Formulation And Control For Dynamic Portfolio Selection And Optimal Execution Problem With Market Constraints

In this work,we study the issues of formulation and control for dynamic portfolio selection and optimal execution problem with market constraints.The studies on dynamic portfolio optimization theory are always based on the assumption that the financial market is complete.However,the encountered situations in real financial market are far from the aforementioned assumption on complete market.Since the strict supervision by the financial supervision institution and the different requirements of the real financial market,the investment behavior of the investor is restricted by some constraints,such as,no-shorting constraints,the upper and lower bound of fund position on some assets,no-bankruptcy constraints,cardinality constraints.For this kind of problem,we propose dynamic mean-variance(MV)portfolio selection problems with market constraints both in discrete-time and continuous-time.We transform this two kind of problems into two class of constrained linear-quadratic(LQ)optimal control problem formulations for the scalar-state stochastic system both in discrete-time and continuous-time.That is to say,the constrained MV portfolio selection problems are just as special cases of the constrained LQ optimal control problems which have various applications,especially in the financial risk management.The linear constraint on both the control and state variable considered in our model destroys the elegant structure of the conventional LQ formulation and has blocked the derivation of an explicit control policy so far in the literature.We successfully derive the analytical control policy for such a class of problems by utilizing the state separation property induced from its structure.We reveal that the optimal control policy is a piecewise affine function of the state and can be computed off-line efficiently by solving two corresponding Riccati equations.Under some mild conditions,we also obtain the stationary control policy for infinite time horizon.We demonstrate the implementation of our method via some illustrative examples and show how to calibrate our model to solve dynamic constrained portfolio optimization problems.However,both theory and practice show that variance is not an effective risk measurement,since the excess returns is also treated as risk by variance which conflict with the fact that risk is the loss caused by uncertainty.Instead of controlling ”symmetric”risks measured by variance of the terminal wealth,more and more portfolio models have shifted their focus to manage ”asymmetric” downside risks that the terminal wealth is below certain threshold.Furthermore,a well-documented notion in the financial literature is that time-variation in investment opportunities can affect portfolio choice,especially the impact by the mean-reverting property of financial asset price.For this kind of problem,we propose dynamic mean-downside risk portfolio selection problem under the mean-reverting market,namely,mean-LPM and mean-CVaR.Due to the complicated constraints in this kind of dynamic optimization model,it is hard to use the stochastic control approach directly.Instead,under our market setting and bankruptcy prohibition,we are able to solve this problem semi-analytically by using the martingale approach.Once the optimal terminal wealth is identified,we can use Feynman-Kac formula and Fourier Inverse Transformation method to find the optimal wealth process and portfolio process.Our method provide investor a simple tool to deal with the sophisticated model for guiding the portfolio management.Recently,more and more academic work considers the impact of market microstructure on trading behavior and asset prices.From the point of microstructure of transactions,electronic transactions have been dominant in the world’s financial markets.It has been evidenced that 53% of U.S.stock was traded by algorithmic trading in2009.This phenomenon arises plenty of new subjects,such as,how to allocate trading volume optimally for investment institutions to reduce the impact of their trading on market prices when they trade a large number of assets.The reason is that buying or selling stocks in large quantities usually cause stock prices to move toward unfavorable direction.More specially,in the classic finance theory,it has a key assumption that the market possesses complete liquidity.That is,the investors are the market recipients and their trading activities will not affect market prices.However,this is not yet the case in real financial market.When the institutional investor trades(buying or selling)large amount of assets,the actual prices of these assets will deviate from the equilibrium price of the current market and usually move toward unfavorable direction.This kind of price impact may cause huge execution cost.In order to reduce this price impact,we study the constrained optimal execution problem with a random market depth in the limit order market.Motivated from the real trading activities,our execution model considers the execution bounds and allows the random market depth to be statistically correlated in different periods.Usually,it is difficult to achieve the analytical solution for this class of constrained dynamic decision problem.Thanks to the special structure of this model,by applying the proposed state separation theorem and dynamic programming,we successfully obtain the analytical execution policy.The revealed policy is of feedback nature.Examples are provided to illustrate our solution methods.Simulation results demonstrate the advantages of our model comparing with the classical execution policy.