No-Arbitrage
No-arbitrage is the requirement that you cannot start at zero wealth and reach a nonnegative future payoff that is positive with positive probability — the axiom that gives you a positive state-price density.
Definition
A free lunch (in the Harrison–Pliska / Delbaen–Schachermayer sense) is ruled out iff there exists an equivalent martingale measure. In incomplete markets, no-arbitrage gives bounds, not a unique price.
Why it matters
It is why butterfly prices cannot go negative and why put-call parity holds for Europeans on frictionless underlyings. Real books have frictions; the axiom is the benchmark.
Case
A quoted butterfly that is through zero is either a data error or a market you can (almost) arb after costs. Most ‘arbs’ die in the costs.
How to read it
State the traded instruments and the constraints (short-sales, funding). NA in the textbook is not NA on a cash desk with locate and balance-sheet.