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

A market satisfies no-arbitrage when it admits no admissible self-financing strategy with zero initial cost, nonnegative terminal wealth almost surely, and a strictly positive gain with positive probability. The admissibility restriction excludes doubling strategies. Identical attainable cash flows then have the same price.