Chapter 4

Arbitrage, Replication, and Cost of Carry

Master arbitrage principles and the cost of carry model for pricing derivatives

4.1

Arbitrage

Arbitrage is the practice of simultaneously buying and selling identical or similar assets in different markets to profit from a price difference. This situation arises when the fundamental economic principle known as the "law of one price" does not hold. An individual who engages in arbitrage is called an arbitrageur.

The actions of arbitrageurs are crucial for market efficiency. By exploiting price discrepancies, they force prices to converge across markets, thereby eliminating the arbitrage opportunity itself.

Illustrative Example

Imagine a stock is trading for $50.00 on the New York Stock Exchange and, at the exact same moment, for $50.05 on the London Stock Exchange. An arbitrageur could simultaneously buy 10,000 shares in New York and sell 10,000 shares in London, locking in a risk-free profit of $0.05 per share—$500 total—minus transaction costs.

4.2

Replication

Replication is the process of creating a portfolio using an underlying asset and borrowing or lending funds to exactly duplicate the cash flows of a derivative. According to the principle of no-arbitrage, the price of the derivative must equal the cost of constructing its replicating portfolio.

If the derivative's market price deviates from the cost of replication, an arbitrage opportunity arises. Traders can exploit this by buying the cheaper alternative and selling the more expensive one, earning a risk-free profit until prices realign.

For example, the cash flows of a long forward contract can be replicated by borrowing money today to purchase the underlying asset. The forward price must then equal the spot price of the asset plus the cost of carrying that asset (e.g., financing cost) until the delivery date.

4.3

The Cost of Carry Model

The cost of carry is a foundational concept in pricing forward and futures contracts. It represents the net total of all costs and benefits associated with holding or "carrying" an asset until a future date.

Components of Holding an Asset

  • Opportunity Cost (Risk-Free Rate, r): This reflects the cost of capital tied up in the asset. Instead of buying the asset, the funds could have been invested at the risk-free rate.
  • Storage and Ownership Costs (C, c): These are direct expenses related to holding physical assets, such as storage, insurance, and maintenance—common for commodities like gold, oil, or wheat.
  • Benefits and Yield (I, i): These are cash flows or non-monetary advantages from owning the asset.
    • Cash Flows: Examples include dividends from stocks or interest payments from bonds.
    • Convenience Yield: A non-monetary benefit of holding the physical asset. For instance, a manufacturer with inventory of a key raw material avoids production disruptions during supply shortages.

The No-Arbitrage Forward Price

The forward price must reflect the spot price adjusted for the future value of the net cost of carry. This ensures no arbitrage opportunities exist.

F₀(T) = (S₀ + PV₀(Costs) – PV₀(Benefits)) × (1 + r)ᵀ

In simpler terms, the forward price equals the spot price compounded at the risk-free rate, plus the future value of any costs, minus the future value of any benefits received during the holding period.

4.4

Forward Contracts on Foreign Currencies (FX)

The principles of no-arbitrage and cost of carry are essential in determining the fair value of currency forward contracts.

Key Concepts

  • Exchange Rate: The price of one currency (the foreign currency) expressed in terms of another (the domestic currency). The spot rate is denoted as Sf/d, meaning units of domestic currency per one unit of foreign currency.
  • Currency Forward Contract: A binding agreement to buy or sell a specified amount of a foreign currency at a predetermined exchange rate on a future date.

Currency Forward Pricing Formula

The no-arbitrage forward exchange rate is determined by the spot rate and the risk-free interest rates of the two currencies involved.

F₀,f/d = S₀,f/d × 1 + rf 1 + rd

Where:

  • F₀,f/d: Forward exchange rate (domestic currency per foreign currency)
  • S₀,f/d: Spot exchange rate
  • rf: Risk-free interest rate of the foreign currency
  • rd: Risk-free interest rate of the domestic currency

Numerical Example

Scenario: The spot exchange rate for EUR/USD is 1.0800. The US one-year risk-free rate (rd) is 5%, and the Eurozone one-year risk-free rate (rf) is 3%.

Calculation:

F₀,f/d = 1.0800 × 1.03 1.05 ≈ 1.0594

Result: The one-year forward rate is approximately 1.0594 USD per EUR.

Relationship Between Forward and Spot Rates

The difference between the forward and spot rates is driven by the interest rate differential between the two countries (rf – rd).

  • Forward Premium: When the foreign interest rate is lower than the domestic rate (rf < rd), the forward rate is lower than the spot rate (Ff/d < Sf/d). In this case, the foreign currency is said to trade at a forward premium, and the domestic currency at a forward discount.
  • Forward Discount: When the foreign interest rate is higher than the domestic rate (rf > rd), the forward rate exceeds the spot rate (Ff/d > Sf/d). Here, the foreign currency trades at a forward discount, and the domestic currency at a forward premium.

Important Note

The terms "premium" and "discount" can be counterintuitive—they refer to whether the forward rate is higher or lower than the spot rate, not the value of the currency itself.