Title LM04 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives IFT Notes.pdf ✅

LM04 Arbitrage, Replication, & Cost of Carry in Pricing Derivatives 2025 Level I Notes © IFT. All rights reserved 1 LM04 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives 1. Introduction ........................................................................................................................................................... 2 2. Arbitrage ................................................................................................................................................................. 2 3. Replication ............................................................................................................................................................. 5 4. Costs and Benefits Associated with Owning the Underlying .............................................................. 6 Summary ...................................................................................................................................................................10 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM04 Arbitrage, Replication, & Cost of Carry in Pricing Derivatives 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction This learning module covers:  How the concepts of arbitrage and replication are used in pricing derivatives.  How the costs or benefits associated with owning an underlying asset affect the forward commitment price. 2. Arbitrage According to the ‘law of one price’ two identical assets should trade at the same price. An arbitrage opportunity exists if the ‘law of one price’ does not hold. In case of derivatives arbitrage opportunities exist when: 1. Two assets with identical future cash flows trade at different prices. 2. An asset with a known future price does not trade at the PV of its future price. Scenario 1 Exhibit 1 from the curriculum illustrates the first scenario. Consider two zero-coupon bonds A and B. Both bonds are from the same issuer, have identical features, and will mature on the same future date. 1. Bond A is trading at a price of EUR99 at time t=0. 2. Bond B is trading at a price of EUR99.15 at time t=0. 3. Both bonds have an expected future price of EUR100. Since two assets with identical future cash flows trade at different prices, the ‘law of one price’ is violated and an arbitrage opportunity exists. At time t = 0, an investor can:  Sell Bond B short to receive EUR99.15 and purchase Bond A for EUR99  This will give him a net cash inflow at t=0 of EUR0.15 At time t = T, when both bonds mature, the investor:  Receives EUR100 for Bond A and uses this to buy Bond B for EUR100 to cover the
LM04 Arbitrage, Replication, & Cost of Carry in Pricing Derivatives 2025 Level I Notes © IFT. All rights reserved 3 short position.  This leaves him with a riskless profit of EUR0.15 at time 0. Other market participants will also exploit this opportunity, selling Bond B, driving its price down, and buying Bond A, driving its price up, until the prices converge. Scenario 2 The second-type of derivative-related arbitrage opportunity arises when an asset with a known future price does not trade at the PV of its future price. In an earlier reading on time-value-of-money, we learnt that the future value of a single cash flow can be calculated based on discrete compounding or continuous compounding. The FV based on discrete compounding is: FVN = PV(1 + r)N The FV based on continuous compounding is: FVT = PVerT Both approaches can be used for pricing derivatives. The curriculum uses the discrete compounding method for individual underlying assets (e.g. a stock); and the continuous compounding method for underlying assets that represent a portfolio (e.g. an equity index) or where the underlying involves foreign exchange. The following example from the curriculum demonstrates how an arbitrage profit can be earned in scenario 2. Example: Spot vs. Discounted Known Future Price of Gold (This is based on Example 1 from the curriculum.) A company entered into a contract to buy 100 ounces of gold at an agreed-upon price of USD1,792.13 per ounce in three months. Assume that today’s spot gold price (S0) is USD1,770 per ounce, and the annualized risk-free interest rate (r) is 2%. Also assume that the company can borrow at the risk-free rate and there are no additional costs or benefits associated with gold ownership. Under these conditions demonstrate how the company can generate a riskless profit. Solution:
LM04 Arbitrage, Replication, & Cost of Carry in Pricing Derivatives 2025 Level I Notes © IFT. All rights reserved 4 At time t = 0,  The company borrows USD177,000 at 2.0% interest for three months and purchases 100 ounces of gold at today’s spot price.  The company enters into a forward contract today to sell 100 ounces of gold at a price of USD1,792.13 per ounce in three months. At time t = T (in three months),  The company delivers 100 ounces of gold under the forward contract and receives USD179,213 (= 100 × USD1,792.13).  The company repays the loan principal with interest: USD177,878.44 = USD177,000(1.02)0.25.  The company’s riskless profit at time T is equal to the difference between the forward sale proceeds and the loan principal and interest: USD1,334.56 = USD179,213 – USD177,878.44. In this example, the spot price of gold is below the PV of the known futures price in three months. Other market participants will also exploit this arbitrage opportunity by purchasing gold in the spot market and selling it in the forward market. This will cause the spot price to increase and the forward price to fall until the arbitrage opportunity is eliminated. Instructor’s Note: The two scenarios demonstrate that arbitrage opportunities will be exploited by market participants and the prices will quickly adjust. Therefore, while pricing derivative we set a price such that no arbitrage opportunities exist. The two key arbitrage concepts used to price derivatives for an underlying with no additional cash flows are:  Identical assets or assets with identical cash flows traded at the same time must have the same price  Assets with a known future price must have a spot price that equals the future price discounted at the risk-free rate The relationship between spot prices, forward commitment prices, and the risk-free rate is