No-arbitrage ($V_0=0,V_T\ge 0\text{ a.s.}⟹V_T=0\text{ a.s.}$)

A financial market satisfies no-arbitrage when there is no admissible trading strategy that is self-financing, starts at zero cost, and has nonnegative terminal wealth almost surely with a strictly positive gain on an event of positive probability:The allowed trading class matters; in continuous time its lower-bound restriction excludes doubling strategies. The law of one price follows for identical cash flows when the offsetting long-short position is itself admissible, as in an unconstrained finite one-period model. Under merely local martingale prices, a required short position may have unbounded possible interim losses and be inadmissible; the unrestricted price-equality conclusion must then be justified separately.