OPTIMAL PORTFOLIO ON TRACKING THE EXPECTED WEALTH PROCESS WITH LIQUIDITY CONSTRAINTS

Available online at www.sciencedirect.comMalhemiqclLientia. ScienceDirect数学物理学报Acta Mathematica Scientia 201 1,31B(2):483- 490http://actams.wipm.ac.cnOPTIMAL PORTFOLIO ON TRACKING THEEXPECTED WEALTH PROCESS WITHLIQUIDITY CONSTRAINTS*Luo Kui (罗葵)Industrial Training Centre, Shenzhen Polytechnic, Shenzhen 518055, ChinaE-mail: luokui666@yahoo. com.cnWang Guangming (王光明)China Merchants Bank, Shenzhen 518040, ChinaHu Yijun (胡亦钧)School of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaAbstract In this article, the authors consider the optimal portfolio on tracking theexpected wealth process with liquidity constraints. The constrained optimal portfolio isfirst fornulated as minimizing the cumulate variance between the wealth process and theexpected wealth process. Then, the dynamic programming methodology is applied toreduce the whole problem to solving the Hamilton-Jacobi- Bellman equation coupled withthe liquidity constraint, and the method of Lagrange multiplier is applied to handle theconstraint. Finally, a numerical method is proposed to solve the constrained HJB equationand the constrained optimal strategy.Especially, the explicit solution to this optinalproblem is derived when there is no liquidity constraint.Key words Portfolio selection; wealth tracking; liquidity constraints; HJB equation; .Lagrange multiplier2000 MR Subject Classification 91 B28IntroductionThe optimal portfolio selection is a very important problem in financial control theory. Theearliest approach to consider the optimal portfolio problem was the mean-variance approach (seeMarkowitz [8] and Bielecki et al. [1], etc.), which makes an one off decision at the beginningof the period and holds on until the end of the period. Merton [9- 11] developed this singleperiod model to continuous time models, which consid， 中国煤化工ximization. Byapplying the stochastic control theory to the optimalsolutions wereTYHCNMHGobtained for some special cases."Received June 8, 2008; revised August 20, 2009. Supported in part by the National Natural ScienceFoundation of China (10671149) and the Ministry of Education of China (NCET-04-0667)484ACTA MATHEMATICA SCIENTIAVol.31 Ser.BIn real markets, there are some restrictions on trade and the investors would not always re-alize their investment strategy. In general, there is some liquidity risk in all financial products.Liquidity becomes more and more important from the practical viewpoint as the variety offinancial products increases. Therefore, more and more researchers studied the portfolio prob-lem with liquidity constraints. Cvitanic and Karatzas [4] used the martingale approach andduality methods (Karatzas and Shreve [7]) to settle the optimal portfolio investment problemwith a generally constraint, in which they treated an example, with borrowing and short- saleconstraints, for a log utility investor with two risky assets having uncorrelated returns of equalvolatility. Grossman and Vila [6], using a stochastic dynamic programming approach, studiedthe optimal intertemporal portfolio policies of a borrowing constrained power-utility investor inthe standard Merton setting with constant coefficient. Tepla [13] characterized optimal portfoliopolicies for CRRA utility investors facing either a borrowing limit on the total wealth investedin the risky assets, or short-sale restrictions on all risky assets or both. All the referencesmentioned above focused on the expected utility maximization of the terminal wealth.However, the investors' investment objectives could be various, not all the investors' re-wards and risks can be interpreted as means and variances, and not all the investors believethere exists utility functions to make their decisions through maximizing the expected utility.In real life, many investors just want to make their wealth grow better than sleeping in thebank. They want to live their life without too many fuctuations. The investment objective ofthis kind of investor can be interpreted as follows: They have a target to track, which might be awealth level or a growth rate better than the counterpart achieved in the bank; they can endurethe temporary undesirable achievements compared with the target, but they don't want therealized wealth or growth rate deviates from the target too much during the whole investmentperiod. In other words, the investor's objective is to approach or track their expected wealthprocess as closely as possible.In this article, we consider the optimal portfolio on tracking the expected wealth process,in which liquidity constraints are also regarded. First, the constrained optimal portfolio isformulated as minimizing the cumulate variance between the wealth process and the expectedwealth process. By the dynamic programming methodology, the above constrained optimalportfolio problem is transformed to solving the Hamilton-Jacobi-Bellman equation [12] coupledwith the liquidity constraint, and the method of Lagrange multiplier [3] is applied to handlethe constraint. Especially, the explicit solution to this optimal problem is derived when there isno liquidity constraint. Finally, a numerical method is proposed to solve the constrained HJBequation and the constrained optimal strategy, also an example is demonstrated to show thedifference between with and without the liquidity constraint.2The Wealth Tracking Problem with Liquidity ConstraintsIn this article, T is a fixed terminal time, (2, FA1tered completeprobability, and W(t) = (W(,.. Wn(t)' is defined中国煤化工n-dimensionalBrownian motion with W(0) = 0, Ft = σ{W(8);s≤t]YHCNM.HGR吵thesetofall Rd-valued, progressively measurable stochastic processes X(:) = {X(t);0≤t≤T} adaptedtoF such that E s" |X()2dt < +o. Here, we denote that X (t) is the wealth during period t,Luo et al: OPTIMAL PORTFOLIO ON TRACKING EXPECTED WEALTH PROCESS485π() = (n().., πn())' is the money investing in n risk assets during period t.We consider a standard model of a dynamically complete financial market defined on thetime interval [0, T]. Suppose that there is a market with n + 1 assets. The risk free asset priceprocess satisfies the following ordinary differential equation (ODE):) dPo() = rPo(t)dt, t∈[0,T],(1)P(0) = po,where the interest rate r is a constant.The n risk assets price processes P(t) = (R(),-,. Pn(t) satisfy the fllowing stochasticdifferential equation (SDE):dP(t) = diag(P())[udt + σdW(t)]，t∈ [0,T],(2)P(0) = (n..,Pn',where μ= (41,..,Hn)' andσ = (σij )nxn are the appreciation and volatility rates, respectively.Then, the wealth process X(t) obeys) dX(t) = [rX(t) + (μ- r)'"(t)]dt + r(t)'σdW(t),(3)X(0) = xo.Definition 1 A portfolio π() is said to be admisible if (X(t), π(t)) satisfies that X(t)∈候([0,T];R) and (t)∈[3(!0,T];R"), for anyt∈(0,T].Without loss of generality, in this article, we supposen= 1, then,dX(t)= [rX(t) + (u - r)r()]dt + r(t)odW(t)，(4)X(0)=xo.Then, the wealth tracking problem with lquidity constraints may be formulated as mini-mizing the cumulate variance between our wealth process and the expected wealth process asfollows:pTminE| (X(8) - y())2ds, .π∈L子.subject toS (t)≤LX(t)， 0≤t≤T,l (),()) satisfying (4),where y(s) is a given function, and then, {y(s);0 ≤8≤T} is the expected wealth process,which denotes the investor's investment objective; L > 0 is a given constant, which shows theliquidity constraint.Remark 1 In fact, for L= 1, the liquidity constraint π(t) < LX(t) tells us the investorcannot borrow anything from the bank and there is nq中国煤化主the bank.For L> 1, the liquidity constraint π(t)≤LX(t)Fing constraint,which means that the investor can borrow the amountMYH.CNMH G the bank.For L < 1, the liquidity constraint π(t)≤LX(t) shows that the investor must keep theamount (1 - L)X(t) at least in the bank.486ACTA MATHEMATICA SCIENTIAVol.31 Ser.B3 The Optimal StrategyIn this section, we apply the dynamic programming technique to solve the optimal portfolioproblem. In fact, π is chosen based on knowing not only time t but also the wealth X(t).Therefore, we use π(x,t) instead of π(t), where π(x,t) is defined as a function of the time tand the wealth x (x is the value of the wealth process X(t) at time t). Because with such πthe corresponding process X(t) is a Markov process, we call π(x, t) the Markov control strategy(see Fleming and Soner [5]).By the dynamic programming methodology, the optimal portfolio problem (5) is equivalentto the problem of finding a solution to the Hamilton-Jacobi-Bellman equation (see Chapter 19of Bjork [2), Chapter 11 of 0ksendal [12).We defineJ(x,t)=_物。[{[()()2dX()=}(6)π(z,t)SLxBy applying Ito's formula to J(x, t), we havedJ(0.,)= [n+ s(X(>+(u -)(,) + e0.0no^t+Jxπ(x, t)odW(t).(7)Then, we obtain the associated HJB equationJt+. min,.({(-1)m(z,DJIa +号r2(,)Ix y+r2J2+(x- ()=0, (9)π(x,t)SLx( J(x,T)=0.In solving the HJB equation, the static optimization problemmin{(u -n(0J。+ 2010,0(9)π(x,t)EL子subject to the liquidity constraintπ(x,t)≤Lx，0≤t≤T,(10)can be tackled separately to reduce the HJB equation to a non-linear partial diferential equationof J only.In the following, we introduce the Lagrange functionV(,)(,)) = (4- r)Jx(x,t)+ 5o2Jxn(x,t) + x(,)(Lx -((,1),，(11)where入(x,t)≤0 is the Lagrange multiplier.The first order necessary condition of the static optimization problem is given by中国煤化工(μ-r)Jx +σ2Jxxπ(x,t).(12)MHCNMHGX(x,t)(Lx - r(x,t))=0,(13)入(x,t)≤0.(14)No.2Luo et al: OPTIMAL PORTFOLIO ON TRACKING EXPECTED WEALTH PROCESS487From (12), we have*(,)= x“(r,t)-(u-r).(15)σ2JxxAlso, from (13)-(14), when *(x,t) < 0,π*(x,)=Lx and *(x,t)= (u-r)Jx + Lxo2Jx. .(16)Hence,( Lxif入*(x,t)<0,π*(x,t)=(μ-r)Jxif λ*(x,t)=0.(17)Substituting (17) into (8) givesJ+ (u- r)Jr*(,t)+ *07+r+(x,1)2 +rxJ2 + (x -y()2 =0，(18)( J(x,T)= 0,which can be solved then for the optimal value function J*(x, t). Thus, the optimal strategyπ*(x,t) and the optimal value function J*(x, t) can be determined by the non-linear system(12)-(14) and (18). Because of the nonlinearity in π* (x, t), the first order conditions togetherwith the HJB equation are a highly non-linear system, and numerical methods are required tosolve for π*(x,t) and J*(x, t).Remark 2 If there is no liquidity constraint, we can derive the exact formula of thesolution to this optimal portfolio problem.In fact, the HJB equation without liquidity constraints becomesJt+. min .{u--r)rJz+> +rxJx +(x-y()2= 0, .x∈L子(19)( J(x,T)=0.Then, the minimurn in (19) can be attained whenμ-r Jxπ*(x,t)= -σ2 Jxo(20)By substituting (20) into (19), the HJB equation can be rewritten as( J4-("=7)”Jxr+rxJx +(x-y())2= 0,(21)J(,T)= 0.To solve this equation, we try the solution as the functional formJ(x,) = f(t)x2 + 9g(t)x + h(t),(22)and substitute for it in (21), Then, we obtain three 0中国煤化工MYHCNMHG( ()+ (2r- (4-2")3(0)+1=0,J2(23)( f(T)=0,488ACTA MATHEMATICA SCIENTIAVol.31 Ser.B8g1+H(t)+(r-“o2F)(t) - 20(t) =0 .(24)( g(T)=0,8hFt(- (μ-r)2q?()+y?()=0,4o2 F(t)(25)h(T)= 0.Solving (23) and (24), we have1f(t)=;2r-(26)(=-ng"o[" er (4+)(-2()ds.(27)Then, we obtain the optimal strategyπ*(x,t)=-g(t)(28)σ2→=σ。2(x+ 2f()Remark 3 Especially, in Remark 2, if we consider the expected wealth process {y(s);0≤8≤T} as y(s)= xoeRs, where R> r is the expected return rate. We verify that f(t) is thesame as (26), and further,2xoeRtg(t)=-.R+r-Cre(elr =D2)(0)-),(29)so we obtaing()π*(x,l)=-J2(x+ 2f()x- xoeRt.(2r- (2(+1=g2)(-)-)(30)σ(R+r- 52(2-5-2)7-)-1)4 Numerical Methods and ExamplesIn this section, we consider the expected wealth process {y(s) = xoeRs,0≤8≤T}, whereR > r is the expected return rate. We will demonstrate how to handle the wealth trackingproblem with the liquidity constraint by numerical methods. Also, some examples will beillustrated.First, we give a numerical algorithm to resolve the highly non-linear system (12)- (14) and(18), then we can get the optimal strategy π*(x,t).The unconstrained solution (22), (26), (29), and (30) will be used as an initial guess to theiterative algorithm. Dividing the computational domain intea orid nf N Y N. mesh points andomitting (x, t) in all variables for simplicity of notation中国煤化工e summarizedas follows:TYHCNMHG(1) λ*(O) =0, π*() and J*() are from the unconstrained solution, which can be seen in .(22), (26), (29) and (30). Set k= 0.No.2Luo et al: OPTIMAL PORTFOLIO ON TRACKING EXPECTED WEALTH PROCESS489(2) For x = [0,0...,NxOx] and t = [(Nt - 1)Ot,.., Ot,0, calculate 入(k+1) andπ+(k+1) from(μ-r)J() +o2Jf()π*(k+1)- x*(k+1)=0,入*<(k+1)(Lx- π*(k+1)= 0,(31)(入(k+1)≤ 0.(3) Forj= (,1,-,N,] and n= [(Nt - 1)，,1,01, by finite diference methods, wesolveJ+(k+1) _ J+(k+1)J+(k+1) - J+(+1)0= j九△tin-1 -+ [(u- r)+*(k+1) + rjOx]in△x .j-1,n.*(k+1)*(k+1)+(k+1)+(1(1)202J+1,n- 2-J7,n+Jj-1.n+ (jOx - y(nD)2,(32)(Ax)2that is,At.]*(k+1)J打小1)= |1+ (u-1r)+*(h+) +rjOx)云-(1(6+1)202 A |Jj,n+(-1r)+*(+) +rjQx)、△△x+ (*(+1)202;J:(k+1)2(Ox)2] j-1,n+(+)20J1(+1) + Ot(jOx - y(nOt)2.(33)2(Ox) "(4) Return to (2) withk= k + 1 until convergence.A Matlab program has been written to implement the above procedure. In the following,we give an example to demonstrate our numerical method.Example 1 The initial wealth is chosen xo = 1 (we can consider the investor's initialwealth as one unit). The terminal time is chosen to be 1 and Nt is fixed at 1000, whichmeans that the horizon period Ot = 1/1000 year if the terminal time 1 denotes 1 year. Thestochastic process is chosen arbitrarily with μ= 0.15,σ = 0.2; the risk-free rate isr = 0.1 andthe expected return rate is R = 0.12. For the liquidity constraint, we take L = 2. Finally,△x = 1/500 and Nx = 1000 are used, which has the rangex∈[0, 2]. In this case, the solutionto the unconstrained problem suggests that, at time t = 0.5,π*(x,t) = -1.25(x - 1.0672xo). .(34)Figure 1 shows the Lagrange multipliers against the wealth X at time t = 0.5, whichshows the exact wealth values when the liquidity constraint becomes active, where the Lagrangemultipliers are negative instead of zeroes.Figure 2 compares the asset distribution for different wealth values with or without theliquidity constraint at time t = 0.5. From the figure, a good control over the investment in therisky asset has been achieved and the portfolio invested in the risky asset has been reducedin order to fulfil the liquidity constraint. In particular, when the constraint is not active, theoptimal portfolio follows the unconstrained solution;中国煤化工the risky assetincreases, the liquidity constraint becomes active and; asset.From the numerical results, we find that the consMH. CNMH Gnvested les inthe risky asset, which is due to the liquidity constraint cutting down the choices of portfoliostrategies.490ACTA MATHEMATICA SCIENTIAVoL.31 Ser.B1.-0.01I .... constraned-0.0200.5心-0.03-0.04-0.5-0.05-0.060.20.40.60.811.21.41.61.82-1.5.51.52WealthXFig.1 The Lagrange multiplier λ shows thatFig.2 the optimal portfolio with or without thethe liquidity constraint becomes active when-liquidity constraint. In this case, the parametersever λ becomes negative.areμ= 0.15;σ= 0.2;r= 0.1;R= 0.12;xo= 1;T= 1;L= 2;t= 0.5. When the liquidity const-raint becomes active, the investment in the riskyasset will be reduced.References[] Bielecki T R, Jin H Q, Pliska S R, Zhou X Y. Continuous-time mean-variance portfolio seletion withbankruptcy prohibition. Mathematical Finance, 2003, 15(2): 213-244[2] Bjork T. Arbitrage Theory in Continuous Time. Oxford: Oxford University Press, 1998[3] Chow G C. Dynamic Economics: Optimization by the Lagrange Method. New York, Oxford: OxfordUniveraity Press, 1997[4] Cvitanic J, Karatzas 1. Convex duality in constrained portfolio optimization. Annals of Applied Probability,1992, 2(4): 767- -8185] Fleming W H, Soner H M. Controlled markov processes and viscosity solutions. Berlin, New York:Springer, 1993[6] Grossman S J, Vila J L. Optimal dynamic trading with leverage constraints. Journal of Financial andQuantitative Analysis, 1992, 27(2): 151-168[7] Karatzas 1, Shreve s E. Methods of Mathematical Finance. New York: Springer Verlag, 1998[8] Markowitz H. Portfolio selection. Journal of Finance, 1952, 7: 77-91[9] Merton R C. Lifetime portfolio selection under uncertainty: the continuous-time case. The Review ofEconomics and Statistics, 1969, 51(3): 247 257[10] Merton R C. Optimal consumption and portfolio rules in a continuous- time model. Journal of EconomicTheory, 1971, 3(4): 373- -413[11] Merton R C. Continuous -Time Finance. Oxford: Blackwell, 1990.[12] Oksendal B. Stochastic Differential Equations: an Introduction with Applications. Fifth Edition. Berlin:Springer Verlag, 200213] Tepla L. Optimal portfolio policies with borrowing and shortsale constraints. Journal of Economic Dy-namics & Control, 2000, 24: 1623- 1639中国煤化工MYHCNMHG