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PORTFOLIO OPTIMIZATION OVER A FINITE HORIZON WITH FIXED AND PROPORTIONAL TRANSACTION COSTS AND LIQUIDITY CONSTRAINTS STEFANO BACCARIN DANIELE MARAZZINA Working paper No. 17 - January 2013 DEPARTMENT OF ECONOMICS AND STATISTICS WORKING PAPER SERIES Quaderni del Dipartimento di Scienze Economico-Sociali e Matematico-Statistiche ISSN 2279-7114 Founded in 1404 UNIVERSIT DEGLI STUDI DI TORINO ALMA UNIVERSITAS TAURINENSIS Portfolio Optimization over a Finite Horizon withFixed and Proportional Transaction Costs andLiquidity ConstraintsStefano Baccarina, , Daniele MarazzinabaUniversit a degli Studi di Torino, Dipartimento di Scienze Economico-Sociali eMatematico-Statistiche, Corso Unione Sovietica 218/bis, 10134 Torino, Italy.bPolitecnico di Milano, Dipartimento di Matematica, Via Bonardi 9, 20133 Milano, Italy.AbstractWe investigate a portfolio optimization problem for an agent who invests in twoassets, a risk-free and a risky asset modeled by a geometric Brownian motion.The investor faces both xed and proportional transaction costs and liquidityconstraints. His objective is to maximize the expected utility from the port-folio liquidation at a terminal nite horizon. The model is formulated as aparabolic impulse control problem and we characterize the value function as theunique constrained viscosity solution of the associated quasi-variational inequal-ity. We compute numerically the optimal policy by a an iterative nite elementdiscretization technique, presenting extended numerical results in the case ofa constant relative risk aversion utility function. Our results show that, evenwith small transaction costs and distant horizons, the optimal strategy is essen-tially a buy-and-hold trading strategy where the agent recalibrates his portfoliovery few times. This contrasts sharply with the continuous interventions of theMertons model without transaction costs.Keywords: Portfolio Optimization, Quasi-variational Inequalities, TransactionCosts, Viscosity Solutions1. IntroductionOptimal portfolio investment strategies have been widely studied in the liter-ature. In his seminal article, Merton (1969) developed a continuous time modelto study the optimal portfolio strategy for an investor managing a portfolioof risky assets, whose prices evolve according to geometric Brownian motions.Since then, research in this area has focused on di erent aspects, aiming tomake the mathematical model closer to the real market, in particular with re-spect to liquidity issues. It is well known that, in the real economy, investorsface nontrivial transaction costs, which in uence their trading policies. It is notpossible to rebalance a portfolio in a continuous way, as assumed by Merton,Corresponding authorEmail addresses: stefano.baccarinunito.it (Stefano Baccarin),daniele.marazzinapolimi.it (Daniele Marazzina)January 28, 2013and solvency constraints and bounds on the amounts of the open short positionsare usually present.In our article we deal with a portfolio optimization problem over a nitehorizon with two assets, a risk-free and a risky asset whose value is modeledby a log-normal di usion. We consider a small agent who does not a ect inany signi cant way the assets prices with his transactions. The investors ob-jective is to maximize his utility from the liquidation of terminal wealth in thepresence of transaction costs and liquidity constraints. More speci cally weformulate the classical Mertons problem over a nite horizon, without inter-mediate consumption, with xed and proportional transaction costs, a solvencyconstraint, and bounds on the open short positions. Most of the literature onportfolio optimization with transaction costs considers the problem of maximiz-ing the cumulative expected utility of consumption over a in nite horizon, withproportional transaction costs. See for instance Akian et al. (1996); Davis andNorman (1990); Kumar and Muthuraman (2006); Shreve and Soner (1994). Thesame in nite horizon problem but with xed and proportional costs has beenstudied in Oksendal and Sulem (2002). A second class of articles studies theproblem of maximizing the long-term growth rate of portfolio value. See Mortonand Pliska (1995) for a problem with transaction costs equal to a xed fractionof the portfolio value (portfolio management fee), (Akian et al., 2001; Assafet al., 1988; Dumas and Luciano, 1991) for models with proportional transac-tion costs, and Bielecki and Pliska (2000) in the more general framework ofrisk-sensitive impulse control. Fewer papers consider a portfolio optimizationproblem with transaction costs over a nite horizon. Liu and Loewenstein (2002)consider proportional transaction costs and approximate the value function bya sequence of optimal analytical solutions for problems with exponentially dis-tributed horizons. This allows to obtain the optimal solution by a sequence ofproblems without the time dimension. However for a given terminal date theoptimal trading strategy is approximated by a stationary policy. In (Easthamand Hastings, 1988; Korn, 1998) both xed and variable transaction costs areconsidered and the model is solved by using impulse control techniques. Theselast articles, which are the closest to our assumptions, use veri cation theoremsto characterize the value function and the optimal policy and apart from somesimple cases only approximate the solution by an asymptotic analysis. In therecent paper by Ly Vath et al. (2007), which has inspired our work, the authorsconsider a portfolio optimization problem over a nite horizon with a perma-nent price impact and a xed transaction cost. The main result in Ly Vathet al. (2007) is a viscosity characterization of the value function, but neither acharacterization of the optimal policy nor a numerical solution of the problemis given.Our portfolio optimization problem is formulated as an impulse control prob-lem, associated by the dynamic programming principle to a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI), as in Bensoussan and Lions(1984) and all the subsequent literature on impulse control. The features of ourstochastic control problem lead to consider a parabolic HJBQVI in two vari-ables and time, and to impose state constraints on the space variables. As thevalue function of our problem is not necessarily continuous and because of thestate constraints, we have considered, as in (Akian et al., 2001; Ly Vath et al.,2007; Oksendal and Sulem, 2002), the very general notion of (possibly) discon-tinuous constrained viscosity solutions. In fact in Section 3 of this paper, by2means of a weak comparison principle, we show that the value function is theunique constrained viscosity solution of the HJBQVI verifying certain bound-ary conditions, and that it is (almost everywhere) continuous. These results aresummarized in Theorem 6.To simplify the numerical solution of the model, in Section 4 we decom-pose our impulse control problem into a sequence of iterated optimal stoppingproblems, as in (Baccarin, 2009; Chancelier et al., 2002). This reduction, rstintroduced in Bensoussan and Lions (1984), has both a theoretical and com-putational interest. It allows to represent the value function by the limit of asequence of solutions of variational inequalities. Moreover it makes it possibleto characterize a Markovian quasi-optimal policy which is arbitrarily close tothe optimal one. We propose an iterative nite element discretization techniqueto solve numerically this sequence of variational inequalities, and therefore tocompute the value function and the optimal policy.In Section 5 of the paper we present extended numerical results for our modelin the case of a constant relative risk aversion (CRRA) utility function, whichis the most commonly used utility function in expected utility maximizationproblems. See, for instance, Akian et al. (1996); Davis and Norman (1990); Liuand Loewenstein (2002); Ly Vath et al. (2007); Kumar and Muthuraman (2006);Shreve and Soner (1994). We describe the form of the optimal transactionstrategy and we investigate how it varies with di erent values of the modelparameters. The article that comes closer to ours in this sense is Liu andLoewenstein (2002), even if authors only considered proportional transactioncosts and stationary policies. We analyze the transaction regions, the targetportfolios, i.e., the portfolios where it is optimal to move from the transactionregions, and how the agents non-stationary optimal strategy varies as time goeson and for di erent horizons. To the best of our knowledge this is the rst paperwhere it is shown explicitly how the transaction regions and the target portfolioschange, as time passes, up to the nite horizon. Sensitivity analysis with respectto the market and agents parameters and a comparison between our optimalstrategy and others suggested in literature is also provided. Our numericalresults show that the transaction costs have a dramatic impact on the frequencyof trading of an optimal policy. This phenomenon has already been noted, ina qualitative way, in Dumas and Luciano (1991); Liu and Loewenstein (2002);Morton and Pliska (1995). We show that the optimal strategy is essentially abuy-and-hold trading strategy where the agent recalibrates his portfolio veryfew times, in contrast with the continuous interventions of the Mertons modelwithout transaction costs.2. The model formulationIn this section we give a precise formulation of the model. We consider aninvestor who holds his wealth in two nancial assets: a risky asset, or stock, anda risk-free security, which we call a bank account. We denote by S(t) the valueof the stocks held by the investor at time t, and by B(t) his amount of moneyin the bank account. The initial wealth in t = 0 is given by (B0;S0): The valueS(t) evolves as a geometric Brownian motiondS(t) = S(t)dt+ S(t)dW(t); S(0) = S0;3whereWt is an adapted Wiener process on the ltered probability space ( ;F;P;Ft),verifying the usual conditions. The bank account grows in a certain way at thexed rate rdB(t) = rB(t)dt; B(0) = B0:At any time the investor can buy ( 0) or sell ( 0; 0 c:i is a Ft stopping timei i+1 8ilimi!+1 i = +1 almost surelyi is F i measurable .(1)Note that condition i!1 a.s. implies that the number of stopping times inany bounded time interval is almost surely nite ( i = +1 for some i 0 and 0 0. This means that the problem is to maximize the expected utility of theportfolio liquidation value at the terminal date T: However we will assume thatour investor will be satis ed if his portfolio reaches a threshold liquidation valueLmax L(B0;S0), at a time t:1 =+1 if S 0 and B 0t otherwise1 =arbitrary if S 0 and B 0S otherwise;8 1is clearly always admissible. Note that V(t;0;0) = 0; 8t2 0;T, because theonly admissible policy is doing nothing, and U(0) = 0 by assumption. MoreoverV(t;B;S) U(Lmaxer(T t) only if (B;S) 2 EFnE, because the points inEFnE can be reached by an admissible policy only after #p, if the initial position(B;S) =2EFnE.6The value function V of our problem veri es the following dynamic program-ming property. See (Fleming and Soner, 1993, Section V.2), or (Ly Vath et al.,2007).Dynamic Programming Property:(a) For any (t;B;S)2Q, p2A(t;B;S) and fFsg-stopping time t we haveV(t;B;S) Et;B;S V(#p ;Bp(#p );Sp(#p ) ; (5)(b) For any (t;B;S) 2Q; and 0, there exists p0( ) 2A(t;B;S) such thatfor all fFsg-stopping time t we haveV(t;B;S) Et;B;SV(#p0 ;Bp0(#p0 );Sp0(#p0 ) + : (6)Combining (a) and (b) we obtain the following version of the dynamic pro-gramming principle, which holds for any (t;B;S)2Q and fFsg-stopping timetV(t;B;S) = supp2A(t;B;S)Et;B;S V(#p ;Bp(#p );Sp(#p ) .Now, we denote by F(B;S) the set of admissible transactions from (B;S)2 F(B;S) = 2R : (B K cj j;S + )2 and by z the subset of where F(B;S)6=;:Remark 2. The set F(B;S) can be empty. For example it is always emptywhen B+S 0 and S + B K cjSj B. In this case we haveV = MV, otherwise V MV. Note that V is upper-semicontinuous but notcontinuous for any point (T;B;0)2Q.For t20;T) the behavior of V depends on which part of we are consid-ering:(a) Along the segments OA and OI in Figure 1 it is not possible to intervenebecause this will bring the process (B;S) outside the admissible region Adr:Actually in the points A and I there is one admissible transaction which leads usto the origin O, but this is certainly unpro table. Therefore we have V MV.Apart from V(t;0;0) = 0, the value of V is not known a priori in this part of Q.(b) Except for the points A and I, along the segments AB and IH it isnecessary to make a transaction, otherwise the process could leave Adr witha positive probability. Moreover the only admissible intervention brings theprocess to O: Consequently it holds V = MV = 0: Note that V is upper-semicontinuous but not continuous in A and I.8(c) In the interior points of the segments BC and HG it is necessary to makea transaction because one of the bounds in the short position is reached. Thevalue of V is not known a priori. We have V =MV.(d) In the upper part of Q, that is along the segments CD;DE, EF andFG, the threshold liquidation value Lmax has already been reached. The valueof V is known. If (B;0) 2 EF then V(t;B;0) = U(B er(T t). If (B;S) 2CDDEfFGnFg then V(t;B;S) = U(Lmax er(T t). It is always optimalnot to intervene, but we also have V =MV; with = S in (7), if S 0 and S + B K cjSj B. Note that V is upper-semicontinuous butnot continuous in the point F, 8t20;T):We give now some bounds on the value function. Since Jp 0; 8p 2A(t;B;S); it is obvious that V(t;B;S) is nonnegative in 0;T : By theproblem de nition we also have V(t;B;S) U(Lmax +K)er(T t), that is thevalue function is nite. Moreover, as it holds U(L(B;S)er(T t) V(t;B;S) U(Lmaxer(T t) when (B;S) =2EF, the value function is also continuous in thesegments CD; fDEnEg, fFGnFg. It is not di cult to show that V is alsobounded from above by the value function of the same problem with U(L) =CL and without transaction costs and liquidity constraints (a Merton problemover a nite horizon without consumption and a CRRA utility function, seeMerton (1969).Proposition 1. We haveV(t;B;S) Ce (T t) (B +S) (9)in 0;T , where= r + ( r)22 2(1 ).Proof. We set Z(t;B;S) := Ce (T t) (B + S) . The inequality (9) is truein T as V(T;B;S) = U(L(B;S) CL(B;S) C (B + S) and in0;T) (0;0) because here we have V(t;0;0) = 0. Moreover, in nf0g; Zveri es Z MZ and Zt LZ 0. Indeed MZ = 1 if (B;S) =2 z,MZ Ce (T t) (B + S K) MZ it follows that, a.s.,Z(#p;Bp(#p);Sp(#p) EZ(#p;Bp(#p);Sp(#p) 8p2A(t;B;S) .9ThereforeZ(t;B;S) supp2A(t;B;S)EZ(#p;Bp(#p);Sp(#p) = supp2A(t;B;S)EC(Bp(#p) +Sp(#p) e (T #p) supp2A(t;B;S)Jp = V(t;B;S) .Bound (9) shows in particular that V(t;B;S) is continuous in (t;0;0), whereV(t;0;0) = 0, 8t 2 0;T. Now we give the precise characterization of thevalue function as a viscosity solution of (8). Since V is not even continuousat some points in Q it is necessary to consider the notion of discontinuousviscosity solutions. Moreover the state constraint (Bp(t);Sp(t) 2 Adr, 8t 20;#p; requires a particular treatment of the lateral boundary conditions when(t;B;S)20;T) 1 and the use of constrained viscosity solutions. We recallnow the de nitions of (possibly discontinuous) constrained viscosity solutions.LetUSC(Q) andLSC(Q) be respectively the sets of upper-semicontinuous (usc)and lower-semicontinuous (lsc) functions de ned on Q. Given a locally boundedfunction u : Q!R+ we will denote by u and u respectively the usc envelopeand the lsc envelope of uu (t;B;S) = lim sup(t0;B0;S0)2Q(t0;B0;S0)!(t;B;S)u(t0;B0;S0) 8(t;B;S)2Qu (t;B;S) = lim inf(t0;B0;S0)2Q(t0;B0;S0)!(t;B;S)u(t0;B0;S0) 8(t;B;S)2Q .We have u u u and u is usc (lsc) if and only if u = u (u = u ). In thefollowing, sometimes we set x (B;S)2 to simplify the notation.De nition 1. GivenO ; a locally bounded function u : Q!R+ is called aviscosity subsolution (resp. supersolution) of (8) in 0;T) O if for all (t;x)20;T) O and (t;x) 2 C1;2(Q) such that (u )(t;x) = 0 (resp. (u )(t;x) = 0) and (t;x) is a maximum of u (resp. a minimum of u )on 0;T) O, we havemint (t;x) L(t;x);u (t;x) Mu (t;x)0 (10)(resp. u and 0) (11)On 0;T) 2 the value function V veri es the Dirichlet boundary conditionV(t;B;S) = U(L(B;S)er(T t). To deal properly with the state constraint(Bp(t);Sp(t)2Adr,8t20;#p; it will be necessary to require that V satis esthe subsolution property also on the 0;T) 1 part of the lateral boundary0;T) (see (Crandall et al., 1992, section 7C), or (Oksendal and Sulem,2002; Ly Vath et al., 2007).De nition 2. We say that a locally bounded function u : Q!R+ is a 1 constrained viscosity solution of (8) in Q = 0;T) if it is a viscosity super-solution of (8) in Q and a viscosity subsolution of (8) in 0;T) f 1 g.10We will need the following properties of the non-local operator M.Lemma 2. Given a locally bounded function u : Q!R+ we have:(a) if u is lower-semicontinuous (resp. usc) then Mu is lower-semicontinuous(resp. usc)(b) Mu (Mu) and Mu (Mu) (c) if u is upper-semicontinuous then there exists a Borel measurable functionu : z!R such that for any (B;S)2zMu(t;B;S) = u(t;B u(B;S) K cj u(B;S)j;S + u(B;S): (12)Proof. (a) and (b) can be proven in the same way as in (Ly Vath et al., 2007,Lemma 5.5). As u is upper-semicontinuous and for (B;S)2z the set F(B;S)is compact the sup in (7) is reached for some values of ,8(B;S)2z. Moreover,as z is -compact, we can select a Borel measurable function u : z!R suchthat (c) holds true (see Fleming and Rishel, 1975, Appendix B, Lemma B). We can now state the viscosity property of the value function.Theorem 3. The value function V(t;B;S) is a 1 constrained viscosity solu-tion of (8) in Q.Proof. Using the dynamic programming property (5-6), and properties (a)and (b) of Lemma

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