FRM Exam Part I · Pricing Conventions, Discounting, and Arbitrage

Law of One Price and Arbitrage for FRM Part I

Updated 11 October 2026 · Fact-checked

The law of one price says identical cash flows must have the same price. If they do not, an arbitrage exists: buy the cheap one, sell the expensive one, and lock in a risk-free profit. To solve questions, replicate the cash flows, compare prices, and trade the gap.

Understand Law of One Price and Arbitrage

The law of one price says two assets with exactly the same cash flows in every state of the world must sell for the same price. If one is cheaper, traders buy it and sell the other. Their trading pushes the prices together.

An arbitrage is a trade that costs nothing (or gives you cash today), carries no risk of loss, and has a chance of a positive payoff. No-arbitrage pricing assumes such trades do not last in liquid markets. This is why you can price a product by building its twin from other assets.

A replicating portfolio is a mix of traded assets that matches the cash flows of a target asset. The target's no-arbitrage price equals the cost of the replicating portfolio. If the market price differs, you buy the cheaper side and sell the dearer side. The difference is your profit today, and the cash flows cancel later.

Bonds are the classic case. A coupon bond is a bundle of zero-coupon bonds, one per cash flow. Stripping means splitting a coupon bond into its separate cash flows and selling them as zeros. Reconstitution means buying the zeros and putting them back together as a coupon bond. If the bond trades below the sum of the zeros, buy the bond and strip it. If it trades above, buy the zeros and reconstitute.

Exam questions assume frictionless markets unless told otherwise: no transaction costs, no bid-ask spread, and you can short freely. In real markets, costs and funding limits mean small gaps can persist.

Key formulas to remember

Law of one price: Same cash flows ⇒ same price; If prices differ, an arbitrage exists in frictionless markets.
Replication price: Price of target = cost of replicating portfolio; Match cash flows at every date, then add up the cost.
Coupon bond as sum of zeros: P = Σ CFₜ × d(t); d(t) is the price of ₹1 or $1 received at time t, i.e. the discount factor.
Discount factor from a spot rate: d(t) = 1 ÷ (1 + z/m)^(m×t); z is the annual spot rate, m is compounding per year. For continuous compounding, d(t) = e^(−z×t).
Arbitrage profit: Profit = |Market price − Replication price| × quantity; Buy the cheaper side, sell the dearer side.
Arbitrage definition: Cost ≤ 0 today, payoff ≥ 0 later in all states, and > 0 in some state; A trade with risk of loss is not an arbitrage.

How to solve Law of One Price and Arbitrage questions

Use the same routine for any law-of-one-price or arbitrage question.

1List the cash flows of the asset being priced, with dates and amounts.
2Find the prices of the building blocks (zeros, discount factors, or other traded assets) that can replicate those cash flows.
3Compute the replication cost by multiplying each cash flow by its building-block price and adding up.
4Compare the replication cost with the market price of the asset.
5If the asset is cheaper than replication, buy it and sell the replicating portfolio. If it is dearer, do the reverse.
6Check that the cash flows cancel at every future date, so the only net amount is today's profit.
7Scale the profit to the trade size asked in the question and state the answer with units and currency.

Quickest way: Compare the asset price with the sum of discounted cash flows

When to use it: Use when the question gives a bond price and zero prices or spot rates and asks whether an arbitrage exists.

Compute the fair price as Σ CF × d(t) using the given discount factors.
Subtract: market price − fair price.
If positive, the asset is rich: sell it, buy the zeros. If negative, it is cheap: buy it, sell the zeros.
Profit per unit is the absolute gap. Multiply by the face amount traded.

Common mistakes in Law of One Price and Arbitrage

Trading the wrong direction
Students mix up which side is cheap when comparing prices.
Fix: Always say it in words: buy whatever costs less for the same cash flows, sell whatever costs more.
Forgetting the final coupon plus principal at maturity
The last cash flow is a single bundle and gets split incorrectly.
Fix: List every date in a table. The final row is coupon + face value.
Calling a risky trade an arbitrage
Students treat any expected profit as arbitrage.
Fix: An arbitrage needs no risk of loss. A trade that profits only on average is a bet, not an arbitrage.
Using a yield instead of a discount factor for each date
A single yield is easy to use, but the law of one price works cash flow by cash flow.
Fix: Value each cash flow with its own spot rate or zero price. Use yield only to summarise.
Ignoring compounding in the discount factor
Semi-annual and annual formulas look alike.
Fix: Read the compounding stated. Use (1 + z/m)^(m×t), or e^(−zt) when continuous.
Scaling profit by the wrong amount
Prices are quoted per 100 of face value but trades are in larger sizes.
Fix: Find the profit per 100 face, then multiply by face traded ÷ 100.

Worked examples

Example 1

The price of a 1-year zero-coupon bond is 96.00 per 100 face. The price of a 2-year zero is 91.00 per 100 face. A 2-year bond with an annual 5% coupon and face value 100 trades at 100.00. What is the arbitrage profit per 100 face, and what trade captures it?

Show the solution

Cash flows of the coupon bond: 5 at year 1, 105 at year 2.
Replication cost = 5 × 0.96 + 105 × 0.91.
5 × 0.96 = 4.80. 105 × 0.91 = 95.55.
Replication cost = 4.80 + 95.55 = 100.35.
Market price = 100.00, below 100.35, so the bond is cheap.
Trade: buy the coupon bond at 100.00 and sell short 5 face of the 1-year zero and 105 face of the 2-year zero, which brings in 100.35.
Profit today = 100.35 − 100.00 = 0.35, and the future cash flows cancel.

Answer: Buy the coupon bond and short the replicating zeros. Profit is 0.35 per 100 face.

Example 2

A 1-year zero costs 0.9500 per 1 face and a 2-year zero costs 0.8900 per 1 face. A 2-year bond paying a 4% annual coupon on 100 face is priced at 96.50. Find the arbitrage profit if you trade ₹10,00,000 face of the bond, ignoring costs.

Show the solution

Cash flows per 100 face: 4 at year 1, 104 at year 2.
Replication cost = 4 × 0.9500 + 104 × 0.8900.
4 × 0.9500 = 3.80. 104 × 0.8900 = 92.56.
Replication cost = 96.36.
Market price = 96.50, above 96.36, so the bond is rich.
Trade: sell the bond at 96.50 and buy the replicating zeros for 96.36.
Profit per 100 face = 96.50 − 96.36 = 0.14.
Scale: ₹10,00,000 ÷ 100 = 10,000 units. Profit = 10,000 × 0.14 = 1,400.

Answer: The bond is overpriced. Sell it and buy the zeros for a profit of ₹1,400.

Exam tips

Write cash flows in a small table by date before touching any formula. It prevents most errors.
Check the direction of the trade last, using one sentence: buy cheap, sell rich.
Watch whether the question gives zero prices, discount factors or spot rates. Convert to discount factors first.
If a question mentions transaction costs or shorting limits, an apparent gap may not be an arbitrage.
Remember that the profit is realised today and cash flows net to zero later.

Practice questions from Pricing Conventions, Discounting, and Arbitrage

Law of One Price and Arbitrage in other exams

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

Law of One Price and Arbitrage: frequently asked questions

What is the difference between the law of one price and no-arbitrage?

The law of one price says identical cash flows have the same price. No-arbitrage is the broader principle that no risk-free profit exists with zero investment. The law of one price follows from it, and it is what lets you price assets by replication.

How do I check if a bond price offers an arbitrage?

Discount each cash flow with the matching zero price or spot rate and add them up. Compare that total with the market price. Any gap means an arbitrage in a frictionless market.

What is bond stripping and reconstitution?

Stripping splits a coupon bond into separate zero-coupon cash flows. Reconstitution rebuilds the bond from those zeros. If the bond is cheaper than its zeros, strip it. If dearer, reconstitute it.

Does arbitrage always exist when two prices differ?

Not always. Real markets have transaction costs, bid-ask spreads and funding limits. The price gap must exceed those costs, and the two positions must have the same cash flows, for it to be a true arbitrage.