The Fundamental Theorems Of Asset Pricing And The Construction Of Arbitrage Strategy

The fundamental theorems of asset pricing have a very important position in financial mathematics.The classical first fundamental theorem of asset pricing shows that there is no arbitrage in the market is equivalent to the existence of an equivalent martingale measure.The classical second fundamental theorem of asset pricing shows that under the condition of no arbitrage,the market is complete is equivalent to the existence of a unique equivalent martingale measure.This thesis discusses the fundamental theorems of asset pricing under the concept of equivalent local martingale measure.The equivalent local martingale measure is the generalization of the equivalent martingale measure.Therefore,compared with the classical fundamental theorems of asset pricing,the results obtained will be more in-depth and the scope of application will be also broader.In this thesis,at first,the concepts and theorems related to asset pricing in stochastic analysis are introduced and then the theory of asset pricing and its related concepts are studied in three probability spaces.In the first probability space[0,](,(),)t t TP?? F,the general market model is analyzed,and the concepts of aggregate excess returns process,information process and market process are introduced.These concepts are the basis of the fundamental theorems of asset pricing and its related properties.In the second probability space[0,](,(),)t t T??XG,we mainly analyze and discuss the normalized excess returns process and its related properties,and then introduce the concepts of trading strategy,tame strategy,arbitrage strategy and guaranteed arbitrage strategy in this space.In the third probability space[0,](,(),)t t T??WA,the concept of equivalent local martingale measure and market completeness are introduced.We mainly discusses an important property of the fundamental theorems of asset pricing,that is,there is an arbitrage strategy in the market is equivalent to? - W(W is the distribution law of one-dimensional standard Brownian motion),and this property is important to detect if there is an arbitrage strategy in the market.Finally,two specific examples of asset pricing are studied,and the existence of arbitrage strategy in the market is detected.The corresponding arbitrage strategy is constructed after analyzing and confirming the existence of arbitrage strategy.