Updated for the 2026-2027 CFA® Level I curriculum.
Arbitrage and replication are the two ideas that let you price a derivative without forecasting the future. For options, replication means building a portfolio of the underlying asset and risk-free borrowing or lending that matches the option's payoff in every possible state.
This note shows why that setup needs more than one static position, unlike a forward commitment, and how the replicating portfolio's cost pins down the option's no-arbitrage price. After reviewing this note, you should be able to build a two-state replicating portfolio for a call and explain why nonlinear payoffs demand state-contingent hedging.
Quick Answer
A replicating portfolio for a call option combines shares of the underlying stock with borrowing or lending at the risk-free rate, sized so the portfolio's payoff matches the call's payoff in every future state. Because a call's payoff bends at the strike price, one fixed stock position cannot match both the up-state and down-state payoffs. No arbitrage then requires the call's price today to equal the cost of building that replicating portfolio.
Key Takeaways About Arbitrage and Replication in Option Pricing
A replicating portfolio matches an option's payoff in every future state, not on average.
Call replication requires shares of the underlying stock plus borrowing or lending at the risk-free rate.
The hedge ratio () equals the spread in option payoffs divided by the spread in stock payoffs across states.
Forward commitments have linear payoffs, so a single static underlying position replicates them; options do not.
No-arbitrage pricing means the option price must equal the replicating portfolio's cost today.
A common error is using the expected option payoff instead of matching payoffs state by state.
Replication prices the option. It does not require forecasting which state will actually occur.
What You Need to Know for CFA Level I
Explain why option pricing relies on no-arbitrage replication rather than probability forecasts.
Build a one-period, two-state replicating portfolio using the underlying asset and risk-free borrowing or lending.
Calculate the hedge ratio () and the bond position needed to replicate a call's payoff.
Contrast option replication with forward commitment replication, and explain why forwards need only a static hedge.
Connect the replicating portfolio's cost today to the option's no-arbitrage price.
Recognize why a kinked payoff requires state-contingent replication instead of one fixed position.
Arbitrage in Option Pricing
No arbitrage means two portfolios with identical payoffs in every future state must cost the same today. If they do not, you buy the cheap portfolio, sell the expensive one, and lock in a riskless profit with no net investment. This logic prices forwards and options alike.
The difference is in the payoff structure. A forward's payoff changes one-for-one with the underlying in either direction. A call's payoff stays at zero below the strike, then increases one-for-one once the underlying rises above it. This nonlinear payoff structure makes option replication more complex than forward replication.
What Is an Option Replicating Portfolio?
An option replicating portfolio holds a position in the underlying asset and a risk-free bond, sized to reproduce the option's payoff in every state the underlying can take. For a one-period call, there are two states: an up move and a down move. Because the call pays zero in the down state and a positive amount in the up state, a single static number of shares cannot match both payoffs at once. You need two instruments, solved together, to match both states.
Replicating a Contingent Claim
The one-period, two-state setup solves for two unknowns: the number of shares and the bond position .
State | Stock payoff | Call payoff | Replicating portfolio payoff |
|---|---|---|---|
Up | |||
Down |
Setting the replicating portfolio's payoff equal to the call's payoff in both states gives:
= call payoff in the up and down states
= stock price in the up and down states
= number of shares held in the replicating portfolio
= amount lent (positive) or borrowed (negative) today
= risk-free rate over the period
The cost of the replicating portfolio today is . Under no arbitrage, this cost equals the call's price today.
Forward Commitment vs Contingent Claim Replication
Feature | Forward Commitment | Contingent Claim (Call Option) |
|---|---|---|
Payoff shape | Linear (moves one-for-one with the underlying) | Kinked (flat below strike, one-for-one above strike) |
Hedge ratio | Fixed at 1 across all states | Changes with the stock price level ( = 0.5 in the example below) |
Replication method | Long the underlying, borrow the present value of the forward price | Solve two equations for and using up and down state payoffs |
Number of instruments needed | Underlying plus one financing leg | Underlying plus one financing leg, but recalculated at each price level |
A forward's constant slope means one static hedge works in every state. An option's slope changes at the strike, so replication must be state-contingent. This is also why multi-period binomial models recompute the hedge ratio at every node as the stock price moves.
How Replication Sets an Option Price
Once h and B are known, the replicating portfolio's cost today is the option's no-arbitrage price. If the option trades above that cost, sell the option and buy the replicating portfolio for a riskless profit. If it trades below that cost, do the reverse. This is the pricing mechanism behind the binomial option pricing model used later in the Level I curriculum.
Worked Example
Setup: A European call has one period to expiration. The stock trades today at ( = $50). In one period, it moves up to ( = $60) or down to ( = $40). The strike price is ( = $50). The risk-free rate over the period is 5%.
Step 1: Find the Call’s Payoffs in Each State
In the up state:
In the down state:
Step 2: Solve for the Hedge Ratio
The hedge ratio tells you how many shares are needed in the replicating portfolio.
The replicating portfolio therefore holds 0.50 shares of stock.
Step 3: Solve for the Bond Position
Use the down-state payoff to determine the amount invested in or borrowed through the risk-free asset.
A negative (B) means the replicating portfolio borrows $19.05 today.
Step 4: Find the Cost of the Replicating Portfolio
The call’s no-arbitrage value equals the cost of creating the replicating portfolio today.
The replicating portfolio costs $5.95, so the call’s no-arbitrage value today is $5.95.
Buying 0.5 shares and borrowing $19.05 reproduces the call's payoff in both states. No arbitrage means the call should trade at $5.95 today. If it trades higher, sell the call and buy the replicating portfolio. If it trades lower, do the reverse.
Contrast with a forward: If this were a forward contract on the same stock instead of a call, replication would only need one share held today and borrowing equal to the present value of the forward price. Because a forward's payoff has the same slope in both states, a single static position works. There is no need to solve two equations, because there is no kink to match.
Common Exam Traps
Using expected payoff instead of matched state payoffs
Averaging and with assumed probabilities does not replicate the option. Replication matches the payoff in each state exactly, without needing probabilities at all.
Assuming one static underlying position alone can replicate a call
A call's payoff is not linear, so a fixed number of shares works in only one state, not both. You need shares plus borrowing or lending to match both states at once.
Confusing replication with forecasting
Replication prices the option today using current no-arbitrage relationships. It says nothing about which state will actually happen.
Forgetting that no-arbitrage requires equal current values when future payoffs match
If two portfolios pay off identically in every state, they must have the same price today. Candidates sometimes forget this equivalence and stop the problem after finding h without pricing the option.
Practice Questions
A European call option has one period to expiration. The underlying stock trades today at $80. In one period, the stock will move up to $100 or down to $64. The call's exercise price is $90. What is the hedge ratio (h) for the replicating portfolio?
0.20
0.28
0.50
Correct Answer: B
Calculation:
The hedge ratio matches the spread in the call's payoffs to the spread in the stock's payoffs across the two states. Only 0.28 shares are needed to replicate this call's payoff pattern.
Option A: 0.20 results from dividing the payoff spread by the wrong base, such as the strike price instead of the stock price spread.
Option C: 0.50 reuses the hedge ratio from a different example without recalculating it for this option's specific payoff and price spread.
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FAQs About Arbitrage and Replication in Option Pricing
Why cannot a call option be replicated with just the underlying stock?
A call's payoff bends at the strike price. It is flat below the strike and rises one-for-one above it. A fixed stock position has a constant slope, so it can only match the call's payoff in one state. Borrowing or lending is needed to match the other state.
Does replication require knowing the probability of an up or down move?
No. Replication matches payoffs state by state using current prices, not probabilities. This is what makes the no-arbitrage price independent of forecasts about which state will occur.
How is option replication different from hedging a stock position?
Hedging a stock position typically offsets risk in an existing holding. Replication builds a new portfolio from scratch that reproduces an option's exact payoff, which is then used to determine the option's fair price today.