An arbitrage is a zero-cost self-financing portfolio whose terminal payoff is nonnegative in every state and strictly positive with positive probability.
An investment-consumption arbitrage starts with zero capital, has nonnegative consumption at every date, and has strictly positive consumption at some date with positive probability.
A pure-investment arbitrage has zero initial cost, no intervening consumption, and a nonnegative terminal payoff that is strictly positive with positive probability.
Consumption in a self-financing portfolio model is wealth withdrawn for use outside the portfolio. With no initial capital, a one-period position of nonpositive initial cost and nonnegative terminal payoff permits nonnegative consumption at both dates.
A terminal-consumption arbitrage has zero initial cost and a nonnegative terminal payoff that is strictly positive with positive probability. Allowing negative initial cost gives the broader one-period notion in which some consumption may occur immediately.
If the terminal price vector is normal with covariance matrix , every portfolio outside has a terminal value with full support on the real line. Arbitrage can therefore arise only from portfolios in , whose terminal values are deterministic.
A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.
A self-financing portfolio is admissible when its wealth satisfies the prescribed lower-bound restriction, typically a deterministic bound below. Nonnegative wealth is sufficient. This condition permits supermartingale bounds on discounted gains and excludes doubling strategies.
