Home › CFA® Level I › Derivatives
CFA® Level I · Derivatives · free lesson
Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives
Derivatives are priced not by forecasting where the underlying will go but by asking what it costs to replicate them with the underlying and risk-free borrowing or lending. This module develops the no-arbitrage principle, uses replication to derive the forward price, and adds the costs and benefits of carrying the underlying, including storage, dividends, coupons, convenience yield and foreign interest rates.
What you will learn
1. Explain arbitrage and the law of one price and why no-arbitrage pricing does not depend on investors' risk preferences or forecasts.
- Arbitrage: risk-free profit, no net investment, exploiting identical cash flows priced differently.
- Law of one price: identical future cash flows must have the same price today.
- A risk-free portfolio must earn the risk-free rate.
- Replication prices a derivative as the cost of a portfolio with the same payoff.
- No-arbitrage prices are independent of forecasts and risk preferences; limits to arbitrage allow small gaps.
2. Use replication to derive the forward price of an underlying with no income or costs, distinguish the forward price from the expected spot price, and describe cash-and-carry arbitrage.
- No income or costs: F_0(T) = S_0(1 + r)^T; the forward is the spot price carried at the risk-free rate.
- Long asset + short forward = risk-free bond; long asset financed by borrowing = long forward.
- The forward price differs from the expected spot price by the risk premium; it is not a forecast.
- Forward too high: cash-and-carry (borrow, buy asset, sell forward).
- Forward too low: reverse cash-and-carry (short asset, lend proceeds, buy forward).
3. Explain how carry costs and carry benefits, such as storage, dividends, coupons and convenience yield, affect forward prices, and calculate forward prices using discrete and continuous compounding.
- Carry costs (storage, insurance) raise the forward price; carry benefits (dividends, coupons, convenience yield) lower it.
- Discrete: F_0(T) = S_0 - PV(I) + PV(C)^T.
- Continuous: F_0(T) = S_0 e^((r + c - i)T).
- Net carry cost > 0 implies contango; large benefits such as convenience yield imply backwardation.
- Arbitrage holds the formula tightly for financial assets, more loosely for hard-to-store commodities.
4. Apply the cost-of-carry model to currency forwards and explain why the higher-interest-rate currency trades at a forward discount.
- The base currency's interest rate is a carry benefit; the price currency's rate is the financing cost.
- Discrete: F = S x [(1 + r_price) / (1 + r_base)]^T; continuous: F = S e^((r_price - r_base)T).
- Lower-rate base currency trades at a forward premium; higher-rate base currency at a forward discount.
- Forward points = F - S; the forward rate is a no-arbitrage price, not a forecast of the future spot rate.
Module at a glance
- Arbitrage is a riskless, self-financing profit; the law of one price rules it out in efficient markets.
- Derivatives are priced by replication, so forecasts and risk preferences do not enter.
- No income or costs: F_0(T) = S_0(1 + r)^T; the forward price is not an unbiased forecast of S_T.
- Forward too high: cash-and-carry; forward too low: reverse cash-and-carry.
- Discrete carry: F_0(T) = S_0 - PV(I) + PV(C)^T; continuous: S_0 e^((r + c - i)T).
- Currency forwards: F = S x [(1 + r_price)/(1 + r_base)]^T; the higher-rate base currency trades at a discount.
- Limits to arbitrage, such as funding and balance sheet costs, can let small deviations persist.
This free module has 4 lessons, 14 practice questions with full explanations.