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Strategic Financial Management · Options

Binomial Option Pricing Model: One and Two-Period Numericals

Updated 11 October 2026 · Fact-checked

The binomial model values an option by assuming the share price moves to only one of two prices each period. Build the price tree, find the option payoff at expiry, then work backward. Either use risk-neutral probability, discounted at the risk-free rate, or build a replicating portfolio of shares and borrowing.

Understand Binomial Option Pricing Model

An option's value today depends on what it can pay at expiry. The binomial model makes this simple. It assumes that in each period the share price can do only two things: go up by a factor u or go down by a factor d. Drawing these moves gives a tree of possible prices.

At expiry, you know the option payoff at every end node. A call pays the greater of (share price − strike) and 0. A put pays the greater of (strike − share price) and 0. The task is to bring these payoffs back to today.

There are two ways to do that, and both give the same answer. In the replicating portfolio method, you buy Δ shares and borrow some money at the risk-free rate so that the portfolio pays exactly what the option pays in both outcomes. The option must then cost the same as the portfolio, otherwise there is an arbitrage gain.

In the risk-neutral method, you find a probability p of an up move that makes the share earn exactly the risk-free rate. You compute the expected option payoff using p and discount it at the risk-free rate. This p is not the real-world probability. It is a pricing device. That is why the real chance of the share rising is not needed in the question.

For two periods, you repeat the one-period step at each node, moving backward from expiry. For European options, you only exercise at the end. For American options, you also compare the value at each earlier node with the value of exercising immediately.

Key rules to remember

Up and down factors
u = Su ÷ S0 ; d = Sd ÷ S0
Su and Sd are the prices after an up and a down move. In a two-period tree: Suu = S0 × u², Sud = S0 × u × d, Sdd = S0 × d².
Call and put payoff at expiry
Call = max(S − K, 0) ; Put = max(K − S, 0)
Apply at each end node of the tree. K is the strike price.
Hedge ratio (delta)
Δ = (Cu − Cd) ÷ (Su − Sd)
Number of shares in the replicating portfolio. Cu and Cd are the option values at the up and down nodes. For a put use Pu and Pd; Δ comes out negative, meaning you short the shares.
Borrowing in the replicating portfolio
Borrowing today = (Δ × Sd − Cd) ÷ (1 + r)
r is the risk-free rate for one period. This holds for a call, where the portfolio is Δ shares financed partly by borrowing.
Option value by replication
C0 = Δ × S0 − Borrowing today
Gives the same value as the risk-neutral method.
Risk-neutral probability
p = ((1 + r) − d) ÷ (u − d) ; probability of down move = 1 − p
Use the rate for one step. This p lies between 0 and 1 only if d < 1 + r < u.
Risk-neutral option value
C0 = [p × Cu + (1 − p) × Cd] ÷ (1 + r)
Same form for a put. Repeat at every node when working backward in a multi-period tree.
Early exercise check (American option)
Value at node = max(continuation value, exercise value)
Continuation value comes from the risk-neutral formula. Exercise value is the immediate payoff.

How to solve Binomial Option Pricing Model questions

Use this order for any binomial question, whether it is one period or two, call or put. Show each step because marks are given for method.

  1. 1Note S0, strike K, u and d (or the up and down prices), the risk-free rate per period, the number of periods, and whether the option is a call or a put and European or American.
  2. 2Draw the price tree. Compute the share price at every node.
  3. 3Compute the option payoff at every end node using max(S − K, 0) for a call or max(K − S, 0) for a put.
  4. 4Choose the method. If the question names replicating portfolio or hedge ratio, use Δ and borrowing. Otherwise the risk-neutral method is faster.
  5. 5For the risk-neutral method, compute p = ((1 + r) − d) ÷ (u − d). Check that it lies between 0 and 1.
  6. 6Work backward one step at a time. At each node, value = [p × up value + (1 − p) × down value] ÷ (1 + r). For an American option, compare with immediate exercise at that node and take the higher.
  7. 7State the option value today in rupees and round only at the end. If asked, verify with the other method or with put-call parity.

Quickest way: Risk-neutral backward induction

When to use it: Use it when the question asks only for the option value and does not specifically ask for the replicating portfolio. It is the shortest route for two-period trees.

  1. Find p = ((1 + r) − d) ÷ (u − d) once. Reuse it at every node.
  2. Write the end-node payoffs on the tree.
  3. At each earlier node, compute [p × up + (1 − p) × down] ÷ (1 + r).
  4. If the option is American, take the higher of that value and the exercise value at the node.
  5. For a European call and put on the same share, check C − P = S0 − K ÷ (1 + r)ⁿ. This confirms your arithmetic.

Common mistakes in Binomial Option Pricing Model

  • Using the real-world probability of an up move in the pricing formula.

    Students assume expected payoff must use the probability given in the question, such as 60% chance of rising.

    Fix: Option value uses only the risk-neutral p computed from u, d and r. Real probabilities do not affect the value.

  • Discounting the expected payoff at the wrong rate, or not discounting at every step in a two-period tree.

    Students jump from the end payoffs straight to today and discount once, or they use a cost of equity.

    Fix: Discount at the risk-free rate for one period at each backward step. Two periods mean two discounting steps.

  • Using an annual rate when the step is shorter than a year.

    The question gives an annual rate but each period is six months or three months.

    Fix: Convert the rate to the period first. For simple per-period rates, use annual rate × period length in years. If the question gives a continuous rate, use e^(rT) in place of (1 + r).

  • Forgetting that the option value at an intermediate node is needed before the final answer.

    Students compute only the end payoffs and apply one formula with probabilities p² and so on, then lose track of discounting.

    Fix: Compute the value at the up node and the down node first, then value the starting node from those two.

  • Taking the sign of Δ and borrowing wrongly, especially for puts.

    Students use the call formulas for the put without noting that Δ is negative.

    Fix: For a put the replicating portfolio shorts shares and lends. If unsure, use the risk-neutral method for the put and the replicating method only for calls.

  • Ignoring early exercise for an American option.

    Students value every option as European by habit.

    Fix: Read the option type. For American options, compare continuation value and exercise value at every node before the end.

Worked examples

Example 1

A share of an Indian company trades at ₹100 today. After one period it will be either ₹120 or ₹80. The risk-free rate is 5% per period. Value a one-period European call option with strike ₹105 using (a) the replicating portfolio and (b) the risk-neutral method.

Show the solution
  1. Payoffs: Cu = max(120 − 105, 0) = ₹15. Cd = max(80 − 105, 0) = ₹0.
  2. (a) Δ = (15 − 0) ÷ (120 − 80) = 15 ÷ 40 = 0.375 shares.
  3. Borrowing today = (0.375 × 80 − 0) ÷ 1.05 = 30 ÷ 1.05 = ₹28.57 (28.5714).
  4. Call value = 0.375 × 100 − 28.5714 = 37.5 − 28.5714 = ₹8.93 (8.9286).
  5. (b) u = 1.2 and d = 0.8. p = (1.05 − 0.8) ÷ (1.2 − 0.8) = 0.25 ÷ 0.40 = 0.625.
  6. Call value = [0.625 × 15 + 0.375 × 0] ÷ 1.05 = 9.375 ÷ 1.05 = ₹8.93 (8.9286).
  7. Both methods agree.

Answer: The call option is worth about ₹8.93.

Example 2

A share is priced at ₹100. In each of two periods it can rise by 20% or fall by 10%. The risk-free rate is 5% per period. Find the value of a two-period European call and a European put, both with strike ₹110. Check the answers with put-call parity.

Show the solution
  1. u = 1.20, d = 0.90. Prices: Su = 120, Sd = 90. Suu = 144, Sud = 120 × 0.9 = 108, Sdd = 81.
  2. p = (1.05 − 0.90) ÷ (1.20 − 0.90) = 0.15 ÷ 0.30 = 0.5.
  3. Call payoffs: Cuu = 144 − 110 = 34. Cud = max(108 − 110, 0) = 0. Cdd = 0.
  4. Cu = [0.5 × 34 + 0.5 × 0] ÷ 1.05 = 17 ÷ 1.05 = 16.1905. Cd = 0.
  5. C0 = [0.5 × 16.1905 + 0.5 × 0] ÷ 1.05 = 8.09524 ÷ 1.05 = 7.7098.
  6. Put payoffs: Puu = 0. Pud = 110 − 108 = 2. Pdd = 110 − 81 = 29.
  7. Pu = [0.5 × 0 + 0.5 × 2] ÷ 1.05 = 0.9524. Pd = [0.5 × 2 + 0.5 × 29] ÷ 1.05 = 15.5 ÷ 1.05 = 14.7619.
  8. P0 = [0.5 × 0.9524 + 0.5 × 14.7619] ÷ 1.05 = 7.85714 ÷ 1.05 = 7.4830.
  9. Check: C − P = 7.7098 − 7.4830 = 0.2268. S0 − K ÷ (1.05)² = 100 − 110 ÷ 1.1025 = 100 − 99.7732 = 0.2268. The values match.

Answer: The European call is worth about ₹7.71 and the European put about ₹7.48.

Exam tips

  • Draw the tree on the page, even for a one-period question. Examiners award marks for the tree and the payoff at each node.
  • If the question says replicating portfolio or hedge ratio, write Δ and the borrowing amount clearly. Do not switch to the risk-neutral method alone.
  • For MCQs, find p first. Many questions can be answered after p and one discounting step.
  • Check that the period rate matches the step length and that p lies between 0 and 1. If it does not, you have used the wrong rate or the wrong u and d.
  • End with a one-line statement of the option value in rupees, and a parity check if time permits.

Practice questions from Options

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

Why does the risk-neutral method give the same value as the replicating portfolio?

Both rely on there being no arbitrage. The replicating portfolio matches the option payoff exactly, so it must cost the same as the option. The risk-neutral probability is the one that makes this cost equal to the discounted expected payoff.

Do I need the real probability of the share going up?

No. The option value does not depend on it. The pricing probability p comes from u, d and the risk-free rate.

How do I value a put in the binomial model?

Use the same tree and method, but the end payoffs are max(K − S, 0). With the risk-neutral method, the formula is the same as for a call. You can also find the put from the call using put-call parity for European options.

What changes for an American option?

At each node before expiry, compare the continuation value from the tree with the payoff from exercising right away. Take the higher value and carry it back. For a European option you do not do this check.

Should I use u and d or the up and down prices?

Either works. If the question gives percentages or factors, apply them to get the prices. If it gives prices, divide by S0 to find u and d.