Economic Modelling · Binomial option-pricing model

One-Period Binomial Option Pricing and Replicating Portfolio

Q: What is a replicating portfolio in the binomial model?

It is a combination of Δ shares and an amount B in cash that gives the same payoff as the option in both the up and down states. Because the payoffs match, the option must cost the same as the portfolio today.

Q: Why does the real-world probability of an up move not matter?

Replication fixes the portfolio using the two possible payoffs only. No-arbitrage then forces the price. Probabilities would only matter for expected returns, not for the price.

Q: What does delta mean in the one-period binomial model?

Delta is the number of shares held in the replicating portfolio. It equals the change in option payoff divided by the change in share price between the two states. The seller hedges by holding the replicating portfolio: Δ shares (short if Δ is negative) plus cash B.

Q: What happens if d < 1 + r < u does not hold?

There is an arbitrage opportunity in the shares and cash alone. If u 1 + r, the share always beats cash, so you could borrow cash at r, buy the share and make a risk-free profit.

Updated 11 October 2026 · Fact-checked

The one-period binomial model lets a share price move to only one of two values, up or down. You build a portfolio of Δ shares and cash B that matches the option payoff in both states. By no-arbitrage, the option price equals the cost of that portfolio today: Δ·S0 + B.

Understand One-Period Binomial Model and Replicating Portfolio

An option has a payoff at expiry that depends on the share price. In the one-period binomial model, the share price today is S0. After one period it becomes either S0·u (up) or S0·d (down), with d < u. Nothing else can happen. This simple world is enough to show how option pricing works.

The key idea is replication. You can buy Δ shares and hold an amount B in cash (B is negative if you borrow). You choose Δ and B so that the portfolio is worth exactly the same as the option in both the up state and the down state. Two states give two equations, and Δ and B are the two unknowns.

If two assets give the same payoff in every state, they must have the same price today. Otherwise you could buy the cheaper one, sell the dearer one and lock in a risk-free profit. This is the no-arbitrage principle. So the option price equals Δ·S0 + B.

Notice what the price does not depend on. It does not depend on the real-world probability of an up move, or on how risk averse investors are. It depends only on S0, u, d, the interest rate and the payoff. That is why the method is powerful.

Delta is the number of shares in the replicating portfolio. For a call it is positive, and for a put it is negative (you short shares). Delta is also the hedge ratio: the seller of the option hedges by holding the replicating portfolio, which is Δ shares (a short position if Δ is negative) plus cash B. This removes all risk.

Key rules to remember

Share price after one period: S_up = S0 × u; S_down = S0 × d; Requires d < u. No-arbitrage needs d < (1 + r) < u.
Option payoffs: Call: max(S_T − K, 0); Put: max(K − S_T, 0); Compute the payoff in the up state (Cu) and the down state (Cd) separately.
Delta (hedge ratio): Δ = (Cu − Cd) ÷ (S0·u − S0·d); Change in option payoff divided by change in share price between the two states.
Cash position: B = (Cu − Δ·S0·u) ÷ (1 + r) = (Cd − Δ·S0·d) ÷ (1 + r); Using discrete interest r for the period. With continuous rate, replace (1 + r) by e^(rT). Negative B means borrowing.
Option price: C0 = Δ·S0 + B; Cost today of the replicating portfolio.
No-arbitrage condition: d < 1 + r < u; If it fails, you can make a risk-free profit from the shares and cash alone.

How to solve One-Period Binomial Model and Replicating Portfolio questions

Use this method for any one-period replication question. State your assumptions first: one period, no dividends, no transaction costs, borrowing and lending at the same risk-free rate.

1Write down S0, u, d, strike K, interest rate r (or continuous rate) and the period length.
2Compute the two share prices: S0·u and S0·d.
3Compute the option payoff in each state: Cu and Cd, using the correct call or put payoff.
4Set up the replicating portfolio: Δ shares and cash B. Write the two equations: Δ·S0·u + B(1 + r) = Cu and Δ·S0·d + B(1 + r) = Cd.
5Subtract the equations to find Δ = (Cu − Cd) ÷ (S0·u − S0·d).
6Substitute Δ back into one equation to find B. Check the other equation as well.
7Price the option as C0 = Δ·S0 + B. State whether you buy or short shares, and whether you borrow or lend.
8Sanity check: a European call price must lie between max(S0 − K/(1 + r), 0) and S0. A European put price must lie between max(K/(1 + r) − S0, 0) and K/(1 + r).

Quickest way: Delta first, then price by difference

When to use it: Use when the exam gives numbers and asks only for the option price or the hedge, and time is short.

Find Cu, Cd and the two share prices.
Compute Δ = (Cu − Cd) ÷ (Su − Sd) at once.
Find B from the down state, which is often zero payoff for a call: B = (Cd − Δ·Sd) ÷ (1 + r).
Compute C0 = Δ·S0 + B.
Check with the up-state equation. If it fails, you made an arithmetic slip.

Common mistakes in One-Period Binomial Model and Replicating Portfolio

Using the real-world probability of an up move to price the option.
Students treat the option like an expected-value problem.
Fix: In replication, probabilities never appear. Solve the two payoff equations for Δ and B.
Forgetting to discount the cash amount B.
Students write Δ·Su + B = Cu instead of Δ·Su + B(1 + r) = Cu.
Fix: B is invested or borrowed for the period. It grows by (1 + r) or e^(rT), so include that factor in both equations.
Getting the sign of Δ wrong for a put.
Students expect Δ to be positive for any option.
Fix: A put gains when the share falls, so Δ is negative. Compute (Pu − Pd) ÷ (Su − Sd) with signs and trust the result.
Taking the payoff as S_T − K without the max with zero.
Rushing through the payoff step.
Fix: Always write max(S_T − K, 0) for a call. A negative payoff is not possible for a holder.
Dividing by S0 instead of (Su − Sd) when finding Δ.
Mixing up the formula with a percentage-change idea.
Fix: Δ is payoff difference over share price difference between the two states.
Not stating whether B is borrowing or lending.
Students stop at the numerical answer.
Fix: Say it in words. Negative B means you borrow. Examiners award marks for interpreting the portfolio.

Worked examples

Example 1

A share is priced at ₹100. After one year it will be either ₹120 or ₹90. The annual risk-free rate is 5% (discrete annual). Find the price of a one-year European call with strike ₹105 using a replicating portfolio.

Show the solution

Payoffs: Cu = max(120 − 105, 0) = 15. Cd = max(90 − 105, 0) = 0.
Delta: Δ = (15 − 0) ÷ (120 − 90) = 15 ÷ 30 = 0.5.
Down state equation: 0.5 × 90 + B × 1.05 = 0, so 45 + 1.05B = 0 and B = −42.857.
Check up state: 0.5 × 120 + (−42.857 × 1.05) = 60 − 45 = 15. Correct.
Price: C0 = 0.5 × 100 − 42.857 = 7.143.

Answer: The call costs about ₹7.14. The replicating portfolio is 0.5 shares bought, financed by borrowing about ₹42.86 at 5%.

Example 2

A share is ₹200 now. After one period it is either ₹250 or ₹160. The one-period risk-free rate is 10% (discrete). Find the price of a European put with strike ₹210 and the replicating portfolio.

Show the solution

Payoffs: Pu = max(210 − 250, 0) = 0. Pd = max(210 − 160, 0) = 50.
Delta: Δ = (0 − 50) ÷ (250 − 160) = −50 ÷ 90 = −0.5556.
Up state equation: −0.5556 × 250 + B × 1.10 = 0, so −138.889 + 1.10B = 0 and B = 126.263.
Check down state: −0.5556 × 160 + 126.263 × 1.10 = −88.889 + 138.889 = 50. Correct.
Price: P0 = −0.5556 × 200 + 126.263 = −111.111 + 126.263 = 15.152.

Answer: The put costs about ₹15.15. The replicating portfolio is a short position of 0.5556 shares with about ₹126.26 lent at 10%.

Exam tips

Show both payoff equations clearly. Method marks are usually given for setting them up even if arithmetic slips.
State assumptions: no dividends, no transaction costs, same borrowing and lending rate, perfectly divisible shares.
Always check the answer using the second state. It takes ten seconds and catches most errors.
Interpret the result in words: how many shares, whether you borrow or lend, and what a seller of the option would hold to hedge.
In multiple-choice questions, check the no-arbitrage condition d < 1 + r < u first. Some questions test whether it holds.

Practice questions from Binomial option-pricing model

One-Period Binomial Model and Replicating Portfolio in other exams

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

One-Period Binomial Model and Replicating Portfolio: frequently asked questions

What is a replicating portfolio in the binomial model?

It is a combination of Δ shares and an amount B in cash that gives the same payoff as the option in both the up and down states. Because the payoffs match, the option must cost the same as the portfolio today.

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

Replication fixes the portfolio using the two possible payoffs only. No-arbitrage then forces the price. Probabilities would only matter for expected returns, not for the price.

What does delta mean in the one-period binomial model?

Delta is the number of shares held in the replicating portfolio. It equals the change in option payoff divided by the change in share price between the two states. The seller hedges by holding the replicating portfolio: Δ shares (short if Δ is negative) plus cash B.

What happens if d < 1 + r < u does not hold?

There is an arbitrage opportunity in the shares and cash alone. If u < 1 + r, cash always beats the share, so you could borrow the share (sell it short), lend the money and make a risk-free profit. If d > 1 + r, the share always beats cash, so you could borrow cash at r, buy the share and make a risk-free profit.