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Economic Modelling · Principles of option pricing

Binomial Tree Option Pricing: Replicating Portfolio and Risk-Neutral Method

Updated 11 October 2026 · Fact-checked

Binomial option pricing models the share price as moving up or down each period. You find the option payoffs, then price by a replicating portfolio of shares and borrowing, or by discounting expected payoff under risk-neutral probability q = (1 + r − d) ÷ (u − d). Work backwards through the tree to time 0.

Understand Binomial Tree Option Pricing

An option's payoff depends on a share price that is uncertain. The binomial model simplifies this. Over one period the price S either rises to S·u or falls to S·d. Only two outcomes are possible, so the model is easy to handle by hand.

The key idea is replication. You can build a portfolio of Δ shares and a cash amount (positive is lending, negative is borrowing) that has exactly the same payoff as the option in both the up and down states. If two things pay the same in every state, they must cost the same today. Otherwise there is an arbitrage. So the option price equals the cost of the replicating portfolio. Real-world probabilities of up and down play no part.

There is a shortcut. Solving the replication equations gives a number q that behaves like a probability. It is the risk-neutral probability. The option price is the expected payoff using q, discounted at the risk-free rate. Do not read q as the real chance of an up move. It is just the weight that makes the share earn the risk-free rate on average.

For several periods, build the tree of share prices, write the option payoffs at the final nodes, then step back one period at a time. At each node apply the one-period rule to the two nodes that follow. For a European option you only need the payoff at expiry. For an American option you also compare with the exercise value at each node and take the larger.

The model needs u > 1 + r > d. If this fails, an arbitrage exists and q falls outside 0 to 1. Unless told otherwise, assume no dividends, no transaction costs and that you can borrow and lend at the risk-free rate r per period.

Key rules to remember

Share price nodes
S(up) = S·u, S(down) = S·d; after j ups and n − j downs: S·u^j·d^(n−j)
For a recombining tree, an up then down equals a down then up.
Risk-neutral probability
q = (1 + r − d) ÷ (u − d)
Here r is the rate per period, and the formula is for discrete compounding. If the question gives a continuous rate δ, use e^δ in place of 1 + r.
One-period option price
V0 = [q·Vu + (1 − q)·Vd] ÷ (1 + r)
Vu and Vd are the option values at the up and down nodes. Use the same step at every node.
Delta (shares held)
Δ = (Vu − Vd) ÷ (S·u − S·d)
Positive for a call, negative for a put (a short share position).
Cash in replicating portfolio
B = [Vu − Δ·S·u] ÷ (1 + r)
Negative B means borrowing. Check with the down state: B = [Vd − Δ·S·d] ÷ (1 + r).
Option price by replication
V0 = Δ·S + B
This must equal the risk-neutral price. Use it as a check.
Multi-period European price
V0 = (1 + r)^(−n) · Σ C(n, j) · q^j · (1 − q)^(n−j) · payoff(S·u^j·d^(n−j))
Valid for a recombining tree with constant u, d and r.
American option node value
V = max[exercise value, continuation value]
Continuation value is the discounted q-expectation of the next two node values.

How to solve Binomial Tree Option Pricing questions

Use this order for any binomial question, whether it asks for a price, a hedge or a check for early exercise.

  1. 1Write down S0, u, d, strike K, the rate per period r and the number of periods n. Check the time step matches the rate given.
  2. 2Draw the tree of share prices. Label each node with its price.
  3. 3Calculate q = (1 + r − d) ÷ (u − d). Confirm 0 < q < 1.
  4. 4At expiry write the option payoff at each node: max(S − K, 0) for a call, max(K − S, 0) for a put.
  5. 5Work backwards. At each earlier node, value = [q × up value + (1 − q) × down value] ÷ (1 + r).
  6. 6For an American option, at every node also compute the immediate exercise value and take the maximum before moving back.
  7. 7If asked for the replicating portfolio, find Δ = (Vu − Vd) ÷ (Su − Sd) and B = V − Δ·S at that node. Check it reproduces both payoffs.
  8. 8State the answer with units (₹) and any assumption such as no dividends.

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

When to use it: Use when the question asks only for the option price, especially in multi-period European questions with small trees.

  1. Compute q once and also 1 − q. Keep them as fractions if they are neat, such as 2/3 and 1/3.
  2. Write the terminal payoffs only. Ignore nodes with zero payoff.
  3. For European options with two periods, use the direct formula: V0 = [q²·Vuu + 2q(1 − q)·Vud + (1 − q)²·Vdd] ÷ (1 + r)².
  4. Discount once at the end. Avoid rounding q early.
  5. Spot-check: a call must be at least max(S0 − K ÷ (1 + r)^n, 0) and at most S0. A put must not exceed K ÷ (1 + r)^n for a European option.

Common mistakes in Binomial Tree Option Pricing

  • Using real-world probabilities of up and down to price the option.

    Expected payoff feels natural, and the question may mention the chance of a rise.

    Fix: Price with q only. Real-world probabilities do not affect the no-arbitrage price.

  • Writing q = (1 + r − d) ÷ (u − d) but swapping u and d, or using r instead of 1 + r.

    The formula is memorised without understanding that the share must grow at 1 + r on average under q.

    Fix: Re-derive: q·u + (1 − q)·d = 1 + r. Solve for q. Always check q lies between 0 and 1.

  • Forgetting to discount at each step in multi-period trees, or discounting twice.

    Students mix the roll-back method with the direct formula.

    Fix: In roll-back, discount by 1 + r at every step. In the direct formula, discount once by (1 + r)^n.

  • Taking Δ with the wrong sign for a put, or treating negative B as lending.

    Sign conventions are not stated clearly.

    Fix: Define Δ as shares held and B as cash lent. A put has negative Δ (short shares) and positive B (lending). Check both payoffs.

  • Not testing early exercise in American options at interior nodes.

    Students only compare at expiry, as for European options.

    Fix: At every node take the maximum of exercise value and continuation value, and use that maximum in the earlier roll-back.

  • Using an annual rate on a period that is not one year.

    The tree step is, say, six months but r is quoted annually.

    Fix: Convert r to the length of one step before computing q and discounting.

Worked examples

Example 1

A share is priced at ₹100. After one year it will be either ₹120 or ₹90. The risk-free rate is 5% per year. Find the price of a one-year European call with strike ₹100 by (a) replication and (b) risk-neutral valuation.

Show the solution
  1. Payoffs: Vu = max(120 − 100, 0) = 20 and Vd = max(90 − 100, 0) = 0.
  2. (a) Delta: Δ = (20 − 0) ÷ (120 − 90) = 2/3.
  3. Up state: Δ × 120 = 80. We need 20, so 80 + 1.05B = 20, giving B = −60 ÷ 1.05 = −57.143. This is borrowing.
  4. Down state check: (2/3) × 90 = 60, and 1.05 × (−57.143) = −60. Total = 0. Matches Vd.
  5. Cost today = Δ × 100 + B = 66.667 − 57.143 = 9.524.
  6. (b) q = (1.05 − 0.9) ÷ (1.2 − 0.9) = 0.15 ÷ 0.30 = 0.5.
  7. Price = [0.5 × 20 + 0.5 × 0] ÷ 1.05 = 10 ÷ 1.05 = 9.524. Both methods agree.

Answer: The call price is about ₹9.52. The replicating portfolio holds 2/3 of a share and borrows about ₹57.14.

Example 2

A share is ₹80. In each of two periods it rises by a factor u = 1.25 or falls by a factor d = 0.8. The risk-free rate is 10% per period. Price a two-period European put with strike ₹85.

Show the solution
  1. Share prices: Su = 100, Sd = 64; Suu = 125, Sud = 80, Sdd = 51.2.
  2. Put payoffs at expiry: Vuu = max(85 − 125, 0) = 0; Vud = 85 − 80 = 5; Vdd = 85 − 51.2 = 33.8.
  3. q = (1.10 − 0.8) ÷ (1.25 − 0.8) = 0.30 ÷ 0.45 = 2/3, so 1 − q = 1/3.
  4. Value at the up node: Vu = [(2/3) × 0 + (1/3) × 5] ÷ 1.1 = 1.6667 ÷ 1.1 = 1.5152.
  5. Value at the down node: Vd = [(2/3) × 5 + (1/3) × 33.8] ÷ 1.1 = (3.3333 + 11.2667) ÷ 1.1 = 14.6 ÷ 1.1 = 13.2727.
  6. Value at time 0: V0 = [(2/3) × 1.5152 + (1/3) × 13.2727] ÷ 1.1 = (1.0101 + 4.4242) ÷ 1.1 = 5.4343 ÷ 1.1 = 4.9403.
  7. Check by direct formula: [(4/9) × 0 + (4/9) × 5 + (1/9) × 33.8] ÷ 1.21 = (2.2222 + 3.7556) ÷ 1.21 = 5.9778 ÷ 1.21 = 4.9403.

Answer: The European put is worth about ₹4.94.

Exam tips

  • Show q and its formula first. Examiners award marks for the method even if arithmetic slips later.
  • Keep q as a fraction when it is neat. It reduces rounding errors in multi-period trees.
  • For written answers, explain in one sentence why real-world probabilities are not used. This is a common reasoning mark.
  • If the question mentions American options, test early exercise at every node, not just at expiry. Puts are the usual case where it matters.
  • Check the answer against simple bounds. A call price above the share price signals an error. Also check that q is between 0 and 1.

Practice questions from Principles of option pricing

Binomial Tree Option Pricing in other exams

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

Binomial Tree Option Pricing: frequently asked questions

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

Use q = (1 + r − d) ÷ (u − d), where r is the risk-free rate per period. It comes from setting q·u + (1 − q)·d = 1 + r. If q is not between 0 and 1, the model allows arbitrage.

Why does the real probability of an up move not matter?

The option can be replicated by shares and cash. Its price must equal the cost of that portfolio to avoid arbitrage. That cost depends only on u, d, r and the payoffs, not on the real chance of up or down.

What does Δ mean in a replicating portfolio?

Δ is the number of shares held. It equals the spread in option payoffs divided by the spread in share prices. The rest of the portfolio is cash, which is borrowing if negative.

How is a multi-period tree different from a one-period tree?

You repeat the one-period step. Write the payoffs at the final nodes, then move back one period at a time, discounting at each step. For American options, you also compare with the exercise value at each node.