Dynamic Mean-CVaR Portfolio Optimization Under The Mean-reverting Market

Proposed by Markowitz in 1952,Mean-Variance portfolio selection model had been the foundation of modern selection theory.Although the approach to express portfolio returns and risk by mean and variance has some shortcomings,the framework of returns and risk established has been accepted worldwide.Based on the framework,scholars brought up more accurate and appropriate lower risk measures such as Lower Partial Moments and Conditional Value at Risk.With the improvement of risk measure,also under processes of two-period and multi-period model,investment portfolio model has been innovated from static model to continuous-time dynamic model.Besides,the description of the market has been developed from certainty to randomness,which is more closed to real market situation.On the basis of former research,this paper is aimed at studying on portfolio op-timization under dynamic mean LPM model and dynamic mean CVaR model.First,we use geometric Brownian motion to describe the process of risk asset value,based on which we build up continuous-time dynamic mean LPM model and CVaR model.The way to find optimal solution of LPM problem is divided into two procedures.The first step is to transform the original problem to a static optimal problem with argument x^*（T）meaning the terminal wealth,then we can find its optimal solution.The second step is to replicate a strategy to identify the portfolio policy which which could result in the optimal terminal wealth solution.As for CVaR issue,we construct an auxiliary func-tion to calculate the risk measure.Since the auxiliary variableαis the benchmark of LPM model,CVaR problem was converted to LPM problem with variable benchmark ingeniously.From what has been mentioned above,both of LPM and CVaR model have similar solutions in spite of the difference of their risk measure.Under circumstance of the confirmation of market parameters,we derive the ana-lytical solution of optimal wealth process and policy through constructing a Lagrange problem.However,we cannot obtain the analytical solution under stochastic mar-ket.Abundant numerical evidence indicates that the stock market shows certain degree of mean-reverting property,but there are rare literatures studying dynamic mean-risk model under mean-reverting market.Because of randomness and complicated con-straints in the model,it is hard to solve the issue directly.Instead,under our market setting,we use martingale method and Monte Carlo simulation method to solve this problem.By generating a large number of random sample paths of market situation z（t）,we deduced the numerical solution of optimal wealth process and policy.After receiving the numerical solution,we work on the simulation experiment upon the model.We analyze the relationship between various parameters and invest-ment strategy by setting different parameters such as drift,volatility,initial time,risk preference and so on,respectively.It is obvious that more allocation of risky assets can increase the returns when market condition is good,and the proportion of risky asset reaches minimum value when wealth is around x₀.In addition,the optimal solution contains some extreme value when volatility rate is relatively high.Thus,we should test stability before fixing those parameters.Although we have done a lot of work of portfolio model and solution,there still lay certain subjects and research questions to be polished in the future.