Asset Liability Management Based On Strochastic Control Theory

Since the financial crisis in 2008,risk management has been attached importance to by financial institutions.In risk management,asset liability management is one of the most important aspects.In recent years,asset liability management has become more and more important in the financial field of Europe and America,while the same is true in the domestic financial field.Asset liability management is how to hold the proper combination of assets and liabilities so as to reduce the risk and achieve the investment goal.Thus,asset liability management is the asset allocation problem,but also the financial optimization problem.Based on stochastic control theory,this thesis studies asset liability management problems under different background risks,i.e.,the asset liability management with double risks of inflation and interest rates,the asset liability management based on asset liability ratio and interest rate risk under liquidity constraints as well as the asset liability management of asset allocation with credit bonds.The main work and achievements of this thesis are as follows:1.Expected utility maximization that is an important objective function of risk decision theory has attracted much attention of many scholars.Moreover,for a portfolio with a longer investment period,the size of the inflation rate will directly affect the real value of the portfolio wealth,the higher the inflation rate,the lower the actual purchasing power of the wealth.Thus,this thesis first studies the asset liability management with double risks of inflation and interest rate under the expected utility maximization criteria,the analytic expressions of optimal asset allocation under the power utility function and exponential utility function are obtained by using the Hamilton-Jacobi-Bellman(HJB)equation method and partial differential equation method.Finally,based on the analytic expressions,the influences of the related parameters on the optimal asset allocation strategies are analyzed by numerical analysis.The results show that the related parameters of inflation risk,interest rate risk and liability will have a greater impact on the optimal asset allocation strategy.To hedge the risk of inflation,the introduction of inflation indexed bond in investment will significantly affect the investor's investment selection.2.The asset liability management with double risks of inflation and interest rate under the mean variance criterion is studied.Compared with the problem based on the expected utility maximization criteria,the problem based on the mean variance criteria allows investors to maximize their earnings in the range of risk that they can take,while the maximization of expected utility only considers the maximization of utility without considering the risks of utility.To obtain the optimal investment strategy of the optimization problem,the Lagrange multiplier method is firstly used to transform the asset liability management problem into a standard mean variance efficient problem.Then,the efficient asset allocation strategy and the analytical expression of the effective frontier are obtained by using the HJB equation method,partial differential equation method and Lagrange duality theorem.Finally,numerical examples are given to analyze the influences of the main parameters of the model on the effective asset allocation strategy and the effective frontier.The results show that the liability,interest rate risk and inflation risk have an essential impact on the effective asset allocation strategy.3.The asset liability management based on asset liability ratio and interest rate risk under liquidity constraints is studied.In actual investment,investors or regulators will limit the amount of transactions of certain assets to control risks,such as restrictions on short selling and the proportion of investment.To obtain the optimal investment strategy with limited investment process,the stochastic control theory and the domain decomposition method are used to transform the optimization problem into a stochastic control problem with piecewise control.Then,the analytic expressions of optimal asset allocation strategy are obtained by using the HJB equation method and partial differential equation method.Under stochastic interest rates,the classical verification theorem is no longer applicable because the stochastic differential equations satisfied by the asset liability ratio do not satisfy the Lipschitz condition and the square growth condition.To construct and prove a new verification theorem which can ensure that the solution of the HJB equation is the value function of the optimization control problem is the main work of this part.Finally,numerical examples are given to analyze the influences of liquidity constraints on the optimal asset allocation strategies.The results show that liquidity constraint has a great influence on the optimal asset allocation strategy.Due to an upper limit on investment,the proportion invested in the risky asset becomes lower,not as he(she)would like to hold the more risky asset.In addition,the investor can't carry on short selling for the investment is subject to a lower limit.4.In the risk assessment of an investment portfolio containing credit bonds,the interest rate risk is another major factor besides default risk.Therefore,the thesis finally studies the asset liability management problem with default risk and interest rate risk under the framework of expectation utility maximization.Unlike many literatures,we assume that the default intensity is a stochastic process,and the stochastic nature of the default intensity is related to the dynamic process of short-term interest rates.Compared with the constant default intensity,the pricing and investment of credit bonds will become very complicated under the stochastic default intensity.To solve this complex problem,the pricing formula of credit bond with stochastic default intensity is firstly obtained by the hedge skills,no arbitrage principle and partial differential equation method,and then the dynamic process of credit bond's price is gotten.Then,the optimal control problem with credit bonds is established by using the theory of dynamic portfolio selection,and the analytic expressions of the optimal asset allocation strategies are obtained by using the HJB equation method and differential equation method.Finally,based on the analytic expression,the influence of the default parameter on the optimal asset allocation strategy is analyzed by numerical analysis.The results show that the default risk has a greater impact on the optimal asset allocation strategy.Any change in the process of default will have a great impact on the optimal asset allocation strategy(especially bond assets).But,it has little effect on the optimal investment strategy of stocks.Moreover,when the default occurs,the recovery value of the credit bond will significantly affect the investor's optimal asset allocation strategy.