CFA Level I Exam · Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Arbitrage and the Law of One Price for CFA Level I

Updated 7 October 2026 · Fact-checked

Arbitrage is a trade that earns a risk-free profit with no net investment. The law of one price says identical assets or cash flows must sell for the same price. Derivative prices are set so no arbitrage exists: build a replicating portfolio, compare prices, and buy the cheap one and sell the expensive one.

Understand Arbitrage and the Law of One Price

Arbitrage exists when you can lock in a riskless profit without putting up your own money. You buy something cheap and sell the same thing, or its equivalent, at a higher price. The profit is certain, and the net cost today is zero.

The law of one price says two assets with identical future cash flows must have the same price today. If they do not, traders buy the cheaper one and sell the expensive one. Buying pushes the cheap price up. Selling pushes the expensive price down. The gap closes, and the market becomes more efficient.

This is the engine behind derivative pricing. You do not guess where a forward or option should trade. You build a replicating portfolio from the underlying asset and borrowing or lending that gives the same payoff as the derivative. The derivative must then cost the same as that portfolio. If it does not, an arbitrage exists.

In general finance usage, arbitrage is a riskless profit that needs no net investment. It is often described in two versions. In the "upfront" version, which this page calls type 1, you receive money today and owe nothing later. In the version this page calls type 2, the trade costs nothing today, has no chance of loss, and has some chance of gain later. These labels are a general classification used on this page, not official CFA curriculum terms. Both break the no-arbitrage rule.

The no-arbitrage principle assumes that markets are frictionless enough for these trades to be done. It also rests on short selling being possible and on borrowing and lending at the risk-free rate. Real frictions such as transaction costs mean prices can sit within a small band around the no-arbitrage value without a profitable trade.

Key formulas to remember

Law of one price: Same payoffs in every state ⇒ same price today; If two portfolios have identical cash flows, any price difference is an arbitrage opportunity.
Replication (derivative price): Price of derivative = Price of replicating portfolio; The replicating portfolio uses the underlying plus risk-free borrowing or lending.
Type 1 arbitrage (general finance usage): Net cash flow today > 0 and net cash flow later = 0 in all states; Riskless profit received up front, with no future liability. A general label, not an official CFA term.
Type 2 arbitrage (general finance usage): Net cash flow today = 0 and later cash flow ≥ 0 in all states, > 0 in some; No cost, no chance of loss, some chance of gain. A general label, not an official CFA term.
Risk-free rate and no-arbitrage: A riskless position with zero investment must earn 0 return; If it earns more, that is arbitrage. A hedged position earns the risk-free rate.

How to solve Arbitrage and the Law of One Price questions

Use this routine for any question on arbitrage, the law of one price or no-arbitrage pricing.

1Identify the two positions that should be equivalent, such as the derivative and a replicating portfolio.
2Check that their payoffs at the end date are the same in every state.
3Compute today's cost of each position, using the risk-free rate to discount or compound where needed.
4Compare the prices. If they are equal, no arbitrage exists. If not, the cheaper side is underpriced.
5Buy the cheaper position and sell (or short) the more expensive one so that the future cash flows cancel.
6Add up the cash flow today. A positive amount with zero later cash flows is a risk-free profit.

Quickest way: Cheap-versus-expensive shortcut

When to use it: Use this when the question gives two prices for equivalent payoffs and asks what a trader does or whether arbitrage exists.

Find the two equivalent positions and their prices.
Buy the lower price, sell the higher price.
If the question asks for profit, subtract the lower from the higher (adjust for compounding if cash flows are at different dates).
If two options contain the wrong direction, eliminate them first. Only one option will buy low and sell high correctly.

Common mistakes in Arbitrage and the Law of One Price

Calling any profitable trade an arbitrage.
Students forget that arbitrage must be riskless.
Fix: Check for risk. If the outcome depends on a price move, it is speculation, not arbitrage.
Treating the type 1 and type 2 labels as exam terms.
The labels appear in some general finance sources, so they look like required vocabulary.
Fix: Learn the core definition: a riskless profit with no net investment. Use the labels only as background, because they are not CFA curriculum terms and you should not expect them in questions.
Buying the expensive asset and selling the cheap one.
Rushing and reading prices in the wrong order.
Fix: Always write 'buy low, sell high' before choosing the option.
Comparing prices at different dates without adjusting for interest.
Forgetting the time value of money.
Fix: Bring both amounts to the same date using the risk-free rate before comparing.
Thinking arbitrage persists in an efficient market.
Confusing the existence of the opportunity with its lasting.
Fix: Arbitrage trading removes the mispricing. Prices move until the opportunity disappears.

Worked examples

Example 1

A risk-free zero-coupon bond pays exactly €100 in one year. The one-year risk-free rate is 4%. A bond Y with the identical, certain €100 payoff in one year trades at €98. Is there an arbitrage, and what is the profit today per €100 payoff?

Show the solution

The no-arbitrage price of a certain €100 in one year is 100 ÷ 1.04 = €96.15.
Bond Y trades at €98, above €96.15, so Y is overpriced relative to the risk-free replication.
Short (sell) bond Y at €98 and buy the risk-free zero-coupon bond, which costs €96.15 and pays €100 in one year.
Cash today is 98 − 96.15 = €1.85.
At maturity, the €100 received from the risk-free bond covers the €100 owed on Y, so the later net cash flow is zero.

Answer: Yes. A type 1 arbitrage exists with a profit of about €1.85 today per €100 payoff.

Example 2

A derivative has the same payoff in every state as a replicating portfolio of the underlying and risk-free borrowing. The portfolio costs $48.00 today. The derivative trades at $50.00. Which statement is correct? (A) Buy the derivative and sell the portfolio, locking in $2.00 today. (B) Sell the derivative and buy the portfolio, locking in $2.00 today. (C) Sell the derivative and buy the portfolio, locking in $48.00 today.

Show the solution

The payoffs are identical in every state, so the law of one price says the two positions must cost the same today.
The derivative costs $50.00 and the portfolio costs $48.00. The derivative is overpriced, so you sell it and buy the cheaper portfolio.
Option A reverses the direction. It buys the expensive position and sells the cheap one, so it loses $2.00 today. Eliminate it.
Cash today is 50.00 − 48.00 = $2.00. Option C states $48.00, which is the cost of the portfolio, not the profit. Eliminate it.
At the end date, the portfolio payoff exactly covers the amount owed on the short derivative, so the later net cash flow is zero and the $2.00 is riskless.

Answer: B

Exam tips

Questions often give a price and ask what trade captures the arbitrage. Decide buy low, sell high first, then match the option.
Know the arbitrage definition: a riskless profit with no net investment. The type 1 and type 2 labels are not CFA terms, so do not expect them in questions.
Remember that arbitrage trades push prices back to the no-arbitrage level, which links to market efficiency questions.
With no penalty for wrong answers, never leave a question blank. Eliminate options that buy the expensive asset to narrow to one.

Practice questions from Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Arbitrage and the Law of One Price in other exams

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

Arbitrage and the Law of One Price: frequently asked questions

What is the law of one price?

It says that assets with identical payoffs must have the same price. If they do not, traders can buy the cheaper and sell the dearer for a riskless gain. That trading pushes the prices together.

What is the difference between type 1 and type 2 arbitrage?

Type 1 gives a positive cash flow today with no liability later. Type 2 costs nothing today and cannot lose money, but has some chance of a gain later. These are general finance labels, not official CFA terms.

How does arbitrage determine derivative prices?

You build a replicating portfolio from the underlying and risk-free borrowing or lending that matches the derivative's payoff. The derivative's price must equal the cost of that portfolio, or arbitrage is possible.

Is arbitrage the same as speculation?

No. Arbitrage is riskless and needs no net investment. Speculation takes risk in the hope of profit and can lose money.