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

About: Two assets with identical payoffs must have identical prices — else arbitrage exists. The bedrock of derivative pricing.Tóm tắt: Hai tài sản có CF giống nhau phải có giá giống nhau. Nguyên tắc nền tảng định giá phái sinh.

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

2. Replication Principle Concept

About: Any derivative payoff can be replicated by combining the underlying with risk-free borrowing/lending. Cost of replication = price of derivative (no-arbitrage).Tóm tắt: Mọi payoff phái sinh có thể tái tạo bằng underlying + lending/borrowing. Chi phí = giá phái sinh.

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

About: Forward = spot × (1+r)^T + PV(costs) − PV(benefits). Captures the cost of holding the underlying versus locking in via forward.Tóm tắt: Forward = spot × (1+r)^T + chi phí − lợi ích. Chi phí nắm giữ vs khóa qua forward.

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

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

About: If forward > theoretical: sell forward, buy spot, finance with borrowing → risk-free profit. Reverse cash-and-carry for the opposite mispricing.Tóm tắt: Forward > lý thuyết: bán forward, mua spot, vay tiền → arbitrage. Ngược lại: reverse cash-and-carry.

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.