  We propose a new definition for tameness within the model of security prices
as It\^o processes that is risk-aware. We give a new definition for arbitrage
and characterize it. We then prove a theorem that can be seen as an extension
of the second fundamental theorem of asset pricing, and a theorem for valuation
of contingent claims of the American type. The valuation of European contingent
claims and American contingent claims that we obtain does not require the full
range of the volatility matrix. The technique used to prove the theorem on
valuation of American contingent claims does not depend on the Doob-Meyer
decomposition of super-martingales; its proof is constructive and suggest and
alternative way to find approximations of stopping times that are close to
optimal.
