= Arbitrage
{title2=$V_0=0,\quad V_T\geq0\ P\text{-a.s.},\quad P(V_T>0)>0$}
{wiki}
An <arbitrage> is an allowed <self-financing portfolio> with zero initial wealth, nonnegative terminal <portfolio wealth> <almost surely> under the physical <probability measure> $P$, and strictly positive terminal wealth with positive <probability>. In a finite model where every state has positive <probability>, nonnegativity means no loss in any state. Positive <expected return> alone does not establish <arbitrage>: losses may still occur.
For the <binomial market> example, suppose the <European call option> sells for 13 while its <replicating portfolio> costs $250/21$. Sell the call, buy its <replicating portfolio>, and lend the surplus $13-250/21=23/21$ in the <bank account>. The initial net cost is zero. At maturity the call obligation and <portfolio> cancel in either state, leaving $(21/20)(23/21)=23/20$. A price below $250/21$ permits the reversed construction, using <short selling> and borrowing. This is the <law of one price> made operational; the assumed trading permissions are essential.
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