Research On Models Of Optimal Investment And Utility Indifference Pricing

More and more people come to realize that financial problems following the global financial crisis, triggered by the American secondary bonds, have impacts on the growth, the prosperity and the stability of the economic and social structures all over the world. It is well-known that the key elements of finance theories are corporation finance and assets pricing theory, in which portfolio selection, risk management and asset pricing in incomplete markets are of great significance for the development of theories and practical applications. Therefore, this dissertation focuses on quantitative researches of some problems about portfolio selection and utility-based indifference pricing of risk assets under incomplete markets. Specifically, the dissertation includes the following several aspects:(I) We study the problem of optimal investment based on the criteria of maximizing the probability of survival. On the assumption that there is a random risk process in the markets, so it is incomplete, and the risk of losses in wealth cannot be completely eliminated -no matter what investment strategy is used. We consider the optimal investment problem with and without the restriction of borrowing. The closed-form expressions of the optimal strategies and the optimal value function are derived by solving the corresponding HJB equation. In addition, the comparative statistical analysis is used to explain the quantitative relations among the parameters, the survival probability and investment strategies. The results indicate that the proportion invested in the risk asset decreases as the wealth level increases. The more the amount of wealth or the less the given goal level is, the greater the survival probability will be.(II) We study the problem of optimal investment with the pattern of linear consumption. There is a positive probability of ruin for external linear consumption. Firstly, we discuss the problem of optimal investment based on the criteria of minimizing probability of bankruptcy, an optimal control problem independent with the time. There are three cases: (i) Loans are permitted with the same deposit and loan interest rates; (ii) Loans are not permitted for investing in the risky asset; (iii) Loans are permitted with the different deposit and loan interest rates. The results indicate that saving and borrowing constraints tend to make the optimal strategy a piecewise linear function and lead to increased bankruptcy risks for investors. Secondly, we discuss the problem of optimal investment based on the criteria of minimizing the lifetime probability of bankruptcy, which is an optimal control problem in a stochastic deadline for the finite of investors'life. We get closed-form expressions of the optimal investment strategies and the optimal value function with wealth-dependent linear consumption rates. The results indicate that the influence of death risk of the investor should not be ignored under the optimal criteria of minimizing lifetime probability of bankruptcy.Finally, we discuss the problem of optimal investment based on the criteria of maximizing expected power or exponential utility from terminal wealth, and compare it with the case based on the criteria of minimizing ruin probability. The results here are different from Merton's problem for external linear consumption. Merton's problem can be seen as a special case when the investor prefers the power utility function. Moreover, the optimal strategies are completely different between the case based on maximizing expected utility criteria and based on minimizing ruin probability criteria.(III) We study the problem of optimal investment with model risk based on the stochastic differential game approach. Suppose that nature is a"fictitious"player of the game, the problem can be represented by a two-player zero-sum stochastic differential game between nature and an investor. Through solving HJBI equations,we derive closed-form expressions of the optimal investment strategies, the optimal value function in complete and incomplete markets with stochastic income by using the stochastic game approach. The results indicate that in complete markets, the amount of optimal investment of risky asset is zero. In incomplete markets, the amount of optimal investment of risky asset is the negative ratio between the income flow's volatility and the risky asset volatility.(IV) We study the problem of continuous-time mean-variance portfolio selection in jump diffusion markets with no-shorting constraints. The goal is to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. We derive the optimal investment strategies and the efficient frontier in closed forms by using the theory of viscosity solutions. The results show that there are no differences between the jump diffusion cases and the pure diffusion cases.(V) We study the problem of optimal investment and reinsurance of insurer. Both, Security investments and reinsurance are effective ways for insurance companies to disperse risks and increase profits. Firstly, we discuss the problem of optimal investment and the proportional reinsurance policy for maximizing the survival probability, in which the optimal investment and reinsurance decrease with increasing wealth. Secondly, we discuss the same problem but based on maximizing the expected utility from terminal wealth, where the investment and retention decrease with increased coefficient of risk aversion, and are independent of the wealth. Finally, we discuss the problem of optimal portfolio selection and reinsurance based on two-player zero-sum stochastic differential game between nature and an insurance company. We derive closed-form expressions for the optimal portfolio selections and the reinsurance strategies of the insurance company and the optimal value function. The results indicate that, if the retention level is zero, the insurance company should not buy the risky asset, but if the retention level is greater than zero, the company should sell short the risky asset. The amount of selling short and the retention level will increase with the correlation efficient between the risky asset and the surplus process of insurer. The amount and the level also increase with the decrease of the remaining time to the maturity but decrease as the risk-free interest rate rises.(VI) We study the problem of asset pricing based on utility indifference. Here the markets are incomplete since the risky asset is not tradable and asset price follows arithmetic Brown motion with or without jump. We obtain the utility indifference price by comparing the maximal expected utility from terminal wealth with and without stochastic risky asset. The results indicate that the utility indifferent price increases with the coefficient of risk aversion and the mean return rate of risky asset, but decreases if the volatility increases.(VII) We study the problem of optimal consumption/investment and option pricing based on maximizing the expected consumption utility in incomplete markets. On the assumption that the underlying asset is not tradable and follows a mean-reverting process, we get the optimal consumption and investment strategies and a partial differential equation satisfied by the European option prices by using stochastic dynamic programming and consumption utility indifference pricing principle. Numerical examples are also presented. The results indicate that risk aversion will decrease option prices, which change with time and depend on the mean reverting level under such model. Corresponding two different cases the option price may increase or decrease with time respectively.