We propose a new definition for tameness within the model of security prices as Itło 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 formulas obtain to price American contingent claims are closer in spirit to a computational approach.