Module 4 · Derivatives

Arbitrage, Replication, and the Cost of Carry

EN: Law of one price, replication, and the cost-of-carry framework.
VN: Quy luật một giá, replication, cost-of-carry.

1. Law of One Price Concept

Two assets with identical future cash flows must have the same price today. Otherwise risk-free arbitrage exists.

2. Replication Principle Concept

Any derivative payoff can be replicated by an appropriate combination of the underlying and a risk-free bond. The cost of the replicating portfolio = price of the derivative.

3. Cost of Carry Core

\[ F_0 = S_0 \cdot (1 + r)^{T} + FV(\text{costs}) - FV(\text{benefits}) \]

Components

S₀ Spot price.
r Risk-free rate.
Costs Storage, insurance (commodities).
Benefits Dividends, coupons, convenience yield.

Practice problem

Spot $50, r = 4%, T = 1 year, cash dividend $2 paid in 6 months. Compute theoretical 1-year forward.

Show solution

PV(div) = 2/1.04^0.5 ≈ 1.961

F = (50 − 1.961)(1.04)

F ≈ $49.96

4. Cash-and-Carry Arbitrage Concept

If observed forward > theoretical: sell forward, buy spot, finance with borrowing. Lock in risk-free profit at maturity.

If observed forward < theoretical: reverse cash-and-carry — buy forward, short spot, lend proceeds.

Practice problem Practice

Practice problem

Spot price of an asset = $100, risk-free rate = 5%, no income or storage cost. The 1-year forward trades at $108 in the market. Identify the arbitrage.

Show solution

Theoretical forward = 100 × 1.05 = $105.

Market forward $108 > theoretical $105.

Cash-and-carry: borrow $100, buy spot, sell forward at $108. At T: deliver, receive $108, repay $105.

Risk-free profit = $3.

Sell overpriced forward, buy spot, finance with borrowing → $3 risk-free profit.