Learning Module 4
Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Explain how the concepts of arbitrage and replication are used in pricing derivatives.
Explain the difference between the spot and expected future price of an underlying and the cost of carry associated with holding the underlying asset.
1 · Arbitrage
Arbitrage = riskless profit opportunity that arises when the law of one price is violated.
Type 1: Identical assets (or assets with identical future cash flows) trade at different prices at the same time → buy the cheap one, sell the expensive one, profit the difference risklessly.
Type 2: An asset with a known future price doesn't trade at the present value of that future price discounted at the risk-free rate → borrow/lend and trade spot vs. forward to lock in riskless profit.
Arbitrageurs' trading pushes prices back toward equilibrium, eliminating the opportunity.
Discrete: F0(T) = S0(1 + r)T | S0 = ST(1 + r)−T
Continuous: F0(T) = S0erT
2 · Replication
Replication recreates a derivative's cash flow stream using a position in the underlying plus borrowing/lending at the risk-free rate — used when no arbitrage exists, to derive the fair (no-arbitrage) forward price.
Long forward replication: borrow S0, buy the underlying at spot; at T, sell at ST, repay loan — replicates the long forward payoff (ST − F0(T)).
Short forward replication: short sell the underlying, invest proceeds at the risk-free rate; at T, cover the short — replicates the short forward payoff.
Long asset + short forward = a fully hedged position that earns exactly the risk-free rate, confirming F0(T) = S0(1 + r)T.
Position to Replicate | Replication Strategy |
|---|---|
Long forward | Sell a risk-free bond (borrow) + buy the underlying in the cash market |
Short forward | Buy a risk-free bond (lend) + sell the underlying short in the cash market |
Long underlying (synthetic) | Long forward + lend PV of forward price |
3 · Cost of Carry — The Core Relationship
Cost of carry = net of costs and benefits of owning the underlying over the contract period; it explains why F0(T) can be above, below, or equal to S0.
Opportunity cost (r) applies to all assets: a positive risk-free rate pushes F0(T) above S0, all else equal — and the effect grows with both r and T.
Other costs (C, c) — storage, insurance, transportation, spoilage (mainly commodities) — push F0(T) higher.
Benefits (I, i) — dividends, coupon interest, convenience yield — push F0(T) lower (forgone by the derivative holder vs. the cash owner).
F0(T) = [S0 − PV0(I) + PV0(C)](1 + r)T
F0(T) = S0e(r + c − i)T (continuous compounding; i = income yield, c = cost yield)
Cost vs. Benefit | Forward vs. Spot |
|---|---|
Opportunity + other costs > benefits | F0(T) > S0 |
Opportunity + other costs < benefits | F0(T) < S0 |
Opportunity + other costs = benefits | F0(T) = S0 |
4 · Cost of Carry by Asset Class
Asset Class | Benefits (i) | Costs (r, c) |
|---|---|---|
No-cash-flow asset (e.g., non-dividend stock) | None | Risk-free rate only |
Equities | Dividend / dividend yield | Risk-free rate |
Foreign exchange | Foreign risk-free rate (rf) | Domestic risk-free rate (rd) — net effect is the rate differential (rf − rd) |
Commodities | Convenience yield | Risk-free rate + storage cost |
Interest rates | Interest income | Risk-free rate |
Credit | Credit spread | Risk-free rate |
Convenience yield — a non-cash benefit of holding the physical commodity (vs. a derivative), arising under conditions such as low inventories; not a cost, and not contingent on cash payment.
A forward price significantly below the no-arbitrage price may signal the presence of a convenience yield not otherwise captured in the standard cost-of-carry formula.
5 · FX Forward Pricing — Interest Rate Parity
FX is a special case: the "cost of carry" is the interest rate differential between the foreign rate (rf) and domestic rate (rd).
Quoting convention S0,f/d: units of price/foreign currency (f) per one unit of base/domestic currency (d).
F0(T) = S0e(r + c − i)T (continuous compounding; i = income yield, c = cost yield)
Interest Rate Differential | Forward vs. Spot | Foreign Currency Trades At |
|---|---|---|
(rf − rd) > 0 | F0,f/d(T) > S0,f/d | Forward discount |
(rf − rd) < 0 | F0,f/d(T) < S0,f/d | Forward premium |
(rf − rd) = 0 | F0,f/d(T) = S0,f/d | Neither premium nor discount |