FRM Exam Part II · Arbitrage Pricing with Term Structure Models

No-Arbitrage Pricing and Replicating Portfolios for Bonds

Updated 11 October 2026 · Fact-checked

No-arbitrage pricing values a security by building a portfolio of other traded securities that has the same payoffs in every state. By the law of one price, the security must cost the same as that replicating portfolio. Solve for the portfolio weights from the payoffs, then price it at today's market prices.

Understand No-Arbitrage Principle and Replicating Portfolios

An arbitrage is a trade that costs nothing (or gives you cash today), has no chance of a loss, and has some chance of a gain. Markets with active traders should not offer such trades for long. The no-arbitrage principle says model prices must not allow one.

The tool that applies this is the law of one price: two portfolios with identical payoffs in every future state must have the same price today. If they did not, you would buy the cheaper one, sell the dearer one and keep the difference with no risk.

So to price an unknown security, you replicate it. You pick traded bonds, usually zero-coupon bonds or a bond and cash, and choose quantities so the portfolio pays exactly what the target pays in each state. In a one-period binomial tree, there are two states (rates up, rates down), so you need two instruments and two equations.

The price of the target is then the cost of the replicating portfolio. Notice what you did not need: the real-world probability of rates going up or down. Replication uses only the payoffs and the current prices. This is why no-arbitrage models fit today's curve exactly and why risk-neutral probabilities work. They are just the probabilities that make the replication price equal to a discounted expected value.

Two cautions. First, the price is only as good as the model of future states. Second, no-arbitrage does not mean the market is right or that the bond has no risk. It means prices are consistent with each other.

Key formulas to remember

Law of one price: If payoff(A) = payoff(B) in every state, then Price(A) = Price(B); Applies to portfolios of traded securities with no frictions. Violation creates an arbitrage.
Replication equations (two states): n1 × V1(up) + n2 × V2(up) = X(up); n1 × V1(down) + n2 × V2(down) = X(down); V1 and V2 are the payoffs of the two instruments next period. X is the target payoff. Solve for n1 and n2.
Price of the target: Price(X) = n1 × P1 + n2 × P2; P1 and P2 are today's prices of the two instruments. Include cash at the amount borrowed or lent.
Risk-neutral probability: p × P(up) + (1 − p) × P(down) = Price × (1 + r), so Price = [p × X(up) + (1 − p) × X(down)] ÷ (1 + r); p is chosen so a traded instrument is priced correctly. Here r is the one-period rate for the step. The same p then prices every other security on the tree.
Discount factor from a zero price: Z(T) = Price of ₹1 (or $1) paid at T, and Price = Σ CF(t) × Z(t); A coupon bond is a portfolio of zero-coupon bonds, so no-arbitrage requires this sum.

How to solve No-Arbitrage Principle and Replicating Portfolios questions

Use this for any question that gives you bond prices or a rate tree and asks for the price of another security, or for an arbitrage.

1Write the states and the payoffs next period of each traded instrument and of the target security. Include any coupon paid at that date.
2Choose as many instruments as there are states. Two states need two instruments, such as a zero-coupon bond and a longer bond.
3Set up one equation per state: portfolio payoff equals target payoff. Solve for the quantities. A negative quantity means a short position.
4Price the replicating portfolio using today's market prices of the instruments.
5By the law of one price, set the target's price equal to that cost.
6If a market price is given for the target, compare it. If the market price is higher, sell the target and buy the portfolio. If lower, do the reverse. The profit today is the gap.
7Check the answer: the payoffs should match in both states, and the price should sit between discounted down and up payoffs.

Quickest way: Risk-neutral shortcut on a one-step tree

When to use it: Use when a question gives a rate tree or a bond price tree and you only need a price, not the portfolio weights.

Find the risk-neutral probability p from a bond already priced, or use the p given (often 0.5).
Compute the expected payoff next period as p × up value + (1 − p) × down value.
Discount at the rate for that step, which is 1 + r with the same compounding as the tree.
Cross-check on an option or bond with a known price if time allows.
Use full replication only if the question asks for the number of units to hold.

Common mistakes in No-Arbitrage Principle and Replicating Portfolios

Using real-world probabilities to price the security.
Students link price with expected value and use the probability given for the economy.
Fix: Replication needs no probabilities. If you use an expectation, use risk-neutral probabilities only.
Forgetting the coupon in next-period payoffs.
The bond's tree shows price, not the cash flow, so the coupon gets dropped.
Fix: Payoff at a date = ex-coupon price + coupon paid at that date. Write it out each time.
Using fewer instruments than states.
Students try to replicate a two-state payoff with one bond.
Fix: You need one instrument per state. Check the equations can be solved before pricing.
Reading a negative quantity as an error.
Students assume weights must be positive.
Fix: A negative quantity is a short position or borrowing. It is valid in replication.
Trading the wrong way when an arbitrage exists.
Mixing up which side is rich and which is cheap.
Fix: Always sell the expensive one and buy the cheap one. Compare the market price with the replication cost first.

Worked examples

Example 1

A one-year zero-coupon bond with face value $100 costs $95. Next year it matures at $100. A one-year bond is the only traded instrument besides cash, so use the following: Bond A, a two-year zero with face $100, costs $88 today and will be worth $94 (rates down) or $92 (rates up) in one year. A risk-free one-year zero at $95 pays $100 in each state. A third security pays $10 if rates are down and $0 if rates up. Find its price.

Show the solution

States: down and up. Instruments: Bond A and the one-year zero Z (payoff $100 in both states).
Let a = units of A and z = units of Z. Down: 94a + 100z = 10. Up: 92a + 100z = 0.
Subtract the up equation from the down equation: 2a = 10, so a = 5.
Up equation: 92 × 5 + 100z = 0, so 460 + 100z = 0, so z = −4.6.
Check down: 94 × 5 + 100 × (−4.6) = 470 − 460 = 10. Correct.
Cost today: 5 × 88 + (−4.6) × 95 = 440 − 437 = 3.
Law of one price: the third security costs $3.

Answer: $3. Buy 5 units of the two-year zero and short 4.6 units of the one-year zero.

Example 2

A one-period tree has a one-year rate of 4% (annual compounding). A bond's price in one year is $97 if rates go up and $99 if rates go down. Risk-neutral probability of the up state is 0.5. What is the bond's price today?

Show the solution

Expected price in one year under risk-neutral probabilities: 0.5 × 97 + 0.5 × 99 = 98.
Discount one year at 4%: 98 ÷ 1.04.
98 ÷ 1.04 = 94.2308 (rounded to four decimals).
Law of one price: any portfolio replicating this payoff must cost the same, so this is the arbitrage-free price.

Answer: About $94.23.

Exam tips

Expect a small tree with two states. Set up two equations and solve quickly; keep the algebra neat to avoid sign errors.
Read the question for the word 'arbitrage'. If a market price is given, compare it with the replication cost and state the direction of the trade.
Remember that the replicated price does not depend on the real-world probability. If a question offers it as a distractor, ignore it.
Check whether the payoff includes a coupon, and whether rates are quoted with annual or semiannual compounding.
If time is short, use the risk-neutral shortcut with the p given, then confirm that the answer lies between the discounted down and up values.

Practice questions from Arbitrage Pricing with Term Structure Models

No-Arbitrage Principle and Replicating Portfolios in other exams

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

CFA Level IPrinciples of No-Arbitrage Pricing

No-Arbitrage Principle and Replicating Portfolios: frequently asked questions

What is the law of one price in bond pricing?

It says two portfolios with the same payoffs in every state must have the same price today. If not, you could buy the cheap one and sell the dear one for a riskless profit. It is the basis for pricing a bond by replication.

Why do replicating portfolios not need real-world probabilities?

The portfolio matches the payoff in each state exactly, so the outcome is the same whichever state occurs. Its cost depends only on today's prices and the payoffs. Probabilities of the states do not enter.

How is a coupon bond replicated?

A coupon bond is a portfolio of zero-coupon bonds, one for each cash flow. Its arbitrage-free price is the sum of each cash flow times the price of a zero-coupon bond maturing on that date. If the market price differs, you can trade the bond against the strip of zeros.

How does replication link to risk-neutral pricing?

The risk-neutral probability is the one that makes the discounted expected payoff equal to the replication cost. It is a pricing tool, not a forecast. Using it gives the same answer as building the replicating portfolio.