Let S=(S_t), t=0,1,...,T (T being finite), be an adapted Rd-valued process. Each component process of S might be interpreted as the price process of a certain security. A trading strategy H=(H_t), t= 1,...,T, is a predictable Rd-valued process. A strategy H is called extreme if it represents a maximal arbitrage opportunity. By this we mean that H generates at time T a nonnegative portfolio value which is positive with maximal probability. Let Fe denote the set of all states of the world at which the portfolio value at time T, generated by an extreme strategy (which is shown to exist), is equal to zero. We characterize those subsets of Fe, on which no arbitrage opportunities exist.