No-arbitrage — why free lunch doesn't last

Derivatives pricing is built on one assumption that sounds tautological until you realize how much it constrains:

If a portfolio costs nothing to set up, has zero risk, and pays out a guaranteed positive amount — someone's going to do that trade until it's gone.

That's the no-arbitrage principle. Free money exists, briefly. Then someone notices and trades it away.

Why this is a strong assumption

On the surface it sounds obvious. In practice it's the load-bearing assumption behind every closed-form pricing formula:

• Forward = spot + carry. If the forward and the spot diverged, you'd cash-and-carry until they reconverge.
• Put-call parity: C − P = (F − K)·e^(−rT). If the equation breaks, build a synthetic forward (long call, short put) at one price and the real forward at the other, lock in the difference risk-free.
• Black-Scholes: the price of an option is the cost of dynamically hedging it. If the market quotes a different price, arb the difference. Hedging cost is the lower bound of pricing.

Without no-arbitrage, none of these formulas would have a unique answer. With no-arbitrage, each one becomes the unique price at which the hedger and speculator agree to clear.

Real-world arbitrage

In real markets, "true" arbitrage (zero risk, zero capital, positive payoff) is rare and short-lived. What exists more often is statistical arbitrage — gaps that should close based on past behavior but might widen first.

Common practical frictions:

• Transaction costs can eat the profit before you reach break-even.
• Funding constraints— you may not be able to borrow at the risk-free rate. If your cost of capital is 6% and the "risk-free" arb is 5.5%, you have no edge.
• Settlement and operational delays. Some arbs only close on specific dates.
• Counterparty risk. Even a riskless trade has the risk that your counterparty defaults before settlement (think 2008).

These frictions are exactly why arbitrage gaps exist — they're what compensates the people who close them. The closer to perfect arbitrage, the smaller the gap and the more capital required to scoop it up.

Why this matters for you

When a structured product term sheet looks "too good to be true", no-arbitrage is the principle that tells you it isn't. There's always a hidden risk you're short, and the "edge" you see is the premium you're collecting for it. Reverse Convertibles, Bonus Certificates, Autocallables — none of them break no-arbitrage. They package the risk so it's less visible.

Your job, as a buyer, is to figure out what you're short. Once you can name the embedded position, the "free yield" or the "free leverage" stops looking free.