r).
7Sanity check: call value < S0; put value < X ÷ (1 + r). Then select the option that matches your computed value.

Quickest way: Compute π once, then plug in payoffs

When to use it: Use when the question gives u, d and r directly and asks for a call or put value. Options are only three, so a rough value often eliminates two.

Get π in one line: (1 + r − d) ÷ (u − d). Store it in the calculator memory.
Find the payoffs. If one payoff is zero, the expected payoff is just π × the other payoff (for a call, π × Vu).
Divide by 1 + r once.
If you have both a call and a put question on the same tree, reuse π. Put-call parity can check your answer: c + X ÷ (1 + r) = p + S0 for a no-income asset.

Common mistakes in Risk-Neutral Probabilities and Valuing Calls and Puts

Using the real-world probability of an up move instead of π.
Problems often give a probability such as 60%, which looks like the one to use.
Fix: For valuation, always compute π from u, d and r. Real-world probabilities are a distractor.
Discounting at the stock's expected return or a required return.
Students are used to discounting cash flows at a risk-adjusted rate.
Fix: Under risk-neutral pricing, discount the expected payoff at the risk-free rate.
Forgetting to convert an annual rate to the period rate.
The tree may cover three or six months while r is quoted annually.
Fix: Match r to the length of the period before computing π and discounting.
Mixing up π and 1 − π when applying them to payoffs.
Both numbers appear in the formula, and the weights are easy to swap.
Fix: π always goes with the up state. Label Vu and Vd clearly.
Using the wrong payoff for a put.
Students reuse Su − X from the call.
Fix: A put payoff is max(0, X − S). Compute each state separately and never allow a negative payoff.
Applying the formula with d ≥ 1 + r or u ≤ 1 + r and not noticing π is outside 0 to 1.
Students plug in numbers without checking.
Fix: Check π lies between 0 and 1. If it does not, the inputs imply an arbitrage.

Worked examples

Example 1

A stock trades at $50. In one year it will either rise to $60 or fall to $40. The risk-free rate is 5% per year. What is the value of a one-year European call with exercise price $50? Options: A) $4.76 B) $5.95 C) $7.14

Show the solution

u = 60 ÷ 50 = 1.20 and d = 40 ÷ 50 = 0.80.
π = (1 + 0.05 − 0.80) ÷ (1.20 − 0.80) = 0.25 ÷ 0.40 = 0.625. So 1 − π = 0.375.
Call payoffs: cu = max(0, 60 − 50) = 10; cd = max(0, 40 − 50) = 0.
Expected payoff = 0.625 × 10 + 0.375 × 0 = 6.25.
Discount: 6.25 ÷ 1.05 = 5.952.

Answer: B) $5.95

Example 2

Using the same stock ($50, up to $60 or down to $40, risk-free rate 5%), what is the value of a one-year European put with exercise price $50? Options: A) $2.38 B) $3.57 C) $4.76

Show the solution

π is unchanged: 0.625, with 1 − π = 0.375.
Put payoffs: pu = max(0, 50 − 60) = 0; pd = max(0, 50 − 40) = 10.
Expected payoff = 0.625 × 0 + 0.375 × 10 = 3.75.
Discount: 3.75 ÷ 1.05 = 3.571.
Check with put-call parity: c + X ÷ (1 + r) − S0 = 5.952 + 47.619 − 50 = 3.571. It matches.

Answer: B) $3.57

Exam tips

Options are only A, B, C and listed smallest to largest. A quick estimate of the expected payoff often removes two choices.
Look out for a given real-world probability. It is usually there to tempt you; the answer uses π.
If a question asks for both call and put values, compute π once and check with put-call parity.
Check the period length. Quarterly or six-month trees need a period rate, not the annual one.
Do the calculation in one pass on your calculator, keeping π unrounded, and round only at the end.

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