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Advanced Financial Management · Derivatives Analysis and Valuation

Binomial Option Pricing Model for CA Final AFM

Updated 5 October 2026 · Fact-checked

The binomial model values an option by assuming the share price moves to only one of two values each period. You find the option payoff at each end node, then roll back using risk-neutral probability p = ((1 + r) − d) ÷ (u − d), discounting at the risk-free rate. A replicating portfolio gives the same value.

Understand Binomial Option Pricing Model

An option's value today depends on what it will pay later. The binomial model breaks the time to expiry into small periods. 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.

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

There are two ways to roll back, and both give the same answer. In the replicating portfolio method, you build a mix of Δ shares and some borrowing that gives exactly the option payoff in both up and down states. Since the mix and the option have the same payoff, they must have the same cost today. In the risk-neutral method, you find a probability p under which the share earns only the risk-free rate. You then take the expected payoff using p and discount at the risk-free rate.

The real-world chance of the share rising does not enter the calculation. This surprises students. The u and d factors already carry the risk, and the no-arbitrage argument does the rest.

For a two-period tree, you work backwards. First value the option at each node of the middle period, treating each as a one-period problem. Then treat those values as payoffs and do one more one-period step to reach today.

Key rules to remember

Up and down factors
u = Su ÷ S0 ; d = Sd ÷ S0
Use when the question gives up and down prices. If given percentage moves, u = 1 + rise and d = 1 − fall.
Call payoff at expiry
C = max(S − K, 0)
S is the share price at the node and K is the exercise price.
Put payoff at expiry
P = max(K − S, 0)
Compute it separately at every end node.
Hedge ratio (delta)
Δ = (Cu − Cd) ÷ (Su − Sd)
Number of shares in the replicating portfolio for a call. For a put, Δ comes out negative, meaning you short shares.
Replicating portfolio borrowing
B = (Δ × Sd − Cd) ÷ (1 + r)
Amount borrowed at the risk-free rate for one period. Check with Δ × Su − Cu, which gives the same loan repayment.
Option value by replication
C = Δ × S0 − B
Cost of shares less the loan.
Risk-neutral probability
p = ((1 + r) − d) ÷ (u − d)
r is the risk-free rate for one period. With continuous compounding use e^(r × t) in place of (1 + r). Probability of a down move is 1 − p.
Option value by risk-neutral method
C = [p × Cu + (1 − p) × Cd] ÷ (1 + r)
Applied at each node while rolling back. For a two-period tree you can also use p², 2p(1 − p) and (1 − p)² on the end payoffs and discount by (1 + r)².
No-arbitrage condition
d < (1 + r) < u
If this fails, p falls outside 0 to 1 and the data is inconsistent.

How to solve Binomial Option Pricing Model questions

This sequence works for one-period and two-period questions, call or put, by either method.

  1. 1Write S0, K, u, d, the risk-free rate per period and the number of periods. If the rate is annual and the period is shorter, convert it to the period rate as the question directs.
  2. 2Draw the tree. Compute the share price at every node by multiplying by u or d. Remember an up-then-down path equals a down-then-up path.
  3. 3Compute the option payoff at each end node using max(S − K, 0) for a call or max(K − S, 0) for a put.
  4. 4Find p = ((1 + r) − d) ÷ (u − d). Check it lies between 0 and 1.
  5. 5Roll back one step at a time. At each node, value = [p × up value + (1 − p) × down value] ÷ (1 + r).
  6. 6If the question is American, compare the rolled-back value with the exercise value at each node and take the higher before moving further back. A European option has no such check.
  7. 7If the question asks for the replicating portfolio, compute Δ and B at the first node and give the value as Δ × S0 − B. State clearly the number of shares and the amount borrowed.
  8. 8Write the final value with the unit in rupees and state the interpretation, such as the fair premium.

Quickest way: Risk-neutral rollback with a calculator

When to use it: Use it when the question asks only for the option value and not the hedge ratio or the borrowing. It is the fastest, especially for two periods.

  1. Compute p once and store it in calculator memory.
  2. Write the end payoffs in a row on the answer sheet.
  3. For a two-period European option, use the direct formula: value = [p² × Cuu + 2p(1 − p) × Cud + (1 − p)² × Cdd] ÷ (1 + r)².
  4. Do not use the direct formula for an American option, because early exercise needs a check at each middle node.
  5. Keep at least four decimals in p and in intermediate values. Round only the final answer.

Common mistakes in Binomial Option Pricing Model

  • Using the real-world probability of an up move in place of the risk-neutral p.

    Students are used to expected value with given probabilities from decision-tree questions.

    Fix: Unless the question tells you to use p from the formula, compute p = ((1 + r) − d) ÷ (u − d). The real-world probability is not needed for pricing.

  • Forgetting to discount, or discounting by the wrong power in a two-period tree.

    Students rush the rollback and treat a two-period payoff as a one-period one.

    Fix: Discount by (1 + r) at every step. If you use the direct formula, discount by (1 + r)² for two periods.

  • Using an annual rate for a six-month or three-month period.

    The question gives an annual risk-free rate and the tree has shorter steps.

    Fix: Convert to the rate per period as the question specifies, then use it in p and in discounting. Write the period rate on the sheet first.

  • Taking the put payoff as max(S − K, 0), or computing the call payoff at a node where it is negative.

    Students copy the call setup for a put and forget the zero floor.

    Fix: Write the payoff formula next to the node values. Put is K − S, call is S − K, and both are floored at zero.

  • Skipping the early-exercise check for an American option.

    Students apply the European rollback to every question.

    Fix: Read whether the option is European or American. For an American option, at each node take the higher of the rolled-back value and the immediate exercise value.

  • Getting the sign wrong in the replicating portfolio for a put, or stating Δ without the borrowing or lending.

    The put hedge involves short selling shares and lending, which is the opposite of the call case.

    Fix: Compute Δ from the formula and keep its sign. A negative Δ means short shares. Compute B with the same formula and interpret a negative B as lending. Check that the portfolio payoff equals the option payoff in both states.

Worked examples

Example 1

A share of Vistara Ltd is priced at ₹100. After one period it will be either ₹120 or ₹80. A one-period European call has a strike price of ₹100. The risk-free rate is 5% per period. Find the call value by (a) the replicating portfolio and (b) the risk-neutral method.

Show the solution
  1. Su = ₹120, Sd = ₹80. Cu = max(120 − 100, 0) = ₹20. Cd = max(80 − 100, 0) = ₹0.
  2. Replicating portfolio: Δ = (20 − 0) ÷ (120 − 80) = 0.5 share.
  3. Borrowing: B = (0.5 × 80 − 0) ÷ 1.05 = 40 ÷ 1.05 = ₹38.0952.
  4. Check: at ₹120 the portfolio is worth 0.5 × 120 − 38.0952 × 1.05 = 60 − 40 = ₹20. At ₹80 it is worth 40 − 40 = ₹0. This matches the call payoffs.
  5. Call value = 0.5 × 100 − 38.0952 = 50 − 38.0952 = ₹11.9048.
  6. Risk-neutral: u = 1.2 and d = 0.8. p = (1.05 − 0.8) ÷ (1.2 − 0.8) = 0.25 ÷ 0.4 = 0.625.
  7. C = [0.625 × 20 + 0.375 × 0] ÷ 1.05 = 12.5 ÷ 1.05 = ₹11.9048.

Answer: The call is worth about ₹11.90. It can be replicated by buying 0.5 share and borrowing ₹38.10 at 5%. Both methods give the same value.

Example 2

A share is priced at ₹100. In each of the next two periods it can rise 10% or fall 10%. The risk-free rate is 5% per period. Value a two-period European put with a strike of ₹100. Then find the value if the put is American.

Show the solution
  1. u = 1.1, d = 0.9. Prices: Su = 110, Sd = 90. Suu = 121, Sud = 99, Sdd = 81.
  2. Put payoffs at expiry: Puu = max(100 − 121, 0) = 0. Pud = max(100 − 99, 0) = 1. Pdd = max(100 − 81, 0) = 19.
  3. p = (1.05 − 0.9) ÷ (1.1 − 0.9) = 0.15 ÷ 0.2 = 0.75. So 1 − p = 0.25.
  4. European, up node: Pu = (0.75 × 0 + 0.25 × 1) ÷ 1.05 = 0.25 ÷ 1.05 = ₹0.2381.
  5. European, down node: Pd = (0.75 × 1 + 0.25 × 19) ÷ 1.05 = (0.75 + 4.75) ÷ 1.05 = 5.5 ÷ 1.05 = ₹5.2381.
  6. European, today: P0 = (0.75 × 0.2381 + 0.25 × 5.2381) ÷ 1.05 = (0.1786 + 1.3095) ÷ 1.05 = 1.4881 ÷ 1.05 = ₹1.4172.
  7. Check with the direct formula: [0.75² × 0 + 2 × 0.75 × 0.25 × 1 + 0.25² × 19] ÷ 1.05² = (0.375 + 1.1875) ÷ 1.1025 = 1.5625 ÷ 1.1025 = ₹1.4172.
  8. American: at the up node, exercise value is 100 − 110 < 0, so keep 0.2381. At the down node, exercise value is 100 − 90 = ₹10, which is more than 5.2381. So take ₹10.
  9. American P0 = (0.75 × 0.2381 + 0.25 × 10) ÷ 1.05 = (0.1786 + 2.5) ÷ 1.05 = 2.6786 ÷ 1.05 = ₹2.5510. Exercise today is worth 100 − 100 = 0, so no gain from exercising at the start.

Answer: The European put is worth about ₹1.42. The American put is worth about ₹2.55, because it is optimal to exercise early if the share falls to ₹90 after one period.

Exam tips

  • Read the first line for European or American. It decides whether you need the early-exercise check at middle nodes.
  • Show the tree neatly with the share price above and the option value below each node. Marks are given for the nodes and the method even if arithmetic slips.
  • If the question says to use the replicating portfolio, do not answer by the risk-neutral method alone. Give Δ, the borrowing and the value. Use the other method only as a check if time allows.
  • Check that the answer is sensible. A call value cannot exceed the share price, and a call with a positive payoff in some state must have a positive value.
  • Convert the annual rate to the rate per period before anything else, and say so in one line on the answer sheet.

Practice questions from Derivatives Analysis and Valuation

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

What is the difference between the replicating portfolio method and the risk-neutral method?

The replicating portfolio method builds a mix of shares and borrowing that matches the option payoff in both states. The risk-neutral method uses a probability p and discounts the expected payoff at the risk-free rate. Both give the same option value.

Why do we not use the actual probability of the share going up?

The option can be hedged using shares and borrowing, so its price is fixed by no-arbitrage. The investor's risk preference and the real-world probability do not change that price. This is why p is derived only from u, d and the risk-free rate.

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

Compute the end payoffs, compute p, and roll back one step at a time. For a European option you can use the direct formula with p², 2p(1 − p) and (1 − p)², and discount by (1 + r)². For an American option, roll back node by node and check early exercise.

What does a negative hedge ratio mean?

It appears for a put. It means the replicating portfolio holds a short position in the share, with the proceeds lent at the risk-free rate. The sign tells you the direction of the position.