CFA Level II Exam · The Arbitrage-Free Valuation Framework

Arbitrage-Free Valuation and the Law of One Price

Updated 7 October 2026 · Fact-checked

Arbitrage-free valuation prices a bond so that no one can earn a riskless profit with no investment. By the law of one price, identical cash flows must have the same price. So you value a bond as a package of zero-coupon bonds, discounting each cash flow at its own spot rate.

Understand Arbitrage-Free Valuation and Law of One Price

An arbitrage opportunity is a way to make a riskless profit with no net outlay. Examples: buy something cheap in one place and sell it dear in another, or buy a bond and sell its identical cash flows for more. Markets with active traders tend to remove such gaps fast, so valuation models assume none exist.

The law of one price says two assets with identical future cash flows, in every state of the world, must trade at the same price. If they did not, you would buy the cheap one and sell the expensive one. You would pocket the difference today and owe nothing later, because the cash flows cancel.

Apply this to a coupon bond. Each coupon and the principal is a separate cash flow at a separate date. Each could be bought as a zero-coupon bond (a strip) maturing on that date. So a bond must be worth the sum of the present values of its cash flows, each discounted at the spot rate for its own maturity. Spot rates come from the zero-coupon yield curve.

The yield to maturity (YTM) is different. It is one single rate that makes the present value of all cash flows equal the price. It is a summary number. It is not the correct rate for any single cash flow unless the spot curve is flat. That is why the spot-rate value is the arbitrage-free value, and YTM is derived from it.

If a bond trades below its spot-rate value, you buy the bond and sell the strips. This is called stripping. If it trades above, you buy the strips and sell the bond, which is reconstitution. Both lock in the gap. Trading like this pushes the price back to the arbitrage-free value.

Key formulas to remember

Arbitrage-free bond value (spot rates): P = C ÷ (1 + z₁) + C ÷ (1 + z₂)² + ... + (C + FV) ÷ (1 + zₙ)ⁿ; z_t is the spot rate for maturity t, annual compounding. Each cash flow uses its own spot rate.
Discount factor from spot rate: DF_t = 1 ÷ (1 + z_t)ᵗ; Bond value = Σ (cash flow_t × DF_t). Discount factors let you value any bond on the same curve.
Bond value using YTM: P = C ÷ (1 + y) + C ÷ (1 + y)² + ... + (C + FV) ÷ (1 + y)ⁿ; One rate y for all cash flows. Solve for y given price. It matches the spot-rate price only by construction.
Arbitrage condition: Market price ≠ Σ (CF_t × DF_t) → arbitrage exists; Price below value: buy bond, sell strips. Price above value: buy strips, sell (short) bond.
Law of one price: Identical cash flows → identical price; Applies to the same cash flows in all states, adjusted for no transaction costs or liquidity differences.

How to solve Arbitrage-Free Valuation and Law of One Price questions

Use this method for any question that asks you to value a bond from a curve, compare it to its market price, or spot an arbitrage.

1Read the vignette and list the bond's cash flows by date: each coupon, plus principal in the last period.
2Find the spot rates (or discount factors) for each date in the exhibit. Check whether the rates are spot, par or forward, and the compounding.
3Discount each cash flow at its own spot rate. Compute the discount factor first, then multiply.
4Sum the present values. This is the arbitrage-free value.
5Compare with the market price. Equal means no arbitrage. If not, say which is cheap.
6If asked for the trade: buy the cheap side, sell the expensive side. Bond cheap: buy bond, sell strips. Bond dear: buy strips, sell bond.
7If asked for YTM, solve the single rate that equates the value to price. Do not use it as the discount rate for individual cash flows.

Quickest way: Discount-factor table method

When to use it: Use when the vignette gives spot rates for several maturities and you must value one or more bonds or test for arbitrage.

Write the cash flow line: coupon, coupon, ..., coupon plus par.
Under it, compute DF = 1 ÷ (1 + z)ᵗ once per date and keep 4 to 5 decimals.
Multiply and add. Use the calculator memory to sum.
Subtract the market price. A positive gap means the bond is cheap.
Check reasonableness: the YTM of a coupon bond typically lies between the shortest and longest spot rates, and it is closer to the later spot rates for a bond with a large final payment.

Common mistakes in Arbitrage-Free Valuation and Law of One Price

Discounting every cash flow at the bond's YTM when asked for the arbitrage-free value.
YTM is familiar from Level I and one rate feels simpler.
Fix: Whenever a spot curve is given, discount each cash flow at its own spot rate. Use YTM only if told or if the curve is flat.
Using par rates or forward rates as if they were spot rates.
Exhibits show several curves with similar labels.
Fix: Read the exhibit heading. Only spot (zero) rates discount a single cash flow directly. Convert par or forward rates first.
Taking the wrong side of the trade.
Students mix up which is cheap, the bond or the strips.
Fix: Compare market price with the spot-rate value. Cheap bond: buy bond, sell strips. Dear bond: sell bond, buy strips. Buy low, sell high.
Forgetting principal in the final cash flow.
The coupon row is built first and the last date is left short.
Fix: The last cash flow is the coupon plus par. Write it out before discounting.
Using the wrong power in the discount factor.
Students reuse the first-year factor or use the wrong period.
Fix: Date t gets exponent t with its own z_t. Check with a table: DF should fall steadily as t rises on a positive curve.

Worked examples

Example 1

A vignette gives annual spot rates: 1-year 2.00%, 2-year 3.00%, 3-year 4.00%. A 3-year, 5% annual-pay bond with par 100 trades at 100.50. Q1: What is its arbitrage-free value? Q2: Is there an arbitrage, and what trade would capture it?

Show the solution

Cash flows: 5 at year 1, 5 at year 2, 105 at year 3.
DF1 = 1 ÷ 1.02 = 0.98039. PV1 = 5 × 0.98039 = 4.9020.
DF2 = 1 ÷ 1.03² = 1 ÷ 1.0609 = 0.94260. PV2 = 5 × 0.94260 = 4.7130.
DF3 = 1 ÷ 1.04³ = 1 ÷ 1.124864 = 0.88900. PV3 = 105 × 0.88900 = 93.345.
Value = 4.9020 + 4.7130 + 93.345 = 102.96.
Market price 100.50 is below 102.96, so the bond is cheap by about 2.46.
Trade: buy the bond and sell its cash flows as strips.

Answer: Q1: The arbitrage-free value is about 102.96. Q2: Yes. The bond is underpriced, so buy the bond and sell the individual zero-coupon strips, locking in about 2.46 per 100 par.

Example 2

Using the same spot curve (1-year 2.00%, 2-year 3.00%, 3-year 4.00%), a 2-year, 4% annual-pay bond with par 100 is priced at 101.50. Q1: What is its arbitrage-free value? Q2: Is the bond overpriced or underpriced? Q3: What trade captures the gap?

Show the solution

Cash flows: 4 at year 1, 104 at year 2.
DF1 = 0.98039. PV1 = 4 × 0.98039 = 3.9216.
DF2 = 0.94260. PV2 = 104 × 0.94260 = 98.030.
Value = 3.9216 + 98.030 = 101.95.
Market price 101.50 is below 101.95 by about 0.45, so the bond is underpriced.
Trade: buy the bond, sell the 1-year and 2-year strips matching its cash flows.

Answer: Q1: About 101.95. Q2: Underpriced, because the market price of 101.50 is below value. Q3: Buy the bond and sell the matching zero-coupon strips (reconstitution is the reverse, not needed here).

Exam tips

Check the exhibit label first: spot, par or forward. Many wrong answers come from using the wrong curve.
Keep discount factors to 5 decimals so rounding does not push you to a wrong option.
For arbitrage questions, work out value first, then compare. The direction of the trade follows from which side is cheap.
YTM is the correct rate for each individual cash flow only when the spot curve is flat. Discounting at YTM reproduces the bond's price by construction, but it does not give the right value for each separate cash flow.
There is no penalty for wrong answers, so answer every question. Move on if a vignette's cash flow table is long, and return later.

Arbitrage-Free Valuation and 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-Free Valuation and Law of One Price: frequently asked questions

What is the law of one price in bond valuation?

It says assets with identical cash flows must have the same price. A coupon bond equals a portfolio of zero-coupon bonds, so its value is the sum of those strips' prices. If the market price differs, an arbitrage exists.

Why is the spot-rate value different from the YTM value?

YTM is the single rate that equates the present value of the cash flows to the market price. It is not the right discount rate for individual cash flows unless the spot curve is flat, so the arbitrage-free value must come from the spot rates.

What is an arbitrage opportunity in fixed income?

It is a riskless profit with no net investment, usually from a bond trading away from the value of its strips. If the bond is cheap, you buy it and sell the strips. If it is dear, you buy the strips and sell the bond.

Does arbitrage-free valuation ignore transaction costs?

The basic framework assumes no transaction costs and no liquidity differences. In practice small gaps can persist because costs and liquidity limit how much trading can remove them.