CFA Level II Exam · Pricing and Valuation of Forward Commitments

Principles of Arbitrage-Free Pricing for CFA Level II

Updated 7 October 2026 · Fact-checked

Arbitrage-free pricing sets a derivative's price so that no one can earn a riskless profit with no investment. It rests on the law of one price: identical cash flows must have the same price. You price by replicating the payoff with the underlying and a risk-free asset, and discounting at the risk-free rate.

Understand Principles of Arbitrage-Free Pricing

Start with the law of one price: two assets or portfolios with identical future cash flows in every state must sell for the same price today. If they do not, a trader buys the cheap one, sells the expensive one and locks in a profit.

That profit is an arbitrage. A true arbitrage needs no net investment and carries no risk of loss. It arises in two forms: one where the same payoff is priced differently, and one where a position has zero cost and a positive payoff in some states and never a loss. Arbitrage opportunities vanish quickly because traders trade on them, which pushes prices back into line.

This gives a way to price a derivative. Build a replicating portfolio from the underlying and a risk-free borrowing or lending position that matches the derivative's payoff. The derivative must cost the same as that portfolio. For a forward, you buy the asset with borrowed money and hold it to expiry. The forward price is the cost of doing this, including financing and carry costs.

The pricing uses risk-neutral valuation. Because the replication argument does not depend on investors' attitude to risk, you can compute the price as if all investors were risk neutral. You use risk-neutral probabilities and discount at the risk-free rate. These probabilities are not real-world probabilities. They are the numbers that make the model price match the no-arbitrage price. Real-world probabilities and risk premiums do not enter the price.

Finally, separate pricing from valuation. Pricing a forward means setting the forward price so the contract has zero value at initiation. Valuation means finding the contract's value at a later date, after the underlying price and interest rates have moved. At initiation a forward has value of zero and no cash changes hands. Later its value is positive for one party and negative for the other. Futures differ in that daily settlement resets value to zero. Also note that carry costs such as storage and benefits such as dividends or convenience yield change the forward price but do not change the logic.

Key formulas to remember

Law of one price: Same payoff in every state ⇒ same price today; If violated, buy the cheaper and sell the dearer to earn an arbitrage profit.
Arbitrage-free forward price (no carry): F₀(T) = S₀ × (1 + r)^T; Uses discrete compounding. With continuous compounding, F₀(T) = S₀ × e^(rT).
Forward price with carry: F₀(T) = (S₀ − PV of benefits + PV of costs) × (1 + r)^T; Benefits include dividends or coupons. Costs include storage and insurance.
Risk-neutral price of a derivative: V₀ = [π × V_up + (1 − π) × V_down] ÷ (1 + r); π is the risk-neutral probability, not the real-world one. Discount at the risk-free rate.
Value of a long forward at initiation: V₀(T) = 0; The forward price is set so no one pays at the start.
Value of a long forward during its life: Vₜ(T) = Sₜ − F₀(T) ÷ (1 + r)^(T − t); With carry, replace Sₜ with Sₜ less PV of remaining benefits plus PV of remaining costs.
Value of a long forward at expiry: V_T(T) = S_T − F₀(T); The payoff of a long forward settled at expiry.

How to solve Principles of Arbitrage-Free Pricing questions

Use this routine for any item-set question on arbitrage-free pricing, whether it asks for a price, a value or an arbitrage profit.

1Read the vignette and list the data: spot price, risk-free rate, time to expiry, dividends or costs, quoted forward price and elapsed time.
2Decide what is asked: a price at initiation (forward price), a value during the life, or an arbitrage profit.
3Adjust the spot for carry: subtract the PV of benefits and add the PV of costs over the contract's life.
4Compute the no-arbitrage price by compounding the adjusted spot at the risk-free rate to expiry.
5For valuation at time t, take the current adjusted spot and subtract the PV of the original forward price over the remaining time.
6For arbitrage, compare the quoted price with the no-arbitrage price. If quoted is higher, sell the forward, buy the asset and borrow. If lower, buy the forward, short the asset and lend.
7State the profit at expiry or today in present value terms, and check the sign.
8For risk-neutral questions, solve for π from the tree, then discount the expected payoff at the risk-free rate.

Quickest way: Compare quoted price with fair price

When to use it: Use this when the vignette gives a quoted forward price and asks whether an arbitrage exists or what the profit is.

Compute fair F₀ = (S₀ − PV benefits + PV costs) × (1 + r)^T.
Subtract: quoted − fair. Positive means the forward is overpriced, so sell it and buy the asset.
The expiry profit is the absolute gap. Its present value is the gap ÷ (1 + r)^T.
For valuation, use Sₜ − F₀ ÷ (1 + r)^(T − t) and ignore the real-world expected return.

Common mistakes in Principles of Arbitrage-Free Pricing

Using real-world probabilities or expected return to price a derivative.
Candidates think the price should reflect the asset's expected growth.
Fix: Use the risk-free rate and risk-neutral probabilities. Expected return does not affect the no-arbitrage price.
Treating the value of a forward at initiation as the forward price.
The words price and value are used loosely.
Fix: The forward price is a contract rate. The initial value is zero. Value only becomes non-zero as time passes.
Forgetting to subtract the PV of dividends or add the PV of storage costs.
Candidates apply S₀(1 + r)^T from memory.
Fix: Scan the vignette for cash flows on the underlying and adjust the spot first.
Discounting the original forward price over the full term when valuing mid-life.
Candidates reuse the initial formula.
Fix: Discount F₀ only over the remaining time, T − t.
Taking the wrong side of the arbitrage trade.
Candidates mix up which price is too high.
Fix: If the forward is overpriced, sell the forward and buy the asset with borrowed money. Do the reverse if it is underpriced.
Calling any profitable trade an arbitrage.
Risky trades with high expected profit look attractive.
Fix: Arbitrage needs zero net investment and no risk of loss. Expected profit with risk is not arbitrage.

Worked examples

Example 1

A share trades at 80 and pays no dividend. The risk-free rate is 5% per year, compounded annually. A dealer quotes a one-year forward on the share at 86. (1) What is the no-arbitrage forward price? (2) Is there an arbitrage and what is the trade? (3) What is the profit at expiry per share?

Show the solution

No carry items, so F₀ = 80 × 1.05 = 84.00.
Quoted price 86 exceeds fair price 84, so the forward is overpriced.
Trade: sell the forward at 86, borrow 80 and buy the share. At expiry deliver the share and receive 86.
Repay the loan: 80 × 1.05 = 84.
Profit = 86 − 84 = 2.

Answer: (1) 84.00. (2) Yes. Sell the forward, buy the share with borrowed money. (3) Riskless profit of 2 per share at expiry.

Example 2

An investor entered a long forward on a share 3 months ago at a forward price of 84, with one-year maturity. The share now trades at 90 and pays no dividend. The risk-free rate is 5% per year, compounded annually. (1) What is the contract's value to the long now? (2) What was its value at initiation? (3) Who holds the liability?

Show the solution

Remaining time is 0.75 years.
PV of forward price = 84 ÷ 1.05^0.75.
1.05^0.75 = e^(0.75 × ln 1.05) = e^(0.75 × 0.04879) = e^0.036593 ≈ 1.03727.
PV = 84 ÷ 1.03727 ≈ 80.98.
Value = 90 − 80.98 = 9.02.
At initiation the forward price was set so that value was zero.
A positive value for the long means a negative value for the short.

Answer: (1) About 9.02 to the long. (2) Zero. (3) The short party holds the liability, about 9.02.

Exam tips

Underline whether the question says price or value. Price means the forward rate at initiation. Value means the contract's worth now.
Check the vignette for dividends, coupons or storage costs before using any formula.
For tree questions, solve π from the up and down returns and the risk-free rate, then discount at the risk-free rate.
Check time units: remaining time for valuation is T − t, not T.
If an option says expected return or risk aversion affects the price, it is almost certainly wrong.

Principles of Arbitrage-Free Pricing in other exams

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

Principles of Arbitrage-Free Pricing: frequently asked questions

What is the law of one price in derivatives?

Assets with identical payoffs in every state must have the same price. If not, traders buy the cheaper and sell the dearer for a riskless gain. This is how forwards and options are priced from replicating portfolios.

What is the difference between pricing and valuing a forward?

Pricing sets the forward price so the contract has zero value at initiation. Valuation measures the contract's worth later, after the spot price and time have changed. The forward price stays fixed while the value changes.

Are risk-neutral probabilities the same as real-world probabilities?

No. Risk-neutral probabilities are the values that make the model reproduce the no-arbitrage price when you discount at the risk-free rate. Real-world probabilities reflect actual likelihoods and risk premiums, and they do not enter the price.

Does the arbitrage-free price depend on investors' risk preferences?

No. The replication argument works for any investor who prefers more to less. That is why you can price as if everyone were risk neutral.