Portfolio Optimization With Stochastic Interest Rate And Stochastic Income

This paper mainly analyzes the problem of portfolio selection with stochastic interest rates and individual investors with stochastic income from two perspectives:complete market and incomplete market.First,under the assumption of a complete market,we consider the optimal consumption-investment problem with the Vasicek stochastic interest rate model.The explicit analytic solution of the value function and optimal consumption and investment strategy are given by using dynamic programming principle and variable substitution method under the framework of utility maximization.The study found that the increase in interest rate volatility will lead to an increase in the interest rate risk of investors' investments in risk-free assets,resulting in a decrease in investor investment in risk-free assets and an increase in investment in risky assets.Secondly,the labor income factor of investors is introduced.Based on the second chapter,the optimal consumption-investment problem with stochastic interest rate and stochastic income is considered simultaneously,in which stochastic labor income dynamic by a log-normal model.Using the principle of dynamic programming and the method of variable substitution,the analytical solution of the optimal consumption and investment strategy with the power utility function is given.Then the numerical analysis of related parameters is given.The study found that when income factors are considered in the model,there will be an additional hedge term representing stochastic income in the optimal investment strategy.And,at the same interest rate level,investors with stochastic income will invest more aggressively in risky assets.Finally,the model is extended to the incomplete market.The optimal portfolio selection problem based on utility hedging of stochastic income in incomplete markets is studied.By assuming that the investor's preference is the CARA utility preference,and using the utility indifference pricing method and the HJB equation to derive the explicit expression of the value function and the indifference price of stochastic income.Then,the first-order approximate solution of indifference price is obtained by Taylor expansion,and the results are used to simulate the residual risk process under the optimal hedging strategy of stochastic income.The study found that the first-order approximation of indifference prices can be expressed by the conditional mean and conditional variance of the discounted value of cumulative stochastic income,and we can use this result to obtain approximate solutions for optimal investment and optimal hedging strategies.At the same time,numerical simulations show that investors will gradually reduce their investment in risky assets to reduce risk as the investment term expires.Moreover,as the correlation coefficient between risk assets and stochastic income increases,income volatility decreases,and investors' degree of risk aversion increases,the standard deviation and error range of the hedging error distribution under the optimal hedging strategy decrease.This means that investors using an optimal hedging strategy have less chance of making a profit or lose when investing in an incomplete market.