本文選題：不完全市場(chǎng) + 資產(chǎn)-負(fù)債管理問(wèn)題　； 參考：《天津大學(xué)》2012年博士論文

【摘要】：連續(xù)時(shí)間投資組合問(wèn)題是數(shù)理金融的重點(diǎn)研究?jī)?nèi)容,是投資人或者投資機(jī)構(gòu)進(jìn)行資產(chǎn)套期保值和風(fēng)險(xiǎn)對(duì)沖的重要理論方法.通過(guò)研究不同投資環(huán)境下的投資組合優(yōu)化問(wèn)題,一方面可以應(yīng)用數(shù)理方法創(chuàng)造性的解決人們?cè)趯?shí)際投資過(guò)程中所遇到的問(wèn)題,為投資人進(jìn)行科學(xué)投資提供理論依據(jù);另一方面可以為投資學(xué)的理論方法有更加廣泛的應(yīng)用提供科學(xué)依據(jù).本文主要對(duì)連續(xù)時(shí)間投資組合優(yōu)化問(wèn)題進(jìn)行了一些擴(kuò)展性研究,取得了一些研究成果.針對(duì)實(shí)際投資環(huán)境的的多樣性和金融市場(chǎng)的不確定性,本文主要側(cè)重于四個(gè)方面的研究: (1)不完全市場(chǎng)下動(dòng)態(tài)資產(chǎn)分配的擴(kuò)展性研究; (2)隨機(jī)環(huán)境下資產(chǎn)-負(fù)債管理問(wèn)題研究; (3)限制性投資組合優(yōu)化問(wèn)題的擴(kuò)展性研究;(4)隨機(jī)環(huán)境下投資-消費(fèi)問(wèn)題研究.具體研究成果詳述如下: 第二章主要對(duì)不完全市場(chǎng)下的動(dòng)態(tài)投資組合優(yōu)化問(wèn)題進(jìn)行了擴(kuò)展性研究.首先,對(duì)不完全市場(chǎng)下基于效用最大化的動(dòng)態(tài)資產(chǎn)分配問(wèn)題進(jìn)行了研究.通過(guò)降低布朗運(yùn)動(dòng)的維數(shù)將不完全市場(chǎng)轉(zhuǎn)化為完全市場(chǎng),并在轉(zhuǎn)化后的完全市場(chǎng)下應(yīng)用鞅方法得到了指數(shù)效用和對(duì)數(shù)效用函數(shù)下最優(yōu)投資策略的解析表達(dá)式.應(yīng)用完全市場(chǎng)與原不完全市場(chǎng)間參數(shù)關(guān)系得到不完全市場(chǎng)下的最優(yōu)投資策略.算例解釋了模型的結(jié)論,分析了從完全市場(chǎng)到不完全市場(chǎng)下最優(yōu)投資策略的變化情況,并把指數(shù)效用和對(duì)數(shù)效用函數(shù)下的最優(yōu)投資策略與冪效用函數(shù)下的最優(yōu)投資策略進(jìn)行了比較.其次,對(duì)不完全市場(chǎng)下基于二次效用函數(shù)的投資組合優(yōu)化問(wèn)題進(jìn)行了研究,應(yīng)用鞅方法得到了最優(yōu)投資組合的解析表達(dá)式.分析了均值-方差模型下不完全市場(chǎng)的最優(yōu)投資策略問(wèn)題,為進(jìn)一步全面探討均值-方差模型提供了理論基礎(chǔ).第三,對(duì)不完全市場(chǎng)下的投資-消費(fèi)問(wèn)題進(jìn)行了研究,應(yīng)用動(dòng)態(tài)規(guī)劃原理和HJB方程方法得到了冪效用、指數(shù)效用和對(duì)數(shù)效用函數(shù)下最優(yōu)投資-消費(fèi)策略的解析表達(dá)式.第四,對(duì)不完全市場(chǎng)下的資產(chǎn)-負(fù)債管理問(wèn)題進(jìn)行了研究,通過(guò)構(gòu)造指數(shù)鞅方法和引入二次優(yōu)化問(wèn)題解決了指數(shù)效用函數(shù)下的最優(yōu)投資組合問(wèn)題.所有這些研究擴(kuò)展了不完全市場(chǎng)下動(dòng)態(tài)投資組合優(yōu)化方面的研究,豐富和發(fā)展了Zhang的研究?jī)?nèi)容. 第三章主要研究隨機(jī)環(huán)境下的資產(chǎn)-負(fù)債管理問(wèn)題,如隨機(jī)利率模型和隨機(jī)波動(dòng)率模型等.首先,在常數(shù)利率環(huán)境下對(duì)效用最大化下的資產(chǎn)-負(fù)債管理問(wèn)題進(jìn)行了研究,應(yīng)用動(dòng)態(tài)規(guī)劃原理和Legendre變換-對(duì)偶解法得到了冪效用、指數(shù)效用和對(duì)數(shù)效用函數(shù)下最優(yōu)投資策略的解析表達(dá)式,并給出算例分析了市場(chǎng)參數(shù)對(duì)最優(yōu)投資組合的影響.其次,假設(shè)無(wú)風(fēng)險(xiǎn)利率、股票收益率和波動(dòng)率均為一致有界隨機(jī)過(guò)程,應(yīng)用向后隨機(jī)微分方程理論和隨機(jī)線性二次規(guī)劃方法得到了最優(yōu)投資策略的解析表達(dá)式.第三,假設(shè)利率是服從Ho-Lee利率模型的隨機(jī)過(guò)程,應(yīng)用動(dòng)態(tài)規(guī)劃原理對(duì)資產(chǎn)-負(fù)債管理進(jìn)行了研究,得到了冪效用和指數(shù)效用函數(shù)下最優(yōu)投資策略的解析表達(dá)式.第四,對(duì)Vasicek利率模型下的資產(chǎn)-負(fù)債管理問(wèn)題進(jìn)行了研究,結(jié)合動(dòng)態(tài)規(guī)劃原理和Legendre變換-對(duì)偶解法得到了冪效用和指數(shù)效用函數(shù)下最優(yōu)投資策略的解析表達(dá)式.將負(fù)債過(guò)程引入到投資組合優(yōu)化問(wèn)題中,并研究了此類問(wèn)題的最優(yōu)投資組合與風(fēng)險(xiǎn)管理的問(wèn)題,是現(xiàn)階段資產(chǎn)-負(fù)債管理的新的研究?jī)?nèi)容.本章的研究?jī)?nèi)容豐富和發(fā)展了資產(chǎn)-負(fù)債管理方面的理論方法,尤其是解決了隨機(jī)利率模型下的最優(yōu)投資組合問(wèn)題,為隨機(jī)利率模型下帶有負(fù)債的投資機(jī)構(gòu)進(jìn)行資產(chǎn)套期保值和對(duì)沖風(fēng)險(xiǎn)提供了理論依據(jù). 第四章主要對(duì)不同借貸利率限制下的動(dòng)態(tài)投資組合優(yōu)化問(wèn)題進(jìn)行了擴(kuò)展性研究.首先,對(duì)效用最大化下的投資組合選擇問(wèn)題進(jìn)行了研究,應(yīng)用動(dòng)態(tài)規(guī)劃原理和HJB方程方法得到了冪效用、指數(shù)效用和對(duì)數(shù)效用函數(shù)下最優(yōu)投資策略的解析表達(dá)式,并給出算例對(duì)不同借貸利率限制下投資人的投資行為進(jìn)行了分析;其次,對(duì)負(fù)債情形下的均值-方差問(wèn)題進(jìn)行了研究,應(yīng)用拉格朗日對(duì)偶定理和動(dòng)態(tài)規(guī)劃原理得到了最優(yōu)投資策略和有效前沿的解析表達(dá)式;最后,將幾何布朗運(yùn)動(dòng)擴(kuò)展至CEV模型,對(duì)CEV模型下的投資組合優(yōu)化問(wèn)題進(jìn)行了研究,得到了最優(yōu)投資策略和有效前沿的解析表達(dá)式.本章的研究工作進(jìn)一步豐富和發(fā)展了不同借貸利率限制下投資組合優(yōu)化問(wèn)題的理論方法,擴(kuò)展了Fu和Lari-Lavassani等人的研究工作,為進(jìn)一步研究負(fù)債和CEV模型下的投資組合優(yōu)化模型提供了理論基礎(chǔ). 第五章主要研究了隨機(jī)環(huán)境下的投資-消費(fèi)問(wèn)題.首先,我們假設(shè)金融市場(chǎng)中存在兩種資產(chǎn),一種資產(chǎn)是無(wú)風(fēng)險(xiǎn)資產(chǎn),其中無(wú)風(fēng)險(xiǎn)利率是服從Ho-Lee利率模型的隨機(jī)過(guò)程,且與風(fēng)險(xiǎn)資產(chǎn)價(jià)格存在線性相關(guān)性,以投資人有限投資周期內(nèi)終端財(cái)富和累積消費(fèi)的期望貼現(xiàn)效用作為目標(biāo)函數(shù),應(yīng)用動(dòng)態(tài)規(guī)劃原理和HJB方程對(duì)冪效用和對(duì)數(shù)效用函數(shù)下的最優(yōu)投資-消費(fèi)策略進(jìn)行了研究,得到了兩種效用函數(shù)下最優(yōu)投資-消費(fèi)策略的解析表達(dá)式.其次,我們將Ho-Lee利率模型擴(kuò)展至Vasicek利率模型,應(yīng)用動(dòng)態(tài)規(guī)劃原理和Legendre變換-對(duì)偶方法得到了冪效用和對(duì)數(shù)效用函數(shù)下最優(yōu)投資-消費(fèi)策略的解析表達(dá)式.我們的這些研究將Merton的投資-消費(fèi)模型擴(kuò)展到隨機(jī)環(huán)境下,并著重研究了隨機(jī)利率模型下的投資-消費(fèi)模型,解決了Ho-Lee利率模型、Vasicek利率模型下的最優(yōu)投資-消費(fèi)策略問(wèn)題.
[Abstract]:The problem of continuous time investment portfolio is the key research content of mathematical finance. It is an important theoretical method of asset hedging and risk hedging by investors or investment institutions. By studying the problem of portfolio optimization under different investment environment, on the one hand, we can creatively solve people's investment process by mathematical method. The problems encountered in this study provide a theoretical basis for investors to invest in scientific investment; on the other hand, we can provide a scientific basis for more extensive application of the theory and methods of investment science. This paper mainly studies the extension of the problem of continuous time portfolio optimization, and has obtained some research results. The uncertainty of diversity and financial market is mainly focused on four aspects: (1) the expansibility of dynamic asset allocation under incomplete market; (2) research on asset liability management in random environment; (3) extensibility research on the problem of restrictive portfolio optimization; (4) research on investment consumption in random environment. The results of the study are described in detail as follows:
The second chapter mainly studies the expansibility of the dynamic portfolio optimization problem under incomplete market. First, the dynamic asset allocation problem based on the utility maximization under incomplete market is studied. By reducing the dimension of the Brown movement, the incomplete market is transformed into a complete market, and the application is applied to the complete market after the transformation. The analytic expression of the optimal investment strategy under the exponential utility and the logarithmic utility function is obtained by the martingale method. The optimal investment strategy under incomplete market is obtained by using the relation between the complete market and the original incomplete market. The calculation example explains the conclusion of the model and analyzes the change of the optimal investment strategy from the complete market to the incomplete market. The optimal investment strategy under the exponential utility and logarithmic utility function is compared with the optimal investment strategy under the power utility function. Secondly, the portfolio optimization problem based on the two utility function under incomplete market is studied. The analytic expression of the optimal portfolio is obtained by the martingale method. The optimal investment strategy of incomplete market under the variance model provides a theoretical basis for further study of the mean variance model. Third, the investment consumption problem under incomplete markets is studied. The optimal investment under the power utility, exponential utility and logarithmic utility function is obtained by using the dynamic programming principle and the HJB equation method. The analytical expression of the consumption strategy. Fourth, the problem of asset liability management under incomplete market is studied. The optimal portfolio problem under the exponential utility function is solved by constructing the exponential martingale method and introducing the two optimization problem. All these studies extend the study of dynamic portfolio optimization under incomplete markets. It enriches and develops the research content of Zhang.
The third chapter mainly studies the management of assets and liabilities under random environment, such as the stochastic interest rate model and the stochastic volatility model. First, the asset liability management problem under the utility maximization is studied under the constant interest rate environment. The power utility, the exponential utility and the Legendre change dual solution are applied to the problem of asset liability management under the constant interest rate environment. The analytic expression of the optimal investment strategy under the logarithmic utility function is given, and an example is given to analyze the effect of the market parameters on the optimal portfolio. Secondly, it is assumed that the risk free interest rate, the stock return rate and the volatility are uniformly bounded random processes, and the optimal investment is obtained by using the backward stochastic differential equation theory and the stochastic linear two programming method. The analytic expression of the capital strategy. Third, assuming that the interest rate is a stochastic process that obeys the Ho-Lee interest rate model, the dynamic programming principle is used to study the asset liability management, and the analytic expression of the optimal investment strategy under the power utility and the exponential utility function is obtained. Fourth, the asset liability management problem under the Vasicek interest rate model is carried out. The analytical expression of the optimal investment strategy under power utility and exponential utility function is obtained by combining the dynamic programming principle and the Legendre transformation dual method. The debt process is introduced into the portfolio optimization problem, and the optimal portfolio and risk management of this kind of problem are studied. The research content of this chapter enriches and develops the theoretical method of asset liability management, especially solving the optimal portfolio problem under the stochastic interest rate model, which provides a theoretical basis for the asset hedging and hedging risk of the investment institutions with liabilities under the stochastic interest rate model.
The fourth chapter mainly studies the expansibility of the dynamic portfolio optimization problem under the different lending rates. First, the portfolio selection problem under the utility maximization is studied. The dynamic programming principle and the HJB equation method are used to obtain the power utility, the exponential utility and the logarithmic utility function to analyze the optimal investment strategy. An example is given and an example is given to analyze the investor's investment behavior under the restriction of different lending rates. Secondly, the mean variance problem under the debt situation is studied. The Lagrange dual theorem and the dynamic programming principle are used to obtain the optimal investment strategy and the analytic expression of the effective frontier. Finally, the geometric Brown movement is carried out. This chapter extends to the CEV model, studies the portfolio optimization problem under the CEV model, and obtains the optimal investment strategy and the analytical expression of the effective frontier. This chapter further enriches and develops the theory square method of the portfolio optimization with different lending rate restrictions, and extends the research work of Fu and Lari-Lavassani. It provides a theoretical basis for further research on portfolio optimization models under liabilities and CEV models.
The fifth chapter mainly studies the investment and consumption problem in random environment. Firstly, we assume that there are two kinds of assets in the financial market, one kind of assets are risk-free assets, and the risk free interest rate is a random process that obeys the Ho-Lee interest rate model, and there is a linear correlation with the risk asset price, and the terminal money in the investor's limited investment cycle is in the end. The expected discounted utility of rich and cumulative consumption is the objective function, the optimal investment consumption strategy under power utility and logarithmic utility function is studied with dynamic programming principle and HJB equation, and the analytic expression of the optimal investment consumption strategy under two kinds of utility functions is obtained. The second, we extend the Ho-Lee interest rate model to Vasicek The interest rate model, using the dynamic programming principle and the Legendre transformation dual method, obtains the analytic expression of the optimal investment consumption strategy under the power utility and the logarithmic utility function. These studies extend the investment and consumption model of Merton to the random environment, and focus on the investment consumption model under the stochastic interest rate model and solve the problem. The Ho-Lee interest rate model and the optimal investment consumption strategy under the Vasicek interest rate model.
【學(xué)位授予單位】：天津大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2012
【分類號(hào)】：F224;F830.59

【引證文獻(xiàn)】

相關(guān)期刊論文 前1條

1 常浩;;不完全市場(chǎng)下基于對(duì)數(shù)效用的動(dòng)態(tài)投資組合[J];數(shù)理統(tǒng)計(jì)與管理;2013年01期

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