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CFA Level II Exam · Valuation of Contingent Claims

Binomial Option Pricing Model for CFA Level II

Updated 7 October 2026 · Fact-checked

The binomial model values an option using a tree of stock prices. Find the option payoffs at expiry, compute the risk-neutral probability π = (1 + r − d) ÷ (u − d), then discount expected values back one step at a time at the risk-free rate. For American options, also compare exercise value at each node.

Understand Binomial Option Pricing Model

An option's value depends on what the stock might do. The binomial model simplifies this: in each period the stock moves to one of two prices, an up price S0 × u or a down price S0 × d. Repeat this over several periods and you get a tree.

You can value the option in two equivalent ways. Replication builds a portfolio of the stock and risk-free borrowing or lending that pays exactly what the option pays in both states. If the option and the portfolio have the same payoff, they must have the same price today, or there is an arbitrage. The risk-neutral method gives the same answer faster. You pretend investors are risk neutral, find the probability π that makes the stock grow at the risk-free rate, take the expected option payoff using π, and discount at the risk-free rate.

Important point: π is not the real-world probability of an up move. It is a number that makes the maths consistent with no arbitrage. The real-world probability never enters the price.

For two periods, work backward. Compute option payoffs at the final nodes, then value the option at each earlier node using only the two nodes that follow it. Continue until you reach time 0.

A European option can be exercised only at expiry, so you only discount. An American option can be exercised at any node. At every node before expiry you compare the exercise value with the continuation value (the discounted expected value) and keep the larger one. American calls on non-dividend-paying stocks are not exercised early, so their value equals the European call. American puts often are worth exercising early, especially when deep in the money.

Key formulas to remember

Up and down factors
u = S(up) ÷ S0; d = S(down) ÷ S0; often d = 1 ÷ u
The exam usually gives u and d, or gives the prices directly. When built from volatility, u = e^(σ√Δt).
Risk-neutral probability of an up move
π = (1 + r − d) ÷ (u − d)
Here r is the risk-free rate per period. Probability of a down move = 1 − π. For a stock paying a dividend yield or an asset with carry, adjust the numerator for the benefit or cost, as the question specifies.
Option value, one period
V0 = [π × V(up) + (1 − π) × V(down)] ÷ (1 + r)
Use at every node when working backward in a multi-period tree.
Hedge ratio (delta)
h = [V(up) − V(down)] ÷ [S(up) − S(down)]
h is the number of shares in the portfolio that replicates one option. A call has 0 < h < 1 and a put has −1 < h < 0, so replicating a put means shorting shares. To hedge a long put, buy |h| shares. To hedge a short call, buy h shares.
Replication value of a call
c0 = h × S0 − PV of borrowing, where borrowing = [h × S(down) − c(down)] ÷ (1 + r)
The replicating portfolio holds h shares and borrows the amount needed to match payoffs in both states.
Terminal payoffs
Call = max(0, S − K); Put = max(0, K − S)
Compute these at every final node first.
American option value at a node
V = max(exercise value, continuation value)
Exercise value is the immediate payoff. Continuation value is the discounted risk-neutral expectation. European options use continuation value only.

How to solve Binomial Option Pricing Model questions

Use this order for any binomial tree question in an item set. It works for calls and puts, European and American.

  1. 1Pull from the vignette: S0, u and d (or the node prices), strike K, risk-free rate per period, number of periods, option type, and whether it is European or American.
  2. 2Check that r matches the period length. If the rate is annual and each step is a quarter, convert it to a per-period rate as the question implies.
  3. 3Compute π = (1 + r − d) ÷ (u − d) and 1 − π. Check that π lies between 0 and 1.
  4. 4Build the stock prices at each node by multiplying by u or d.
  5. 5Compute the option payoffs at the final nodes: max(0, S − K) for a call, max(0, K − S) for a put.
  6. 6Work backward one node at a time: V = [π × V(up) + (1 − π) × V(down)] ÷ (1 + r).
  7. 7If the option is American, at each node before expiry compare this value with the immediate exercise value and keep the larger. Carry that larger value back.
  8. 8If the question asks for the hedge ratio or replication, use h = [V(up) − V(down)] ÷ [S(up) − S(down)] at the relevant node.

Quickest way: Risk-neutral roll-back with a payoff check

When to use it: Use when the question asks only for the option value or an early-exercise decision, not for the hedge ratio or an arbitrage profit.

  1. Calculate π once and keep it on your calculator memory.
  2. Write the final-node payoffs next to the tree on your scratch paper.
  3. Roll back using the same π at every node. Do not recompute π.
  4. For a put, check the lowest node first. If the exercise value there exceeds the roll-back value, early exercise is likely at that node for an American put.
  5. Do a sanity check: a call cannot be worth more than S0, and a European put cannot be worth more than the PV of K.

Common mistakes in Binomial Option Pricing Model

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

    The vignette may give a probability such as 60% up, and it looks like the one to use.

    Fix: Option values use only the risk-neutral π = (1 + r − d) ÷ (u − d). Real-world probabilities are a distractor.

  • Forgetting to compare with exercise value for an American option.

    Candidates roll back the tree the European way and stop.

    Fix: At each node before expiry, write both numbers (exercise and continuation) and take the larger. Then use that larger number in the next step back.

  • Using the annual risk-free rate in a multi-period tree with shorter steps.

    The rate is quoted once in the vignette and is copied straight into the formula.

    Fix: Use the rate that applies per step. Confirm what period each step represents before computing π and discounting.

  • Discounting at the wrong point, such as discounting the terminal payoff twice at one step or skipping a step.

    A two-period tree has three node valuations to discount: two middle nodes when stepping back from expiry, then one at time 0. Each is discounted once at (1 + r), and it is easy to lose count.

    Fix: Discount once at every step back. Value each middle node first, using only its two following nodes, then value time 0 from the two middle nodes.

  • Getting the sign or direction of the hedge ratio wrong for a put.

    Candidates apply the call intuition of buying shares.

    Fix: Compute h from the formula. A negative h means the replicating portfolio of a put is short shares. A long put is hedged by buying shares.

  • Mixing up π and 1 − π when applying them to the up and down values.

    Candidates compute π correctly but attach it to the wrong payoff.

    Fix: π always multiplies the up-node value. 1 − π always multiplies the down-node value.

Worked examples

Example 1

Vignette: A stock trades at 50. Over one period it will either rise to 60 or fall to 40. The risk-free rate is 5% per period. An analyst values a European call option with a strike of 50. Questions: (1) What is the risk-neutral probability of an up move? (2) What is the call value using the risk-neutral method? (3) How many shares does the replicating portfolio hold, and how much does it borrow?

Show the solution
  1. u = 60 ÷ 50 = 1.20 and d = 40 ÷ 50 = 0.80.
  2. (1) π = (1 + 0.05 − 0.80) ÷ (1.20 − 0.80) = 0.25 ÷ 0.40 = 0.625. So 1 − π = 0.375.
  3. Call payoffs: up = max(0, 60 − 50) = 10; down = max(0, 40 − 50) = 0.
  4. (2) c0 = (0.625 × 10 + 0.375 × 0) ÷ 1.05 = 6.25 ÷ 1.05 = 5.9524.
  5. (3) h = (10 − 0) ÷ (60 − 40) = 0.5 shares.
  6. In the down state the 0.5 shares are worth 0.5 × 40 = 20, and the option pays 0, so the loan repayment must be 20. Amount borrowed today = 20 ÷ 1.05 = 19.0476.
  7. Check: cost of replicating portfolio = 0.5 × 50 − 19.0476 = 25 − 19.0476 = 5.9524, which matches the risk-neutral value. In the up state: 0.5 × 60 − 20 = 10, which matches the call payoff.

Answer: (1) π = 0.625. (2) The call is worth about 5.95. (3) Hold 0.5 shares and borrow about 19.05 today.

Example 2

Vignette: A stock is at 100. Each period it rises by a factor of 1.25 or falls by a factor of 0.80. There are two periods. The risk-free rate is 5% per period. An analyst values put options with a strike of 110. Questions: (1) What is the risk-neutral probability of an up move? (2) What is the value of a European put? (3) What is the value of an American put, and at which node is early exercise optimal?

Show the solution
  1. u = 1.25, d = 0.80. (1) π = (1.05 − 0.80) ÷ (1.25 − 0.80) = 0.25 ÷ 0.45 = 5/9 = 0.5556. Then 1 − π = 4/9 = 0.4444.
  2. Stock prices: up = 125, down = 80. After two periods: uu = 156.25, ud = 100, dd = 64.
  3. Put payoffs at expiry (K = 110): uu = max(0, 110 − 156.25) = 0; ud = 110 − 100 = 10; dd = 110 − 64 = 46.
  4. European put, node up (S = 125): (5/9 × 0 + 4/9 × 10) ÷ 1.05 = 4.4444 ÷ 1.05 = 4.2328.
  5. European put, node down (S = 80): (5/9 × 10 + 4/9 × 46) ÷ 1.05 = (5.5556 + 20.4444) ÷ 1.05 = 26 ÷ 1.05 = 24.7619.
  6. (2) European put at time 0: (5/9 × 4.2328 + 4/9 × 24.7619) ÷ 1.05 = (2.3516 + 11.0053) ÷ 1.05 = 13.3569 ÷ 1.05 = 12.72.
  7. American put, node up (S = 125): exercise value = max(0, 110 − 125) = 0, continuation 4.2328, so value 4.2328.
  8. American put, node down (S = 80): exercise value = 110 − 80 = 30, continuation = 24.7619. Exercise is better, so value = 30.
  9. (3) American put at time 0: continuation = (5/9 × 4.2328 + 4/9 × 30) ÷ 1.05 = (2.3516 + 13.3333) ÷ 1.05 = 15.6849 ÷ 1.05 = 14.94. Exercise now = 110 − 100 = 10, which is less, so do not exercise at time 0.

Answer: (1) π = 5/9, about 0.556. (2) European put is about 12.72. (3) American put is about 14.94, with early exercise optimal at the down node (S = 80) after the first period.

Exam tips

  • Every tree question sits inside a vignette. Underline u, d, r per period, K and the exercise style before you calculate anything.
  • Ignore real-world up and down probabilities if they are given. Only π enters the option value.
  • For American puts, test the down nodes first. That is where early exercise is most likely, and where the answer usually differs from the European value.
  • Know the qualitative points too: an American option is worth at least as much as the European option with the same terms, and an American call on a non-dividend-paying stock has the same value as the European call.
  • Use the hedge ratio and replication logic for questions about arbitrage or how to hedge. Both give the same value as the risk-neutral method, so use one as a check if time allows.

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

How do I calculate the risk-neutral probability in a binomial tree?

Use π = (1 + r − d) ÷ (u − d), where r is the risk-free rate per period. The probability of a down move is 1 − π. Check that π is between 0 and 1, otherwise there is an arbitrage in the inputs.

What is the difference between European and American option valuation in a binomial tree?

For a European option you only roll back the discounted expected values and apply the payoff at expiry. For an American option, at each node before expiry you also compare the immediate exercise value with the continuation value and keep the larger. This can only raise the value or leave it unchanged.

Does the real-world probability of the stock rising matter?

No. The option price depends only on the risk-neutral probability, which is built from u, d and the risk-free rate. The real-world probability does not change the arbitrage-free value.

How do I solve a two-period binomial tree quickly?

Compute π once, write the three terminal payoffs, then value the two middle nodes, then time 0. Use the same π and the same one-period discount at each step. For American options, compare with exercise value at each middle node and at time 0.