On The First Fundamental Theorem Of Asset Pricing

Managing risk and maximizing utility are the key contents in mathematical finance. And the study of the first fundamental theorem of asset pricing is the base of the study of the above two subjects. The first fundamental theorem of asset pricing is the one of two most fundamental theorems in mathematical finance. It mainly describes the fairness of a market which is one of the two most important properties of the market. A market is fair is that the market exists the equivalent martingale measure. Consequently, the study on the existence of the equivalent martingale measure is equivalent to the research on the fairness of the market; and the corresponding result is called the first fundamental theorem of asset pricing. Now the most prevalent description of the first fundamental theorem of asset pricing is: there are the equivalent martingale measures in a market if and only if the bounded semi-martingale S(t) satisfies no free lunch with vanishing risk.Firstly, in this paper the author discusses the relation between arbitrage and strategy, and constructs that non-tame portfolio can bring arbitrage. Secondly, the author summarizes the descriptions on NFLVR for several decade years, and gets four descriptions which are the most representative and comprehensive. We make out mathematical analytic of the four descriptions, and prove the relations between them. Then we get some satisfied results. Finally, further more, there is a main theorem in the paper, it is that the bounded semi-martingale S(t) satisfiesNFLVR if and only if V - L+∞ ∩ L+∞ {0} Thus we construct a new topology description on NFLVR.