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CFA Level I · CFA Level I Exam

Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Derivatives are priced by no-arbitrage. If you can replicate a payoff with the underlying and borrowing or lending, the derivative must cost the same as the replication, or arbitrage profits appear. For a forward, the price is the spot price grown at the risk-free rate, adjusted for the costs and benefits of holding the underlying.

What this chapter covers

This chapter gives you the logic behind all derivative pricing at Level I. You start with arbitrage: a way to earn a riskless profit with no net investment. The law of one price says two assets with identical payoffs must trade at the same price. If they do not, traders buy the cheap one, sell the expensive one, and prices move back.

Next comes replication. You build the payoff of a derivative from the underlying asset plus a risk-free loan or deposit. The cost of building it is the fair value of the derivative. Risk-neutral pricing is the shortcut: you can value the derivative as if investors were risk-neutral, discounting expected payoffs at the risk-free rate, using risk-neutral probabilities instead of real-world ones. The probabilities are a tool, not a belief about what will happen.

Then you add cost of carry. Holding the underlying can bring benefits (dividends, interest, convenience yield) and costs (storage, insurance). These adjust the forward price. The chapter ends with pricing and valuing forward contracts. This sets up the later modules on futures, swaps and options, which reuse the same no-arbitrage logic. It also links to fixed income (discounting) and equities (dividends).

Derivatives and Risk Management carries a topic weight of 6-9%, and this chapter is the base for the rest of it. Questions are short, formula-driven and often numerical, which suits the 90-second pace per question. The same ideas come back in swaps, options and portfolio questions, so a firm grasp pays off more than once. With no penalty for wrong answers, you should still answer every item, and clear no-arbitrage logic helps you eliminate two of the three options fast.

Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives: topics in the order to study them

  1. 1Arbitrage and the Law of One PriceEverything else rests on this idea, so learn the definition and the buy-low, sell-high logic first.
  2. 2Replication and Risk-Neutral PricingIt shows how a payoff is built from the underlying and a risk-free loan, which gives you the price.
  3. 3Cost of Carry: Benefits and Costs of Holding the UnderlyingYou need to know which items raise or lower the forward price before you use the formula.
  4. 4Pricing and Valuation of Forward ContractsIt pulls the earlier ideas together into formulas for the forward price and the value of a position over time.

How to prepare Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

Aim to understand why each price must hold, then drill the arithmetic. Short daily sessions work well on a phone, with calculator practice at a desk.

  1. Read the definition of arbitrage and explain it aloud in two sentences: no net investment, no risk, positive profit.
  2. Work one replication example by hand: buy the underlying with borrowed money and compare it to the forward. Write down each cash flow at time 0 and at expiry.
  3. Learn the forward price formula in its basic form, F0 = S0 × (1 + r)^T, then add carry: subtract the future value of benefits and add the future value of costs.
  4. Practise both discrete compounding and continuous compounding, F0 = S0 × e^((r − q) × T) for a continuous dividend yield q. Use the exponent key on your TI BA II Plus (press 2ND then ex) or your HP 12C.
  5. Separate the forward price at initiation from the forward value after initiation. At time t, the value to the long is Vt = [Ft(T) − F0(T)] ÷ (1 + r)^(T − t): the difference between the current forward price and the original forward price, discounted at the risk-free rate over the remaining time to expiry.
  6. Do timed sets of three-option questions. For each miss, note whether the error was logic, a sign or a compounding mistake.
  7. On the last day, reread the quick revision list and redo two numerical examples without notes.

Common mistakes in Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

  • Adding dividends to the forward price instead of subtracting them.

    Fix: Ask who gets the dividend. The spot holder does, not the forward holder, so the forward is cheaper. Benefits subtract, costs add.

  • Confusing the forward price with the forward value.

    Fix: The price is the agreed delivery price fixed at initiation. The value is what the contract is worth to the long or short at a given date, and it starts at zero.

  • Treating risk-neutral probabilities as true probabilities.

    Fix: Remember they are constructed so the risk-free rate discounts payoffs correctly. Use them for valuation only.

  • Mixing compounding methods or time units.

    Fix: Convert T to years first and match the formula to the rate type stated in the question.

  • Calling any price gap an arbitrage.

    Fix: Check the definition: no net investment, no risk and a certain profit. If the position carries risk, it is not an arbitrage.

  • Dropping the present value step when valuing a forward after initiation.

    Fix: Discount that difference from expiry back to today at the risk-free rate over the remaining time, T − t.

Last-day revision: Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

  • Arbitrage: riskless profit with no net investment.
  • Law of one price: identical payoffs must have the same price.
  • Arbitrage keeps prices in line, so derivative prices are set by no-arbitrage, not by expectations.
  • Replication: underlying plus risk-free borrowing or lending reproduces the derivative payoff.
  • Risk-neutral probabilities are used for pricing only; they are not real-world forecasts.
  • Forward price (no carry): F0 = S0 × (1 + r)^T.
  • Benefits of holding the underlying (dividends, interest, convenience yield) lower the forward price.
  • Costs of holding (storage, insurance) raise the forward price.
  • With carry: F0 = (S0 − PV of benefits + PV of costs) × (1 + r)^T.
  • Continuous form: F0 = S0 × e^((r − q) × T) when the benefit is a continuous yield q.
  • A forward has zero value at initiation, because the forward price is set so that neither side pays.
  • Value of a long forward at time t: Vt = [Ft(T) − F0(T)] ÷ (1 + r)^(T − t), the difference in forward prices discounted over the remaining time to expiry; it can be positive or negative.

Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives practice questions

Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives in other exams

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

Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives: frequently asked questions

What is the difference between arbitrage and speculation?

Arbitrage earns a certain profit with no net investment and no risk. Speculation takes on risk in the hope of profit. If a trade can lose money, it is not an arbitrage.

Why is the value of a forward contract zero at initiation?

The forward price is set so that the present value of the payoff is zero for both sides. Neither party pays anything upfront. The value moves away from zero as the spot price and interest rates change.

Do I need to memorise both discrete and continuous formulas?

Yes. The discrete form with no carry is F0 = S0 × (1 + r)^T. The continuous form is F0 = S0 × e^(rT) with no benefits or costs, or F0 = S0 × e^((r − q) × T) when the underlying pays a continuous yield q. Questions state which rate type applies, so check the wording before you choose.

Can I use my calculator for these questions?

Yes, the TI BA II Plus and HP 12C are both approved. Use the yx key for discrete compounding and the ex key for continuous compounding. Practise the keystrokes before exam day so you do not lose time.