| In 1952, Harry M. Markowitz published his famous paper"PortfolioSelection" in the Journal of Finance, which studied the problem of investment pro-folio based on the average value-variance analytic method. It was the first time thismethod was used investment prolio. And it developed the history which is doing re-search in financial property relations between income and risk based on the methodof mathematics analysis and has laid the theory and methodology foundation for thetheory research and development of the modern investment profolio. Consideringthe theory evolution and the development of the average value-variance investmentprofolio in recent fifty years as a clue, this paper discusses the selection action ofinvestment profotio, and studies the relation between the investment income andrisk. The theoretical research and empirical analysis methods are employed.The main achievements and the innovation of this article are as follows:In chapter 2, the standard Black-Schoes option price is introduced firstly. Sec-ondly, the general differential equation and its solution of the western-style optionprice are derived. The formula and the fair price relations for western-style's pricingwhich is expected to rise and fall are presented. After revising and analysising on themodel, we can apply it to the pricing for the stock certificate which the Europeanoption develops, the value which the wrap time guarantees, and the sign propertypayment middle dividend and so on. The results show that the main characteristicfor the B-S model comes in the following: under the conditions of the arbitrage,equilibrium and the complete market, any undecided rights and interests marketvalue can be determined by the bond or the stock market value, and its establish-ment is dependent on the calculation for the standard missesσof the stock incomerate. Generally, the price of the option which buys or sells is a increased functionwith the increasement ofσ. Based on the B-S model,σcan be calculated precisely.In chapter 3, the average value-variance model given by Harry M. Markowitzhas solved the selection problem of the investment profolio and established its pos-sible boundary, however, in practical application, it is sufficient to estimate theexpectation income rate of N's investments, calculate the N step income rate vari-ance-covariance matrix and solve the problem which is produced by the variance -covariance matrix. When the number of N is very large, the algorithm is ex-tremely tedious. Therefore, after the model proposed by Markowitz, the successordedicated themselves to doing research in its simplication model, where, the follow-ing are rather representative: the diagonal model proposed by Willian F.Sharpe,the property choice model given by Edwin J.Elton, Martin J.Gruber and ManfredW.Padberg. the diagonal model is named by the fact that the variance-covari-ante matrix is a N+1 step diagonal matrix, which is also called the single indexmodel. The above models are called the index models. Taking the factor that theproperty income rate correlation coefficient same as a premise, the value correlationsimplification model dependent on the mean value of various properties income ratecorrelation coefficient. If the deviation of the correlation coefficients is small, thenthe model is applicable. In fact, the deviation of the correlation coefficients may berather big, and there has some special rules. Then the similar model is applicabletoo. For instance, according to the deviation of the coefficient value, the coefficientmatrix is devided to several sub-matrices, and the coefficiences of the suv-matrixswhich are symmetrical are rather close. Similarly, we can utilize the expansion formof the value correlation model to carry on the choice of investment profolio.In chapter 4, considering the expected rates of security returns are difficultyto estimate precisely due to many uncertainties in securities markets, fuzzy num-bers is used. A new portfolio selection model with fuzzy coefficients is proposed.Recently, fuzzy set theory have been used to the portfolio selection problem exten-sively. Tanaka used possibility distribution to model uncertainty in the returns.In this paper, an alternative model is proposed based on the fuzzy expected rateof return, in which the objective function is the maxmum of fuzzy returns and therestriction condition is that all of the risks are not larger than the given fuzzy num-bers. For the given cut set, the fuzzy expected rate of return is tranformed to severalinterval numbers, and the objective function is tranformed to the maximum of thelower bounds of the interval numbers. The comparison between trigangular fuzzynumbers can be carried out using two criteria: dominance possibility and strictdominance possibility. The two criteria are often used for fuzzy programming. Sothe fuzzy mathematics model can be transformed to the linear programming model.The maximum solution can be solved by using Matlab. The last but not the least,an example is given to illustrate that the new model can be used efficiently to solveprotfolio selection problems.In chapter 5, the problem of investment profolio was analysised theoretically and empirically. In the financial investment domain, the most problem cared byinvestors is the following: in order to realize income rate which the investmentbrings as far as possible in a big way and the investment risk is as far as possiblesmall simultaneously, how to take action on the chocie of investment profolio? Somescholars have studied the problem of investment profolio which onlyhave the riskproperty or non-risk property. Generally speaking, the more dispersible the invest-ment is, the small the the risk degree is. Since the anticipated income rate presentedprecisely is very hardly, and the risk is unknown. This chapter utilizes fuzzy mathe-matics to analysis on the selection problem of investment profolio including non-riskproperty, according to the clue that the anticipated income rate is represented byfuzzy number. On each fiduciary level, considering the degree of deviation of thecentral value as the measure of the risk, the chapter takes the central value as theweight which anticipates income rate target. Owing to the effection of the investorrisk, the optimized model is constructed, in which the objective function is the onesuch that the total risk is the smallest one and the future income rate is biggest.Last but not the least, the existence of the optimal solution is proved and the mostsuperior investment profolio is derived too. |
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