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CFA Level I · CFA Level I Exam

Valuing a Derivative Using a One-Period Binomial Model

A one-period binomial model assumes the underlying price moves to only one of two values at expiry. You value the derivative by building a replicating portfolio, or by taking the expected payoff under risk-neutral probabilities and discounting it at the risk-free rate. Both methods give the same price.

What this chapter covers

This chapter teaches you to price a derivative with the simplest possible model. The underlying asset can move up or down once. That gives two possible payoffs for the derivative. With two outcomes you can build a portfolio of the underlying and a risk-free asset that matches the derivative exactly, and its cost is the derivative's value.

You will meet two routes to the same answer. The first is the replicating portfolio (or hedge) approach, which uses the hedge ratio h = (Vu − Vd) ÷ (Su − Sd). The second is the risk-neutral approach, where π = (1 + r − d) ÷ (u − d) and the value is [π × Vu + (1 − π) × Vd] ÷ (1 + r). The pricing logic rests on no-arbitrage. If the market price differs from the model value, you can lock in a riskless profit.

The chapter sits in Derivatives and Risk Management and builds on pricing and valuation of forwards, futures and options, and on time value of money and discounting. It also gives you the intuition behind option pricing in general. Later ideas such as put-call parity and the idea that option value does not depend on investors' risk preferences become much easier once you understand this model.

Derivatives and Risk Management carries a modest weight (6-9%) in the 2027 curriculum, but this chapter is mechanical and predictable. Questions are short, numerical and usually solvable in under 90 seconds once you have the formulas. Each question counts the same as any other, and there is no penalty for a wrong answer, so a reliable method here is cheap marks. The risk-neutral idea also helps you reject wrong options in conceptual questions on option valuation.

Valuing a Derivative Using a One-Period Binomial Model: topics in the order to study them

  1. 1Binomial Model Basics and AssumptionsStart here to learn the tree, u and d factors, up and down payoffs and the no-arbitrage setting that every later calculation uses.
  2. 2Replicating Portfolio and Hedge RatioNext, because it explains why the price is what it is: you build a riskless position from the underlying and a loan, and compute its cost.
  3. 3Risk-Neutral Probabilities and Valuing Calls and PutsOnce the replication logic is clear, this faster shortcut makes sense, and you can apply it to both calls and puts.
  4. 4Arbitrage and Extension to Other DerivativesFinish with mispricing and arbitrage trades, and extend the same method to other derivatives, as this tests your understanding of the whole framework.

How to prepare Valuing a Derivative Using a One-Period Binomial Model

Treat this chapter as a short procedure you can repeat without thinking. Aim for fluency, not reading time.

  1. Draw the one-step tree for every problem: S0 on the left, Su and Sd on the right, then the derivative payoffs Vu and Vd below them.
  2. Compute the payoffs at expiry first. For a call, Vu = max(Su − X, 0) and Vd = max(Sd − X, 0). For a put, use max(X − S, 0).
  3. Solve one problem with the hedge ratio method: find h, build the riskless portfolio, discount its value at 1 + r, and back out the derivative value.
  4. Solve the same problem with risk-neutral probabilities and confirm you get the same value. This is the best self-check you have.
  5. Practise with your BA II Plus or HP 12C, storing π in memory so you can reuse it. Keep the order of operations clear: compute π, then the expected payoff, then discount.
  6. Do mixed sets of calls, puts and mispriced options. For mispricing, state which side is cheap and what the arbitrage trade is.
  7. Finish with timed three-option MCQs. For numerical options, estimate the value first and eliminate the options that fall outside the possible range.

Common mistakes in Valuing a Derivative Using a One-Period Binomial Model

  • Using the real-world probability of an up move instead of the risk-neutral probability.

    Fix: Use π = (1 + r − d) ÷ (u − d) for valuation. A real-world probability is a distractor in pricing questions.

  • Discounting the expected payoff at the wrong rate.

    Fix: Under risk-neutral valuation, always discount at the risk-free rate for one period: divide by (1 + r).

  • Computing the payoff of a put with the call formula.

    Fix: Write the payoff formula next to the tree before filling it in: call max(S − X, 0), put max(X − S, 0).

  • Mixing up u, d and π, for example putting (u − 1 − r) in the numerator.

    Fix: Remember that π is the weight on the up state, so its numerator measures how far 1 + r is above d. Check that π lies between 0 and 1.

  • Getting the sign or direction of the hedge wrong in arbitrage questions.

    Fix: Decide first whether the derivative is overpriced or underpriced. Sell the expensive side and buy the cheap side, and always check that the future payoffs net to zero.

  • Forgetting to apply the up and down factors to the underlying price before computing payoffs.

    Fix: Apply u and d only to the underlying price S0. The strike stays fixed.

Last-day revision: Valuing a Derivative Using a One-Period Binomial Model

  • The one-period binomial model has one up move (u) and one down move (d) for the underlying.
  • Up and down prices: Su = S0 × u and Sd = S0 × d.
  • Hedge ratio h = (Vu − Vd) ÷ (Su − Sd).
  • A call has a positive h. A long call is replicated by long h units of the underlying plus borrowing. A put has a negative h. A long put is replicated by short |h| units of the underlying plus lending. To hedge, take the opposite position in the derivative.
  • Risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d), and the down probability is 1 − π.
  • Derivative value = [π × Vu + (1 − π) × Vd] ÷ (1 + r).
  • Discount at the risk-free rate, not at a required return on the underlying.
  • The risk-neutral probability is not the real-world probability of an up move.
  • Option value does not depend on investors' risk preferences.
  • If the market price is above the model value, sell the derivative and buy the replicating portfolio; if below, buy the derivative and sell the replicating portfolio.
  • Payoffs at expiry: call = max(S − X, 0); put = max(X − S, 0).
  • No arbitrage requires d < 1 + r < u.

Valuing a Derivative Using a One-Period Binomial Model practice questions

Valuing a Derivative Using a One-Period Binomial Model in other exams

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

Valuing a Derivative Using a One-Period Binomial Model: frequently asked questions

What is the hedge ratio in a one-period binomial model?

It is the number of units of the underlying that replicate the derivative, h = (Vu − Vd) ÷ (Su − Sd). For a call, holding h units of the underlying long against a short position in the derivative gives the same value in both states, so the combined portfolio is riskless. For a put, h is negative, so the offsetting position in the underlying has the opposite direction.

Why do we use risk-neutral probabilities if investors are not risk neutral?

Because the derivative can be replicated, its price does not depend on investors' risk preferences. The risk-neutral probabilities are a mathematical device that gives the same value as replication. They are not forecasts of what will happen.

Do I get a different answer from replication and from risk-neutral valuation?

No. If the inputs are the same, both methods give the same value. If your two answers differ, there is an arithmetic error, so use one method as a check on the other.

How should I handle these questions in the exam?

Draw the tree, compute the payoffs, then use the method you are fastest with, usually risk-neutral. With about 90 seconds per question and three options, a quick estimate can also remove one or two wrong answers.