A Study For The Fundamental Theorem Of Asset Pricing With Transaction Costs And For Simple Arbitrage

The existence of arbitrage opportunities is one of the big reasons for such acharming real financial world, but the nature of society requires objectively no ar-bitrage for its stability and harmony. Financial Mathematics establishes on the theoryof no arbitrage pricing, so the Fundamental Theorem of Asset Pricing plays the role ofcornerstone. The frame of fundamental theorem of asset pricing is the equivalence be-tween a suitable notion of absence of arbitrage and the existence of pricing functionals,e.g. equivalent martingale measures. This paper firstly constructs a numerical exam-ple to show the cases for contingent claim pricing in the incomplete market. Thenwe consider the no-arbitrage conditions when introducing the transaction costs, andshow that the income of S with higher transaction cost k applying H is worse thanthe one of S? with 0≤k? < k applying H. Then if a price process S? without transac-tion costs lies within the bid-ask spread of a process S with transaction costs k > 0,thus the claims attainable with S are dominated by the claims attainable with S?, andso any no-arbitrage condition which applies to S? will also extend S with transactioncosts k, including (NFLV R) with transaction cost k. Moreover, from this theoremwe can obtain that for a larger class of processes satisfy no-arbitrage with transactioncosts, including fractional Brownian Motion. And we give a coherent and clear illus-tration for the fundamental theorem of asset pricing with transaction costs respect tothe consistent price system relating to the theorem above from both mathematical andfinancial aspects. In this paper we also study a more real situation that in the marketan investor can only make finite tradings. Then this situation may lead to a simple ar-bitrage. We study the financial intentions of simple arbitrages, and from the (EMM)and properties of the process aspect to analyze the reasons of appearing simple arbi-trage. Throughout the whole paper we use many examples to illustrate the existence of arbitrage opportunities and no-arbitrage conditions.