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FRM Exam Part I · Binomial Trees

One-Step Binomial Tree Option Pricing for FRM Part I

Updated 11 October 2026 · Fact-checked

A one-step binomial tree lets the stock move to only two prices, up or down, over one period. You value a European option by building a portfolio of delta shares and borrowing that matches its payoffs in both states. By no-arbitrage, the option price equals that portfolio's cost today.

Understand One-Step Binomial Tree Option Pricing

A one-step binomial tree is the simplest option model. Today the stock price is S0. After one period it is either S0 × u (up) or S0 × d (down). The option payoff is known in both states: fu in the up state and fd in the down state.

The key idea is replication. Two assets (the stock and a risk-free bond) can be combined to give exactly the same payoffs as the option in both states. If two positions pay the same in every state, they must cost the same today. Otherwise you could buy the cheap one, sell the expensive one and lock in a risk-free profit. That is the no-arbitrage argument.

The number of shares in the replicating portfolio is the delta: the change in option payoff divided by the change in stock price across the two states. A call has positive delta (you buy shares and borrow). A put has negative delta (you short shares and lend).

The same price comes out of risk-neutral valuation. You find the probability p that makes the stock earn the risk-free rate, compute the expected payoff using p, and discount at the risk-free rate. This p is not the real-world probability. The real-world probability of an up move never enters the price.

Key formulas to remember

Delta of the replicating portfolio
Δ = (fu − fd) ÷ (S0·u − S0·d)
Shares held per option. Negative for a put, meaning a short stock position.
Option value by replication
f = Δ·S0 − B
B is the amount borrowed today. If B is negative, you are lending.
Borrowing amount
B = (Δ·S0·u − fu) ÷ (1 + r) for discrete rates, or Δ·S0·u − fu discounted by e^(−rT)
Set so the portfolio payoff equals the option payoff in the up state; check with the down state.
Risk-neutral up probability (continuous compounding)
p = (e^(rT) − d) ÷ (u − d)
With a dividend yield q, use e^((r−q)T). Needs d < e^(rT) < u.
Risk-neutral value
f = e^(−rT) × [p·fu + (1 − p)·fd]
Use 1 ÷ (1 + r) instead if the rate is given as a simple one-period rate.
Payoffs at expiry
Call: max(S − K, 0). Put: max(K − S, 0)
Compute both states before anything else.

How to solve One-Step Binomial Tree Option Pricing questions

This method works for any one-step call or put on a non-dividend stock, using replication or risk-neutral valuation. Do both if time allows, as a check.

  1. 1Write down S0, K, u and d (or the up and down prices), r, and the period length T.
  2. 2Compute the two stock prices at expiry: Su = S0·u and Sd = S0·d.
  3. 3Compute the option payoffs fu and fd using max(S − K, 0) for a call or max(K − S, 0) for a put.
  4. 4Find delta = (fu − fd) ÷ (Su − Sd). Keep the sign.
  5. 5Find the borrowing: B = (Δ·Su − fu) discounted at the risk-free rate. The replicating portfolio is Δ shares less B borrowed.
  6. 6Option price = Δ·S0 − B.
  7. 7Check with risk-neutral valuation: p = (e^(rT) − d) ÷ (u − d), then f = e^(−rT)[p·fu + (1 − p)·fd]. The two answers must match.

Quickest way: Risk-neutral shortcut

When to use it: Use it when the question asks only for the option price and not for delta or the borrowing amount.

  1. Compute fu and fd from the expiry stock prices.
  2. Compute p = (e^(rT) − d) ÷ (u − d). Confirm 0 < p < 1.
  3. Compute p·fu + (1 − p)·fd.
  4. Multiply by e^(−rT) (or divide by 1 + r for a simple rate).
  5. If delta is requested, compute (fu − fd) ÷ (Su − Sd) separately. It is one subtraction and one division.

Common mistakes in One-Step Binomial Tree Option Pricing

  • Using the real-world probability of an up move to price the option.

    The tree often gives a probability, and it looks like it should be used.

    Fix: Price with the risk-neutral p computed from r, u and d. Real-world probabilities do not affect the price.

  • Discounting the expected payoff at the wrong rate or forgetting to discount.

    Students focus on finding p and stop at the expected payoff.

    Fix: Always finish with the factor e^(−rT), or 1 ÷ (1 + r) for a simple rate.

  • Using e^(rT) in the formula for p but then 1 + r in the discounting, or the reverse.

    The question mixes conventions and students switch halfway.

    Fix: Choose one compounding convention at the start. Use it for both p and the discounting.

  • Getting the sign of delta wrong for a put.

    Students take absolute values of the payoff difference.

    Fix: Keep fu − fd as it is. A put has fu < fd, so delta is negative: you short the stock. You hedge by lending, not borrowing.

  • Computing the borrowing from the current price, Δ·S0, instead of the expiry payoffs.

    Confusing the portfolio cost today with the payoff match at expiry.

    Fix: Match the payoff in one state first (Δ·Su − B·growth = fu), then solve for B. Price = Δ·S0 − B.

  • Treating a call's payoff as negative when the option expires out of the money.

    Writing S − K without the max function.

    Fix: The payoff is never below zero for a long option. Apply max(·, 0) in each state.

Worked examples

Example 1

A stock is at $50. In one year it will be either $60 or $40. The risk-free rate is 5% per year, continuously compounded. Price a one-year European call with strike $52 using a replicating portfolio.

Show the solution
  1. Payoffs: fu = max(60 − 52, 0) = 8. fd = max(40 − 52, 0) = 0.
  2. Delta = (8 − 0) ÷ (60 − 40) = 0.4 shares.
  3. Match the down state: 0.4 × 40 = 16, and the option pays 0, so the loan repays 16 at expiry. Borrowing today B = 16 × e^(−0.05) = 16 × 0.951229 = 15.2197.
  4. Check the up state: 0.4 × 60 = 24, minus loan repayment 16 = 8. This matches fu.
  5. Option price = 0.4 × 50 − 15.2197 = 20 − 15.2197 = 4.7803.
  6. Risk-neutral check: u = 1.2, d = 0.8, e^0.05 = 1.051271. p = (1.051271 − 0.8) ÷ 0.4 = 0.628178. f = 0.951229 × (0.628178 × 8) = 0.951229 × 5.025424 = 4.780.

Answer: The call is worth about $4.78. The replicating portfolio is long 0.4 shares financed by borrowing about $15.22.

Example 2

A stock is at $80. In six months (T = 0.5) it will be $88 or $72. The risk-free rate is 4% per year, continuously compounded. Find the value of a six-month European put with strike $80 and its delta.

Show the solution
  1. Payoffs: fu = max(80 − 88, 0) = 0. fd = max(80 − 72, 0) = 8.
  2. Delta = (0 − 8) ÷ (88 − 72) = −0.5. You short 0.5 shares per put.
  3. u = 88 ÷ 80 = 1.1 and d = 72 ÷ 80 = 0.9.
  4. e^(rT) = e^0.02 = 1.020201. p = (1.020201 − 0.9) ÷ 0.2 = 0.601007.
  5. 1 − p = 0.398993. Expected payoff = 0.601007 × 0 + 0.398993 × 8 = 3.19194.
  6. Discount: e^(−0.02) = 0.980199. f = 3.19194 × 0.980199 = 3.1287.
  7. Replication check: the portfolio is short 0.5 shares plus a loan. Value = −0.5 × 80 + L = −40 + L. Down state: −0.5 × 72 = −36, plus loan proceeds L × 1.020201 must equal 8, so L × 1.020201 = 44, giving L = 43.1287. Value = −40 + 43.1287 = 3.1287. This matches.

Answer: The put is worth about $3.13 and its delta is −0.5.

Exam tips

  • Compute the two option payoffs first. Many questions are solved in a few lines once fu and fd are on paper.
  • Use the risk-neutral shortcut when only the price is asked. Use replication when the question asks for delta, shares held or amount borrowed.
  • Check that 0 < p < 1. If it is not, you have made an arithmetic error or the tree allows arbitrage.
  • Watch the rate convention. 'Continuously compounded' means use e^(rT). A plain 'annual rate' with one period often means 1 + r.
  • Wrong answer options often come from using real-world probabilities or skipping the discount. Sanity-check that a call price is below S0 and above the discounted intrinsic value.

Practice questions from Binomial Trees

One-Step 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.

One-Step Binomial Tree Option Pricing: frequently asked questions

Why does the real-world probability not matter in the binomial price?

The option can be replicated exactly using the stock and a bond, so its price is fixed by no-arbitrage. Replication works in both states whatever their likelihoods. The risk-neutral p is just a convenient way to get the same number.

What does delta mean in a one-step tree?

Delta is the number of shares needed to replicate the option, or to hedge a short option position. It equals the spread of option payoffs divided by the spread of stock prices between the two states. It is the slope of the option payoff across the tree.

Do I have to calculate both replication and risk-neutral value in the exam?

No. Either method gives the same price. Use risk-neutral valuation for speed when only a price is asked, and replication when delta or borrowing is requested.

What condition must hold for the tree to be free of arbitrage?

The risk-free growth factor must lie between the down and up factors: d < e^(rT) < u. This is the same as requiring the risk-neutral probability p to be strictly between 0 and 1.