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CFA Level I Exam · Pricing and Valuation of Futures Contracts

Principles of No-Arbitrage Pricing for Derivatives

Updated 7 October 2026 · Fact-checked

No-arbitrage pricing sets a derivative's price so nobody can earn a risk-free profit with no net investment. Two assets with identical payoffs must have the same price (law of one price). You find the price by replicating the payoff with the underlying and a risk-free asset, not by forecasting the spot price.

Understand Principles of No-Arbitrage Pricing

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

That trade is an arbitrage. It has two features: it needs no net investment of your own money, and it earns a risk-free profit. If both conditions hold, you gain with certainty and risk nothing. Speculation is different. A speculator takes risk, hoping to earn a return. A speculator can lose money. An arbitrageur cannot, in theory.

Arbitrage is the engine of derivative pricing. You build a replicating portfolio from the underlying asset and a risk-free bond (borrowing or lending) that pays exactly what the derivative pays. The derivative must then cost the same as the replicating portfolio. If it did not, arbitrageurs would trade until the gap closed. Buying pressure raises the cheap asset and selling pressure lowers the expensive one.

This is why the expected spot price does not set a forward price. A forward price depends on today's spot price, the risk-free rate, and the costs and benefits of holding the asset. It does not depend on what investors think the asset will be worth later, or on how risk-averse they are.

Risk-neutral valuation is a calculation shortcut that follows from this. Because replication makes the price independent of investors' risk preferences, you can compute the price as if all investors were risk neutral. You use risk-neutral probabilities and discount the expected payoff at the risk-free rate. These probabilities are not real-world probabilities. They are the ones that make the model price match the no-arbitrage price.

Two more ideas matter. Arbitrage pricing assumes frictionless markets: no transaction costs, free short selling, and the ability to borrow and lend at the risk-free rate. And an arbitrage that exists in the real world disappears quickly once traders act on it.

Key formulas to remember

Law of one price
If payoff(A) = payoff(B) in every state, then price(A) = price(B)
If prices differ, buy the cheaper and sell the more expensive for a risk-free profit.
Replication
Derivative value = value of replicating portfolio (underlying + risk-free bond)
Price comes from the replicating cost, not from expected spot prices.
Arbitrage conditions
No net investment AND risk-free profit
Both must hold. A trade that needs capital or carries risk is not arbitrage.
Risk-neutral value
Value today = [π × payoff(up) + (1 − π) × payoff(down)] ÷ (1 + r)
π is the risk-neutral probability of the up state and r is the risk-free rate for the period. π is not the real-world probability.
Risk-neutral probability, one-period binomial
π = [(1 + r) − d] ÷ (u − d)
u and d are the up and down gross factors (1 + return). It requires d < 1 + r < u so that no arbitrage exists.

How to solve Principles of No-Arbitrage Pricing questions

Use this method for any question on arbitrage, the law of one price or risk-neutral valuation.

  1. 1Identify what is being compared: two assets with the same payoffs, or a derivative and its underlying.
  2. 2Write down the payoff of each position in every state (or at expiry).
  3. 3Check whether the payoffs are identical. If so, the prices must match; if the prices differ, an arbitrage exists.
  4. 4Build the trade: buy the cheaper, sell the more expensive, and borrow or lend at the risk-free rate to leave zero net investment today.
  5. 5Confirm the two arbitrage tests: no money out today, and a profit with certainty. If risk remains, it is not arbitrage.
  6. 6If the question asks for a model price, compute the replicating cost or the risk-neutral expected payoff discounted at the risk-free rate.
  7. 7Check the answer against the options. Reject any option that uses expected spot prices or investor risk preferences to set the price.

Quickest way: Two-test shortcut and risk-neutral price

When to use it: Use this for conceptual MCQs and for one-period binomial pricing when time is short.

  1. Ask: does the trade need money today, and can it lose? If either answer is yes, it is not arbitrage.
  2. For concept items, pick the option that links price to replication and the risk-free rate. Reject options based on expected spot price, risk aversion or forecasts.
  3. For numbers: compute π = (1 + r − d) ÷ (u − d), then value = [π × up payoff + (1 − π) × down payoff] ÷ (1 + r).
  4. Sanity check: the value must lie between the discounted down and up payoffs.

Common mistakes in Principles of No-Arbitrage Pricing

  • Calling any profitable trade an arbitrage

    Students focus on the profit and forget the conditions.

    Fix: Test both conditions: zero net investment and risk-free profit. A trade that can lose money is speculation.

  • Thinking the forward price equals the expected future spot price

    It sounds logical that a contract on a future price should reflect forecasts.

    Fix: The forward price comes from the spot price and the cost of carrying the asset. Replication, not expectation, sets it.

  • Treating risk-neutral probabilities as real probabilities

    Both are called probabilities and both sum to 1.

    Fix: Risk-neutral probabilities are a pricing device. They are backed out from the no-arbitrage condition and need not match actual odds.

  • Assuming risk-neutral valuation means investors are actually risk neutral

    The name suggests an assumption about people.

    Fix: It is a technique. Because replication removes risk, the price is the same as in a risk-neutral world, whatever investors' real attitudes.

  • Discounting the risk-neutral expected payoff at a risky rate

    Students carry over the idea of a risk-adjusted discount rate from equity valuation.

    Fix: Under risk-neutral probabilities, always discount at the risk-free rate.

  • Ignoring the frictionless-market assumptions

    Textbook arbitrage looks effortless.

    Fix: Remember the model assumes no transaction costs, free short selling and risk-free borrowing and lending. Real prices may stay within a band around the no-arbitrage level.

Worked examples

Example 1

A stock trades at $100. Next period it will be either $120 or $90. The one-period risk-free rate is 5%. What is the risk-neutral probability of the up move? (A) 0.40 (B) 0.50 (C) 0.60

Show the solution
  1. u = 120 ÷ 100 = 1.20 and d = 90 ÷ 100 = 0.90.
  2. 1 + r = 1.05.
  3. π = (1.05 − 0.90) ÷ (1.20 − 0.90).
  4. π = 0.15 ÷ 0.30 = 0.50.

Answer: (B) 0.50. Options A and C do not follow from π = (1 + r − d) ÷ (u − d) = 0.50.

Example 2

Using the same stock, what is the value today of a one-period call option with a strike of $100? (A) $8.57 (B) $9.52 (C) $10.00

Show the solution
  1. Payoff if up: 120 − 100 = 20. Payoff if down: max(90 − 100, 0) = 0.
  2. Risk-neutral π = 0.50 from the previous example.
  3. Expected payoff = 0.50 × 20 + 0.50 × 0 = 10.
  4. Discount at the risk-free rate: 10 ÷ 1.05 = 9.5238.
  5. Check by replication: hold 2/3 of a share and borrow. Delta = (20 − 0) ÷ (120 − 90) = 0.6667. Cost = 0.6667 × 100 = 66.67. Down payoff of portfolio = 0.6667 × 90 − B × 1.05 = 0 gives B = 60 ÷ 1.05 = 57.14. Call = 66.67 − 57.14 = 9.52.

Answer: (B) $9.52. Option C forgets to discount the expected payoff of 10. Option A (8.57 = 9 ÷ 1.05) implies an expected payoff of 9, which would need π = 0.45, not the risk-neutral probability of 0.50.

Exam tips

  • Questions on this topic are usually conceptual. Look for the option that ties price to replication and the risk-free rate.
  • Reject any option that sets a derivative price from expected spot prices, forecasts or investors' risk aversion.
  • When an item asks whether a trade is arbitrage, check for zero net investment and no risk before anything else.
  • In binomial items, compute π first, then discount at the risk-free rate. Never use real-world probabilities unless the question asks for an expected value.
  • With three options and no penalty for wrong answers, eliminate two on principle and always answer.

Practice questions from Pricing and Valuation of Futures Contracts

Principles of No-Arbitrage 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 No-Arbitrage Pricing: frequently asked questions

What is the difference between arbitrage and speculation?

Arbitrage earns a risk-free profit with no net investment. Speculation takes on risk in the hope of a return and can lose money. Many trades called arbitrage in practice carry some risk and are really speculation.

Why doesn't the forward price depend on the expected spot price?

A forward can be replicated by buying the asset with borrowed money. That cost depends on the spot price and the interest rate, not on forecasts. If the forward price differed, arbitrageurs would trade until it matched the replication cost.

What is risk-neutral pricing?

It values a derivative by using risk-neutral probabilities and discounting the expected payoff at the risk-free rate. It works because replication makes the price independent of investor risk preferences. The probabilities are a tool, not real-world odds.

Are risk-neutral probabilities the same as real probabilities?

No. Real probabilities describe how likely each outcome is. Risk-neutral probabilities are the ones that make the model price equal the no-arbitrage price. The two usually differ.