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CFA Level I Exam · Valuing a Derivative Using a One-Period Binomial Model

Replicating Portfolio and Hedge Ratio in the Binomial Model

Updated 7 October 2026 · Fact-checked

A replicating portfolio holds h units of the underlying plus risk-free borrowing or lending so that it matches the option's payoff in both the up and down states. The hedge ratio is h = (c⁺ − c⁻) ÷ (S⁺ − S⁻). By no-arbitrage, the option's value equals the portfolio's cost today.

Understand Replicating Portfolio and Hedge Ratio

An option payoff depends on the underlying price. In a one-period binomial model, the underlying can move to only two prices: an up price S⁺ or a down price S⁻. This gives two option payoffs, c⁺ and c⁻ for a call.

You can copy these two payoffs with a mix of the underlying and a risk-free bond. You buy h units of the underlying and borrow or lend cash at the risk-free rate. If this mix pays the same as the option in both states, it is a replicating portfolio.

The hedge ratio (delta) is the number of units of the underlying in that portfolio. It equals the spread of option payoffs divided by the spread of underlying prices. It tells you how much the option value moves for each one-unit move in the underlying.

The law of one price then applies. Two positions with identical payoffs in every state must cost the same today. If the option trades at a different price, you can buy the cheap one, sell the expensive one and lock in a risk-free profit. That is arbitrage.

A hedge uses the same numbers the other way round. Being long h units of the underlying and short 1 call gives a position with the same value in both states. The payoff is riskless in both states. It earns the risk-free rate only if the option is fairly priced.

Key formulas to remember

Hedge ratio (delta) for a call
h = (c⁺ − c⁻) ÷ (S⁺ − S⁻)
Positive for a call. Use put payoffs p⁺ and p⁻ for a put; the result is negative.
Up and down prices
S⁺ = S × u ; S⁻ = S × d
u and d are gross factors, for example u = 1.25 means a 25% rise.
Call payoffs at expiry
c⁺ = max(0, S⁺ − X) ; c⁻ = max(0, S⁻ − X)
X is the exercise price. Put payoffs are max(0, X − S).
Replicating portfolio value
c = h × S + B
B is the bond position today. B is negative when you borrow.
Bond position from the down state
B = [c⁻ − h × S⁻] ÷ (1 + r)
Same B results from the up state: [c⁺ − h × S⁺] ÷ (1 + r).
Hedged position
Long h units, short 1 call: value today = h × S − c
The payoff is riskless in both states; it earns the risk-free rate only if the option is fairly priced.

How to solve Replicating Portfolio and Hedge Ratio questions

Use this order for any replicating portfolio or hedge ratio question in a one-period tree.

  1. 1Write down S, X, u and d (or S⁺ and S⁻), the risk-free rate r and whether the option is a call or put.
  2. 2Compute S⁺ = S × u and S⁻ = S × d.
  3. 3Compute the option payoffs at expiry in both states using max(0, ...).
  4. 4Find the hedge ratio h = (payoff up − payoff down) ÷ (S⁺ − S⁻).
  5. 5Find the bond position B = (payoff down − h × S⁻) ÷ (1 + r). A negative B means borrowing.
  6. 6Option value today = h × S + B. Check by testing the up state: h × S⁺ + B × (1 + r) should equal the up payoff.
  7. 7If asked about arbitrage, compare this value with the market price. Sell the overpriced one and buy the replicating portfolio, or the reverse.

Quickest way: Delta first, then the cash leg

When to use it: Use when the question gives prices and asks for the option value or the number of shares to hold. It takes about a minute.

  1. Compute both payoffs and the price spread. Get h in one division.
  2. Compute B from the down state only. If the down payoff is 0, then B = −h × S⁻ ÷ (1 + r).
  3. Value = h × S + B. On a calculator, keep full decimals until the end.
  4. Check that your h is reasonable: a call's delta lies between 0 and 1 inclusive. If your h falls outside that range, recheck the payoffs and prices. The call value must also be positive and below S.
  5. Check with the up state if time allows.

Common mistakes in Replicating Portfolio and Hedge Ratio

  • Dividing by the wrong spread, such as using S instead of S⁺ − S⁻.

    Students remember the idea of 'ratio' but not the exact terms.

    Fix: Delta is payoff difference over price difference. Both differences are up minus down.

  • Forgetting to discount the bond position by (1 + r).

    The payoff is at expiry, but the bond leg is valued today.

    Fix: Divide the future bond amount by (1 + r) to get B. Check that h × S⁺ + B × (1 + r) equals the up payoff.

  • Treating borrowing as positive cash.

    The sign of B is easy to lose when it comes out negative.

    Fix: A call's payoff is below the share payoff by up to X, so the replicating portfolio is long h shares financed partly by borrowing. B is therefore negative (for X > 0), and the call value is below h × S.

  • Using the put's formula with a positive delta.

    Students memorise call delta only.

    Fix: For a put, put payoffs are larger in the down state, so h is negative. You short the underlying and lend.

  • Computing the payoff as S⁻ − X when the option is out of the money.

    Skipping the max(0, ...) step.

    Fix: A call payoff is never negative. Set it to 0 when S⁻ < X.

Worked examples

Example 1

A stock trades at $50. In one period it will rise to $60 or fall to $40. A European call has exercise price $50. The risk-free rate is 5% for the period. Find the hedge ratio and the call value using a replicating portfolio.

Show the solution
  1. Payoffs: c⁺ = max(0, 60 − 50) = 10; c⁻ = max(0, 40 − 50) = 0.
  2. Hedge ratio: h = (10 − 0) ÷ (60 − 40) = 0.5.
  3. Bond position: B = (0 − 0.5 × 40) ÷ 1.05 = −20 ÷ 1.05 = −19.0476.
  4. Call value: c = 0.5 × 50 + (−19.0476) = 25 − 19.0476 = 5.9524.
  5. Check up state: 0.5 × 60 − 19.0476 × 1.05 = 30 − 20 = 10. This matches c⁺.

Answer: The hedge ratio is 0.5 and the call value is about $5.95. The replicating portfolio buys 0.5 shares and borrows $19.05.

Example 2

Using the same stock and call as above, the call trades in the market at $7.00. Describe the arbitrage and the profit today.

Show the solution
  1. The no-arbitrage value is $5.95, so the market call at $7.00 is overpriced.
  2. Sell the call at $7.00 and build the replicating portfolio at a cost of $5.9524.
  3. Replicating portfolio: buy 0.5 shares for $25 and borrow $19.0476 (about $19.05, rounded up).
  4. Net cash today: 7.00 − 5.9524 = 1.0476, which is about $1.05.
  5. At expiry the portfolio payoff equals the call payoff in both states, so the short call is fully covered.

Answer: Sell the call, buy 0.5 shares and borrow about $19.05. You gain about $1.05 today with no risk at expiry.

Exam tips

  • Questions are three-option MCQs with numbers in ascending order. Compute h first. A wrong delta usually gives a wrong value that still appears as an option.
  • Do a sanity check: call value must be positive, below S, and below h × S in the usual case where you borrow.
  • If the down payoff for a call is zero, the bond leg is just −h × S⁻ ÷ (1 + r). This saves time.
  • Read whether the question asks for the number of shares, the borrowing amount or the option value. They are different numbers from the same working.
  • Check the sign: a put's hedge ratio is negative, meaning a short position in the underlying.

Practice questions from Valuing a Derivative Using a One-Period Binomial Model

Replicating Portfolio and Hedge Ratio in other exams

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

Replicating Portfolio and Hedge Ratio: frequently asked questions

What is the hedge ratio in the binomial model?

It is the number of units of the underlying in the replicating portfolio. It equals (c⁺ − c⁻) ÷ (S⁺ − S⁻). It also shows how much the option value changes per unit change in the underlying.

Do I need probabilities to find the replicating portfolio?

No. The replicating method uses only the two payoffs, prices and the risk-free rate. Real-world probabilities do not appear, which is why the method is called no-arbitrage pricing.

Why is the call replicated by borrowing?

The call's payoff is below the share's payoff by up to X, so the replicating portfolio is long shares financed partly by borrowing. You hold h shares, which cost more than the call is worth, and fund part of that cost by borrowing at the risk-free rate. When both states are in the money, h = 1 and you borrow the present value of the exercise price.

How do I replicate a put?

Compute h from put payoffs. It is negative, so the replicating portfolio is short the underlying and long the risk-free bond. The net cost of this portfolio, h × S + B, equals the put value, which is positive.