Study Of Some Questions Relative To Theoretics Of European Option Pricing

Along with the trade of option developing the important content of internationalmoney market, the theoretical and technological study of option pricing becomes one focusof finance research. Although in resent years, people strive to walk out the perfect generalequilibrium frame, considering asymmetry information, nonsense behavior, etc., while itcan be seen that those proposed new theories don't quite pull down the intrinsic generalequilibrium frame or classical financial economics. In fact, by this time, if the question,which is how to price financial goods at some time, have to be answered finally, thensome kind of steady equilibrium state is still needed. In this paper, based on the classicalfinancial economics, a series of questions related to the option pricing are discussed andstudied and some theoretical fruits are achieved. The main work is provided as follows:1. Two basic properties of a market, which are no arbitrage and completeness,are considered from the existence of a continuous linear pricing system. It is presentedthat there are one-to-one correspondences between no arbitrages of di?erent degrees andthe existence of sign measures satisfying di?erent characters. Meanwhile, the necessarycondition of the market completeness is proved from the point of view of the law of linearpricing which casts o? the supposition of no arbitrage with strongest degree.2. The completeness of market is considered through the correlation of basic assetsprice processes and filtration. When the objective probability measure is not always anequivalent martingale measure, the proof of Jacod lemma is presented and the concreteconstruction method of predictable process in the lemma is given with an example. Ifthe given filtration is generated finitely, a complete financial market can be constructed.Meanwhile, an example is given to demonstrate how to construct and the method hasgreat meaning of theoretical guidance in the design and study of the market model.3. whereas the importance of the growth optimal portfolio in financial market, hereits properties in discrete time and continuous time market are studied further. In discretetime financial market with simple model, the economic meanings of the change in theproportion distribution of the growth optimal portfolio is analyzed intuitionally,and thisquestion is surveyed from the angle of pure math. In continuous time financial market, it isproved that the growth optimal portfolio doesn't change with the choice the numeraire, andthe multiple of its proportion and the volatility vector of primary assets is a fixed vectorstochastic process, which is the so-called the market price process of risk if some additionalconditions satisfied. The result embodies the special status of the growth optimal portfolio in financial market again.4. The economic meanings of the generator g in backward stochastic di?erentialequation is discussed with classical Black ? Scholes model. It is proposed for the firsttime that there is the notion of discount in the expression of g , and showed that di?erentgenerator g can re?ect di?erent forms of the same price process of contingent claimwith particular analysis. At the same time, it is pointed out that, owing to the propertythat the generator g has to satisfy to define the g? expectation, when g? expectation isapplied to the financial market, the market is discounted simultaneously, and the numerairedepends on the concrete expression of g .5. By virtue of a simple market model, the individual risk preference and the marketrisk preference are discussed. The mathematical definition of each preference which hasnatural and intuitive economic meaning is given. The di?erence in both preferences isthat the former is up to personal measure, while the latter depends on the price processof primary asset. With concrete computation, the obvious conclusion is validated that therisk aversion of a investor (or a market) is stronger, the price process of a contingent claimgiven is lower.6. Illuminated by F¨ollmer,Sondermann and Schweizer etc., some questions aboutstochastic analysis, which are related with the measure and the inducted conditional ex-pectation produced by increasing process, are proposed for the first time. These questionsadvanced here will help to perfect the theory of stochastic analysis and have great realisticsignificance.