9.4 Arbitrage, Pricing, and No-Arbitrage Intuition

Arbitrage as a pricing anchor

Arbitrage is the ability to earn a riskless profit with no net investment, after transaction costs and practical constraints. In competitive markets such opportunities are short-lived because traders buy the cheap leg and sell the rich leg, pushing prices back toward fair value. The core principle: two strategies with identical future payoffs must have the same current cost; otherwise an investor buys the cheaper package and shorts the dearer one. This is the law of one price.

Derivative pricing follows from replication. If a derivative's payoff can be reproduced with the underlying plus borrowing or lending, the derivative must cost the same as that replicating portfolio. CFA Level I uses this as intuition rather than as a trading manual.

Carry-arbitrage forward pricing

For an asset with no income, storage cost, or convenience yield, the no-arbitrage forward price equals the spot price grown at the risk-free rate to delivery: F_0 = S_0 (1 + r)^T. If the forward price is too high, an arbitrageur runs a cash-and-carry: buy the asset now, finance the purchase, and sell it forward, locking the spread. If the forward is too low, a reverse cash-and-carry: short the asset, invest the proceeds, and buy it forward.

The full carry-arbitrage model adjusts for benefits and costs of holding the underlying: forward price = (spot + present-value of costs - present-value of benefits) compounded at the risk-free rate. Income (dividends, coupons) and convenience yield are benefits that lower the forward price; storage and insurance costs are carrying costs that raise it.

Worked example: spot = 100, one-year risk-free rate = 5%, no income or storage. The no-arbitrage one-year forward price is 100 x 1.05 = 105. If the quoted forward is 110, buy at 100, borrow 100 at 5% (owe 105 at maturity), and sell forward at 110, capturing 5 of riskless profit per unit before costs.

Value versus price

The forward price is the delivery term that makes a new contract fair at initiation. The value of an existing forward is what it is worth after conditions change. A long forward struck at 100 gains value if the current fair forward price rises to 106. The same split applies elsewhere: an option premium is the entry price, while option value drifts with the underlying, volatility, time, and rates; a swap's fixed rate is set so the swap has near-zero value at initiation, then moves as rates change.

Put-call parity intuition

For European options on the same underlying with the same X and expiration, put-call parity links a call, a put, the underlying, and a risk-free bond:

• A fiduciary call = long call + a risk-free bond that pays X at expiration.
• A protective put = long underlying + long put.

Both payoffs equal max(S_T, X) at expiration, so their current costs must be equal: C + X/(1+r)^T = S_0 + P. If the equality is violated, an arbitrage relation exists. This identity also lets you create a synthetic position, for example building a synthetic call from a put, the underlying, and borrowing.

Structured aid: no-arbitrage decision tree

1. Identify two strategies with the same future payoff.
2. Compare their current costs.
3. If one is cheaper, buy it and short the dearer one.
4. Lock the difference, subject to costs and execution risk.
5. Expect arbitrage pressure to restore equality.

Replication and the two-state intuition

The deepest no-arbitrage idea on the exam is replication: any derivative payoff can be reproduced by a portfolio of the underlying and a risk-free position, so the derivative must cost the same as that replicating portfolio. The clearest illustration is a one-period, two-state world where the underlying can move up or down. A call's payoff in each state can be matched exactly by holding a specific fraction of the underlying (the hedge ratio) financed partly with borrowing.

Because that portfolio reproduces the call in both states, it is riskless relative to the call, and the law of one price fixes the call's value as the cost of building the portfolio. The same logic produces the carry-arbitrage forward price: a forward is replicated by buying the asset today with borrowed money, so the forward price must equal spot compounded at the financing rate net of carry. If you remember that a derivative price is just the cost of copying its payoff, you can reason through most Level I pricing items without memorizing formulas.

Exam focus

Do not confuse arbitrage with a good trade idea. Arbitrage needs matched future payoffs and minimal risk, not a belief that a price will rise; a risky expected profit is speculation. When a forward looks too high relative to spot-plus-carry, think cash-and-carry: buy the asset, finance it, sell forward. When it looks too low, think reverse cash-and-carry: short the asset, lend the proceeds, buy forward.

Then adjust the intuition for income, storage, and convenience yield, and remember that more income or a higher convenience yield pushes the no-arbitrage forward price down, while higher storage costs or a higher financing rate push it up. On put-call parity items, rearrange the identity to isolate whichever instrument the stem asks for, because the exam often gives three of the four components and tests whether you can solve for the fourth.