A Numerical Scheme for A Singular control problem: Investment-Consumption Under Proportional Transaction Costs
AA NUMERICAL SCHEME FOR A SINGULAR CONTROL PROBLEM:INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COSTS
WAN-YU TSAI AND ARASH FAHIMA
BSTRACT . This paper concerns the numerical solution of a fully nonlinear parabolic double ob-stacle problem arising from a finite portfolio selection with proportional transaction costs. We con-sider optimal allocation of wealth among multiple stocks and a bank account in order to maximizethe finite horizon discounted utility of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a numericalmethod which is composed of Monte Carlo simulation to take advantage of the high-dimensionalproperties and finite difference method to approximate the gradients of the value function. Nu-merical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitativeproperties proved in Dai et al. (2009) and Chen and Dai (2013).
Keywords:
Hamilton-Jacobi-Bellman equation, stochastic control, Monte Carlo approximation,backward stochastic differential equations, portfolio optimization, transaction costs.1. I
NTRODUCTION
This paper presents the numerical solution of an optimal investment-consumption problem inthe presence of proportional transaction costs during a finite time period. Given a known initialwealth, the objective of an investor is to decide the best consumption and investment strategy whichmaximizes the expected discounted utility of consumption over the finite investment period. In theabsence of transaction costs and for specific utility functions, the solution can be exactly obtainedand an investor’s optimal trading strategy is to maintain a constant proportion of wealth investedin risky stocks, which is called the
Merton proportion shown by Merton (1971). This constantproportion depends on the investor’s risk preference and also the market parameters. Merton’sstrategy, simply stated, is to continuously rebalance portfolio holdings in order to keep the fractionof investment in risky assets constant. However, in the presence of transaction costs, a continu-ous portfolio rebalancing process may incur infinite costs. Thus, the question arises: what is theoptimal strategy if there are transaction costs in the market?
Transaction cost appears in different ways, as a fixed commission or a proportion to the size oftrade. This paper deals with the case where there is only proportional transaction costs; for a reviewof constant cost or a mixture of both, see Altarovici et al. (2016) and references therein. Magilland Constantinides (1976) are the first to introduce proportional transaction costs into Merton’smodel. They provide a valuable insight on the optimal strategy; i.e. an investor should maintainthe fraction of wealth in risky assets inside a so-called no-trading region and trading only takesplace along the boundary of the no-trading region. As a consequence, the crucial question is: howto identify the optimal no-trading region which corresponds to the optimal trading strategy?
Under certain restricted settings, this question has been partially answered. When the market isconfined to consist of a single risky asset and a bank account, Davis and Norman (1990) give a
Date : November 6, 2017.A. Fahim is partially supported by the NSF (DMS-1209519). a r X i v : . [ q -f i n . P M ] N ov UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 2 rigorous analysis of the classical Merton’s problem with proportional transaction costs over infi-nite time horizon. The optimal policy is formulated as a nonlinear free boundary problem whichseparates the buying and the selling regions from the no-trading one. Their paper contains detailedcharacterization, both theoretical and numerical, of the value function and optimal policies undercertain assumptions. Shreve and Soner (1994) relax assumptions of Davis and Norman (1990)’sproblem, and apply the viscosity solution approach to provide regularity and existence results.Many other papers have carried out an asymptotic analysis including Janeˇcek and Shreve (2004),Goodman and Ostrov (2010), and Kallsen et al. (2010). A thorough convergence proof for generalutility functions is studied by Soner and Touzi (2013), and an extension to several risky assets isconsidered by Possamaï et al. (2012). Other numerical schemes have been proposed by Tourin andZariphopoulou (1994) and Tourin and Zariphopoulou (1997) for general utility functions, and byMuthuraman and Kumar (2006) for a model with more than one risky asset. Nevertheless, thesepapers only deal with the infinite horizon scenario where the no-trading region does not evolve intime, and are based on finite difference/element method which are not efficient in higher dimen-sions.Theoretical analysis on the finite-time problem has been studied recently and is restricted to theno consumption case with a single risky asset. Liu (2004) first shows analytical properties of theoptimal investment problem with a deterministic finite horizon. Dai and Yi (2009) establish a linkbetween the singular control problem and the obstacle problem, and completely characterize thebehaviors of the resulting free boundaries. Numerical solution of this optimal investment problemis proposed by Arregui and Vázquez (2012). More recently, there is a plethora of literature devotedto the characterization of optimal investment-consumption strategy. Dai et al. (2009) consider theinvestment and consumption optimization decision in finite time horizon, and characterize thebehaviors of free boundaries for a single risky asset case. Dai and Zhong (2008) propose thepenalty method to demonstrate the numerical solution to a singular control problem arising fromportfolio selection with proportional transaction costs. Bichuch (2012) provides a proof to thesame problem with power utility function by expanding the value function into a power series, andobtains a “nearly optimal” strategy.In the present paper, we propose a numerical scheme based on Monte Carlo simulation for theoptimal investment-consumption problem with proportional transaction costs and deterministictime horizon. As discussed in the next section, the value function of such control problem ischaracterized by a
Hamilton-Jacobi-Bellman (HJB) equation . The existing numerical schemesfor this HJB equation in the literature including Tourin and Zariphopoulou (1994), Tourin andZariphopoulou (1997) and Muthuraman and Kumar (2006) are based on finite difference/elementmethod, which are only practical in low dimensional problems. Moreover, the dimension canbe higher in many applications, especially in finance problems. Thus, we propose a numericaltechnique that combines Monte Carlo simulation with finite difference discretization so as to solvethe nonlinear double obstacle problem, and aim to characterize the free boundaries and qualitativeproperties of the solution.Our numerical scheme is strongly motivated by the aforementioned work of Fahim et al. (2011)who introduce the backward probabilistic numerical scheme combined with Monte Carlo and fi-nite difference method for high-dimensional fully nonlinear partial differential equations. Theydecompose the scheme into two steps. First, the Monte Carlo step includes isolating the lineargenerator of some underlying diffusion process to split the PDE into this linear part and a remain-ing nonlinear one. Then, a projection method is employed to evaluate the remaining nonlinear partof the PDE. In this paper, we will modify the numerical method to incorporate the free boundaries
UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 3 on the no-trading region. Moreover, we will show that the proposed method can work in the caseof correlated stocks. It is worth noticing that the type of free boundaries in this current problem isdifferent from the obstacle problem such as the one in Bayraktar and Fahim (2014) and thereforethe scheme developed in this paper is not in the same nature of Monte Carlo scheme. We believethe motivation behind this proposed method can be extended to various HJB for singular controlproblems.This paper is organized as follows. In Section 2 the optimal investment and consumption prob-lem with proportional transaction costs is presented. Section 3 is dedicated to some simplificationsof the control problem in Section 2. The numerical scheme composed of Monte Carlo simulationand finite difference discretization is proposed in Section 4. In Section 5, we show that the imple-mentation of the proposed numerical scheme is compatible with the theoretical results in Dai et al.(2009) and Chen and Dai (2013) in a single risky asset or two risky assets cases. Several examplesthat illustrate performances of the proposed numerical method are also presented in this section.And the last section draws some conclusion.2. T
HE OPTIMAL INVESTMENT - CONSUMPTION PROBLEM
We consider an optimal investment-consumption problem in finite time horizon T ∈ (0 , ∞ ) with proportional transaction costs, the model being the same in Dai and Zhong (2008) and Chenand Dai (2013).Suppose a continuous time market consisting of one risk-free asset and multiple risky assetsavailable for investment. The risk-free asset (bank account), denoted by S t , pays an interest rate r > continuously and thus can be expressed as dS t = rS t dt. (2.1)Let N be the number of available risky investments, called “stocks” hereafter. The N stockshave constant mean rates of return α , α , · · · , α N . We denote the vector of N stock prices by S t = ( S t , S t , · · · , S Nt ) (cid:48) and the mean rates of return by α = ( α , α , · · · , α N ) (cid:48) . The evolution ofstocks can be written as dS t = diag ( S t )( αdt + σdB t ) , (2.2)where diag ( S t ) is the N × N matrix formed with elements of S t as its diagonal, σ denotes the N × N positive definite covariance structure, and { B t : t ∈ [0 , T ] } is a standard N -dimensionalBrownian motion defined on a filtered probability space (Ω , F , {F t } ≤ t ≤ T , P ) .Assume that an investor holds a portfolio ( X t , Y t ) (cid:48) = ( X t , Y t , · · · , Y Nt ) (cid:48) , where X t and Y it aredollar amount invested in the bank account and in the i th stock at time t . His problem is to choosea consumption and investment strategy over the deterministic horizon in order to maximize hisobjective: the discounted utility of consumption during the investment period. We require that theconsumption c t must be non-negative and occur from cash in the bank, and its process c t should beadapted to F t and integrable for any finite t , that is, (cid:90) t c s ds < ∞ ∀ t ≥ . (2.3)Now we introduce two F t -adapted processes L t = ( L t , · · · , L Nt ) (cid:48) and M t = ( M t , · · · , M Nt ) (cid:48) which are non-negative, non-decreasing, and right continuous with left limits (RCLL). L it repre-sents the cumulative dollar value spent for the purchase of stock i before incurring transactioncosts, whereas M it represents the cumulative amount of money obtained from the sale of stock i . Denote the transaction costs for buying and selling stocks by λ = ( λ , λ , · · · , λ N ) (cid:48) ≥ and UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 4 µ = ( µ , µ , · · · , µ N ) (cid:48) ≥ respectively. To be more precise, buying a unit of stock i will cost (1 + λ i ) in cash from the bank and selling a unit of stock i will receive (1 − µ i ) in cash added intothe bank. We assume that λ i + µ i > , i = 1 , , · · · , N . With transaction costs and consumption,the controlled evolution of X t and Y t can be described by the following equations dX t = ( rX t − c t ) dt − ( e + λ ) · dL t + ( e − µ ) · dM t , (2.4) dY t = diag ( Y t ) [ αdt + σdB t ] + dL t − dM t . (2.5)Here, “ · ” is the standard dot product and e is a vector of ones with appropriate length.We require the investor’s net wealth at any time to be positive because he would not be bankruptif he is forced to liquidate his position. If taking transaction costs into consideration, the investor’snet wealth at time t is given by X t + (cid:80) Ni =1 min [(1 + λ i ) Y it , (1 − µ i ) Y it ] . Therefore, we define the solvency region S λ,µ as S λ,µ = (cid:40) ( x, y ) ∈ ( R , R N ) : x + N (cid:88) i =1 min [(1 + λ i ) y i , (1 − µ i ) y i ] ≥ (cid:41) . (2.6)Given an initial position ( X , Y ) (cid:48) = ( x, y ) (cid:48) ∈ S λ,µ , an investment-consumption strategy ( c t , L t , M t ) is called admissible if and only if the portfolio position ( X t , Y t ) lies in S λ,µ for all t ∈ [0 , T ) . Let A t ( x, y ) be the set of admissible strategies. The investor’s objective consists of choosing an ad-missible strategy so as to maximize the expected discounted utility of accumulative consumptionand the terminal wealth, that is, sup ( c t ,L t ,M t ) ∈A ( x,y ) E x,y (cid:20)(cid:90) T e − βt U ( c t ) dt + e − βT U ( W T ) (cid:21) , (2.7)where β > is the discount factor, E x,yt denotes the conditional expectation at time t given aninitial endowment X t = x , Y t = y , W T is the terminal net wealth given by W T = X T + (cid:80) Ni =1 min [(1 + λ i ) Y iT , (1 − µ i ) Y iT ] , and U is the utility function which belongs to the class of Constant Relative Risk Aversion (CRRA) utility functions, i.e. U ( c ) = c γ γ if γ < , γ (cid:54) = 0 , log( c ) if γ = 0 . (2.8)Here γ is the relative risk aversion coefficient that describes the investor’s risk preference. Theseutility functions are well-known and have been used very wildly in modelling the risk preferenceof an investor. Then we define the value function by V ( x, y, t ) = sup ( c t ,L t ,M t ) ∈A t ( x,y ) E x,yt (cid:20)(cid:90) Tt e − β ( s − t ) U ( c s ) ds + e − β ( T − t ) U ( W T ) (cid:21) , (2.9)for ( x, y ) ∈ S λ,µ , t ∈ [0 , T ) . 3. T HE HJB
EQUATION AND SCALING
By applying the dynamic programming arguments [cf. Section IV.3, Fleming and Soner (2006)],the value function V of the stochastic control problem (2.9) satisfies the following Hamilton-Jacobi-Bellman (HJB) equation:
UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 5 (cid:26) − ∂ t V − N (cid:88) i,j =1 a ij y i y j ∂ y i y j V − N (cid:88) i =1 α i y i ∂ y i V − rx∂ x V + βV − U ∗ ( ∂ x V ) , min i (cid:2) − (1 − µ i ) ∂ x V + ∂ y i V (cid:3) , min i (cid:2) (1 + λ i ) ∂ x V − ∂ y i V (cid:3)(cid:27) , (3.1)with the terminal condition V ( x, y, T ) = U (cid:32) x + N (cid:88) i =1 min (cid:2) (1 + λ i ) y i , (1 − µ i ) y i (cid:3)(cid:33) , (3.2)where a = [ a ij ] Ni,j =1 = σσ (cid:48) and U ∗ ( ν ) = sup c ≥ { U ( c ) − cν } = − γγ ( ν ) γγ − if γ < , γ (cid:54) = 0 , − log( ν ) − if γ = 0 . In this paper, we focus on the computational scheme to solve equation (3.1) with terminal condition(3.2).
Remark . Equation (3.1) can be interpreted in the variational inequality sense, i.e.(1) The value function V satisfies all three following inequalities ≤ − ∂ t V − N (cid:88) i,j =1 a ij y i y j ∂ y i y j V − N (cid:88) i =1 α i y i ∂ y i V − rx∂ x V + βV − U ∗ ( ∂ x V ) , ≤ min i (cid:2) − (1 − µ i ) ∂ x V + ∂ y i V (cid:3) , ≤ min i (cid:2) (1 + λ i ) ∂ x V − ∂ y i V (cid:3) . (2) If < min i (cid:2) − (1 − µ i ) ∂ x V + ∂ y i V (cid:3) and < min i (cid:2) (1 + λ i ) ∂ x V − ∂ y i V (cid:3) , we must have − ∂ t V − N (cid:88) i,j =1 a ij y i y j ∂ y i y j V − N (cid:88) i =1 α i y i ∂ y i V − rx∂ x V + βV − U ∗ ( ∂ x V ) Following Dai and Zhong (2008), we use the homothetic property of the value function to reducethe dimensionality of the problem for further numerical analysis. For any constant ρ > , the“homothetic property” of the value function is as follows: V ( ρx, ρy, t ) = ρ γ V ( x, y, t ) if γ < , γ (cid:54) = 0 , (cid:0) − e − β ( T − t ) β + e − β ( T − t ) (cid:1) log( ρ ) + V ( x, y, t ) if γ = 0 . This property allows us to reduce the dimension of the original problem from N + 1 to N byadopting the wealth fraction as state variables. Indeed, we define a new function ϕ ( y, t ) = V (1 − e · y, y, t ) , (3.3)where e is a vector of ones with length N , and y represents the vector of the fraction of wealthinvested in each stock when the total wealth w is one ( w = 1 ). It is clearly sufficient to compute UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 6 ϕ ( y, t ) since the original value function is then given by V ( x, y, t ) = ϕ (cid:18) yx + e · y , t (cid:19) ( x + e · y ) γ . The derivation of the HJB equation and the computational procedure for both the log utilityand the power utility functions are the same. Therefore we provide a detailed description of thepower utility case only. In terms of ϕ ( y, t ) , the HJB equation in (3.1) for the power utility function( U ( c ) = c γ /γ ) becomes (cid:110) − ∂ t ϕ − ˆ L ϕ, min i ˆ S i ϕ, min i ˆ B i ϕ (cid:111) , (3.4)with the terminal condition ϕ ( y, T ) = γ − (cid:16) N (cid:88) i =1 min {− µ i y i , λ i y i } (cid:17) γ for y ∈ Θ N , where Θ N = { ( y , y , · · · , y N ) ∈ R N : 1 + N (cid:88) i =1 min {− µ i y i , λ i y i } ≥ } , and ˆ L ϕ = 12 N (cid:88) i,j =1 η ij ∂ y i y j ϕ + N (cid:88) i =1 b i ∂ y i ϕ − ϑϕ + 1 − γγ (cid:32) γϕ − N (cid:88) i =1 y i ∂ y i ϕ (cid:33) γγ − , (3.5) ˆ S i ϕ = (cid:34) µ i γϕ − N (cid:88) k =1 ( − δ ik + µ i y k ) ∂ y k ϕ (cid:35) , (3.6) ˆ B i ϕ = (cid:34) λ i γϕ − N (cid:88) k =1 ( δ ik + λ i y k ) ∂ y k ϕ (cid:35) , (3.7)with η ij = y i y j N (cid:88) k =1 N (cid:88) (cid:96) =1 a k(cid:96) ( δ ik − y k )( δ j(cid:96) − y (cid:96) ) , (3.8) b i = 12 N (cid:88) k =1 N (cid:88) (cid:96) =1 a k(cid:96) y k y (cid:96) ( γ − δ ik + δ i(cid:96) − y i ) + N (cid:88) k =1 y k ( δ ik − y i )( α k − r ) , (3.9) ϑ = β − γ (cid:32) r + 12 N (cid:88) k =1 N (cid:88) (cid:96) =1 a k(cid:96) y k y (cid:96) ( γ −
1) + N (cid:88) k =1 ( α k − r ) y k (cid:33) . (3.10)Here δ ij represents the Kroneker index with δ ij = 1 if i = j and δ ij = 0 otherwise. The abovedimension reduction technique has been wildly used; see, for example, Dai et al. (2009) for N = 1 ,and Muthuraman and Kumar (2006) for N = 2 without the time variable.We consider a portfolio which consists of a risk-free asset and two risky assets ( N = 2 ) forillustration purpose. Before adopting the homothetic property, this problem is three dimensions,and the polygon cone in Figure 1(a) is the no-trading region. When we apply the homotheticproperty and rewrite the value function V ( x, y, t ) by ϕ ( y, t ) defined in (3.3), we can reduce the UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 7 x y y (a) Homotheticity property y y − λ µ µ − λ N ∩ N N ∩ S N ∩ B S ∩ S S ∩ N S ∩ B B ∩ B B ∩ N B ∩ S Θ (b) Θ region and Trading strategy F IGURE
1. Trading and no-trading regions along the w = 1 cut at time t problem to two dimensions. The red region shown in Figure 1(a) represents the no-trading regionwith the wealth equals one cut at time t after dimension reduction.Now we define the following representations for later use. Let B i = { ( y, t ) ∈ Θ N × [0 , T ) : ˆ B i ϕ = 0 } , (3.11) S i = { ( y, t ) ∈ Θ N × [0 , T ) : ˆ S i ϕ = 0 } , (3.12) N i = Θ N × [0 , T ) \ ( B i ∪ S i ) , (3.13)where B i , S i and N i represent the buying region, selling region, and no-trading region with respectto the i th stock. For illustration, we consider the case of N = 2 as well. Figure 1(b) shows thatthe domain Θ at time t is partitioned into different regions along the wealth equals one cut. In thearea filled with gray, the investor should buy or sell one stock just enough to push the fraction backto the no-trading region N ∩ N . In the area filled with blue, it is not possible to trade only onestock to make the fraction reaching the no-trading region. Thus, two stocks should be transactedsimultaneously to reach the corner of inaction region N ∩ N .Although we can formulate the value function as the HJB equation presented in (3.4), the com-plete analytical solution cannot be obtained. Also, the standard numerical methods, such as finitedifference method and finite element method, work only for low dimensional cases. Due to thecurse of dimensionality and the lack of an exact solution, the development of appropriate numer-ical methods are highly desirable to approximate the solution and provide qualitative propertiesunder different model parameter settings. Therefore, in the next section we propose a numericalscheme that combines Monte Carlo simulation with finite difference discretization in order to solvethis nonlinear variational inequality problem. UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 8
4. N
UMERICAL M ETHOD
As mentioned before, the main goal of this paper is to propose an appropriate numerical methodin order to approximate the solution of the optimal investment-consumption problem and conse-quently obtain the trading strategies. We first notice that the main difficulties associated with thenumerical solution of the HJB in (3.4) are twofold:(1) the free boundary feature related to the double obstacle problem,(2) the equation presents a nonlinear term in (3.5), (cid:16) γϕ − (cid:80) Ni =1 y i ∂ y i ϕ (cid:17) γγ − .4.1. The two-step procedure.
In order to overcome the free boundary feature, we will use a two-step procedure which extends the idea proposed by Muthuraman and Kumar (2006). Step 1 solvesthe nonlinear second order PDE in the first part of the HJB equation while step 2 updates the valuefunction in different regions of the domain Θ N . To be more precise, we begin by finding the values ϕ ( y, t ) for all y in the domain Θ N such that the first part of HJB equation in (3.4) holds true, thatis, − ∂ t ϕ − ˆ L ϕ = 0 for y ∈ Θ N , (4.1)with the boundary condition ϕ ( y, t ) = (cid:40) if γ > , −∞ if γ < , for y ∈ ∂ Θ N , t ∈ [0 , T ) . The reason that we set the boundary condition in (4.1) to be zero or negative infinity is because onthe boundary of solvency region, the investor is forced to liquidate his position at any time t andhis net wealth on the boundary is zero. Since we consider the power utility function, the utility ofconsumption is zero for γ > and negative infinity for γ < because of zero net wealth. For thepower utility function, we can set ϕ ( y, t ) = (cid:40) if γ > , −∞ if γ < , for y ∈ ∂ Θ N ∪ ( R N \ Θ N ) , t ∈ [0 , T ) because the investor’s position never exits the solvencyregion. This will later become useful in the numerical implementation where we have to define thevalue function at the discrete points outside the region Θ N .We also require that the other two formulas in (3.4) should be satisfied in the domain Θ N . Hence,in the next step our procedure deals with the free boundary terms ˆ S i ϕ and ˆ B i ϕ in (3.6) and (3.7) forall i = 1 , , · · · , N . We have to find the point y ∗ where ˆ S i ϕ ( y ∗ , t ) and/or ˆ B i ϕ ( y ∗ , t ) are negative,and then adjust the value at the point such that the variational inequalities hold true. Denote thetrading strategy for buying and selling stocks at ( y ∗ , t ) by υ ( y ∗ ,t ) = ( { ˆ B ( y ∗ ,t ) < } , · · · , { ˆ B N ( y ∗ ,t ) < } ) (cid:48) , and (cid:37) ( y ∗ ,t ) = ( { ˆ S ( y ∗ ,t ) < } , · · · , { ˆ S N ( y ∗ ,t ) < } ) (cid:48) , where is an indicator function. We update the value function at ( y ∗ , t ) by ϕ ( y ∗ , t ) = ϕ (¯ y, t ) (cid:32) (cid:80) Ni =1 λ i y ∗ i υ ( y ∗ ,t ) i − (cid:80) Ni =1 µ i y ∗ i (cid:37) ( y ∗ ,t ) i (cid:80) Ni =1 λ i ¯ y i υ ( y ∗ ,t ) i − (cid:80) Ni =1 µ i ¯ y i (cid:37) ( y ∗ ,t ) i (cid:33) γ . (4.2)Here ¯ y is the point that satisfies the following conditions: UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 9 − λ µ − λ µ Θ Buy y Sell y Buy y Buy y Sell y Sell y Sell y Buy y Buy y Sell y B u y y S e ll y ˆ y ˇ y ¯ yy ∗ F IGURE
2. Illustration for the two-step procedure with N = 2 at time t (1) ¯ y is the point that is closest to y ∗ along the characteristic curves in the region which includes y ∗ ,(2) ¯ y is the point on the boundary of the no-trading region facing the region to which y ∗ be-longs.This procedure will be repeated backward in time until a sequence of no-trading regions and tradingstrategies are obtained at each time step.For illustration purpose, we use Figure 2 to convey the idea of the two-step procedure when N = 2 at a given time t . The region inside the blue diamond refers to the domain Θ . First, wehave to solve ϕ ( y, t ) for y ∈ Θ satisfying the nonlinear second order partial differential equationin (4.1). Once the values ϕ ( y, t ) are known, we check the gradient constraints ˆ S i ϕ and ˆ B i ϕ for i = 1 , . The no-trading region filled with red meets the conditions ˆ S i ϕ ≥ and ˆ B i ϕ ≥ for i = 1 , , and thus the value ϕ ( y, t ) in this region do not need to be changed. However, forexample, if y ∗ = ( y ∗ , y ∗ ) marked green in Figure 2 is the point such that ˆ B ϕ ( y ∗ , t ) < and ˆ S ϕ ( y ∗ , t ) < , y ∗ is classified as an element in the set B ∩ S and also its value ϕ ( y ∗ , t ) shouldbe adjusted to meet the conditions ˆ B ϕ ( y ∗ , t ) = 0 and ˆ S ϕ ( y ∗ , t ) = 0 . In Figure 2, ¯ y = ( ¯ y , ¯ y ) marked green is the point which is closest to the point y ∗ along the characteristic curves and also onthe boundary of the no-trading region facing the region to which y ∗ belongs. Therefore we updatethe value function by ϕ ( y ∗ , t ) = ϕ (¯ y, t ) (cid:18) λ y ∗ − µ y ∗ λ ¯ y − µ ¯ y (cid:19) γ . It means that the investor should buy the first stock and sell the second one to rebalance his positionso as to reach the corner of no-trading region. For another example, ˆ y = (ˆ y , ˆ y ) marked orange in UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 10
Figure 2 is the point such that ˆ B ϕ (ˆ y, t ) < , and then we have to adjust its value function by ϕ (ˆ y, t ) = ϕ (ˇ y, t ) (cid:18) λ ˆ y λ ˇ y (cid:19) γ , where ˇ y is the point marked orange in Figure 2 that is closest to ˆ y along the characteristic curve inthe B region.4.2. The computational scheme for solving the nonlinear second order PDE.
Due to the curseof dimensionality and the lack of an analytical solution, an appropriate numerical method for solv-ing high-dimensional fully nonlinear PDEs is highly desirable. Our numerical method is mainlymotivated by the recent work of Fahim et al. (2011). The computational scheme they providedconsists of two parts. First, the Monte Carlo step includes isolating the linear generator of someunderlying diffusion process to split the PDE into this linear part and a remaining nonlinear one.Next, discrete-time finite difference approximation is applied to evaluate the remaining nonlinearpart of the PDE along the underlying diffusion process. The first part takes the advantage of thehigh-dimensional property of Monte Carlo method while the second part deals with the nonlin-ear term of the equation. In this paper, we modify the numerical method to incorporate the freeboundaries on the no-trading region.4.2.1.
Notation . We shall first introduce some notations. The collection of n × d matrices withreal entries is denoted by M ( n, d ) . For a matrix A ∈ M ( n, d ) , A (cid:48) represents its transpose and √ A returns square root of each element in the matrix. For A, B ∈ M ( n, d ) , we define A · B := T r [ A (cid:48) B ] . In particular, A and B are vectors of R n when d = 1 and A · B reduces to the standarddot product. D and D are the gradient and the Hessian matrix defined by Dϕ = (cid:16) ∂ϕ∂y , ∂ϕ∂y , · · · , ∂ϕ∂y N (cid:17) (cid:48) and D ϕ = ∂ ϕ∂y ∂ ϕ∂y ∂y · · · ∂ ϕ∂y ∂y N ∂ ϕ∂y ∂y ∂ ϕ∂y · · · ∂ ϕ∂y ∂y N ... ... . . . ... ∂ ϕ∂y N ∂y ∂ ϕ∂y N ∂y · · · ∂ ϕ∂y N . (4.3)Let b = ( b , b , · · · , b N ) (cid:48) be a vector of R N where b i is the coefficient of the first derivative of ϕ with respect to the variable y i in (3.9), and η ∈ M ( N, N ) be a matrix with elements of η ij at row i and column j in (3.8). The diagonal matrix ξ ∈ M ( N, N ) is defined by ξ := diag ( η ) . Next, wedetermine the linear operator L ˆ Y ϕ := ∂ t ϕ + b · Dϕ + 12 ξ · D ϕ. Then the remaining nonlinear parts are represented as F ( y, t, ϕ, Dϕ, D ϕ ) := 12 N (cid:88) i,j =1 ,j (cid:54) = i η ij ∂ y i y j ϕ − ϑϕ + 1 − γγ (cid:32) γϕ − N (cid:88) i =1 y i ∂ y i ϕ (cid:33) γγ − . UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 11
Hence, the problem we have to deal with becomes − L ˆ Y ϕ ( y, t, ϕ, Dϕ, D ϕ ) − F ( y, t, ϕ, Dϕ, D ϕ ) for y ∈ Θ N , t = [0 , T ); (4.4) ϕ ( y, T ) = γ − (cid:16) N (cid:88) i =1 min {− µ i y i , λ i y i } (cid:17) γ for y ∈ Θ N . (4.5)4.2.2. Discretization . As with any numerical scheme, the first step is to discretize the time spaceand the domain of state variables. Let h := T /n be the time step, and t k = kh , k = 0 , , · · · , n fora positive integer n . Suppose we have a uniform grid, denoted by G t k ∆ y , for the domain Θ N withthe grid size ∆ y = (∆ y , ∆ y , · · · , ∆ y N ) (cid:48) in each state variable direction. Denote a discretizedpoint with y = ( y , y , · · · , y N ) (cid:48) ∈ G t k ∆ y at time t k by ( y, t k ) .Let B t be an R N -dimensional standard Brownian motion defined in section 2. Consider theone-step-ahead Euler discretization of the diffusion ˆ Y corresponding to the linear operator L ˆ Y ˆ Y y,t k +1 h := y + b ( y, t k ) h + (cid:112) ξ ( y, t k ) ( B t k +1 − B t k ) . (4.6)If we assume that the nonlinear PDE in (4.1) has a solution, we follow from Itô’s formula andreplace the process ˆ Y by its Euler discretization to get E t k ,y (cid:104) ϕ ( ˆ Y y,t k +1 h , t k +1 ) (cid:105) = ϕ ( y, t k ) + E t k ,y (cid:20)(cid:90) t k +1 t k L ˆ Y ϕ ( ˆ Y s , s, ϕ, Dϕ, D ϕ ) ds (cid:21) , (4.7)where E t k ,y := E [ ·| ˆ Y t k = y ] is the conditional expectation, and D κ is the κ th order partial differ-ential operator with respect to the space variable y defined in (4.3). By approximating the integral,the value function ϕ ( y, t k ) can be evaluated as follows: ϕ ( y, t k ) = E t k ,y (cid:104) ϕ ( ˆ Y y,t k +1 h , t k +1 ) (cid:105) − h L ˆ Y ϕ ( y, t k , D ϕ, D ϕ, D ϕ ) + O ( h ) , (4.8) D κ ϕ := E t k ,y [ D κ ϕ ( ˆ Y y,t k +1 h , t k +1 )] , κ = 0 , , . (4.9)Since ϕ is also a solution to the PDE in (4.4) which means L ˆ Y ϕ ( y, t k , D ϕ, D ϕ, D ϕ ) = − F ( y, t k , D ϕ, D ϕ, D ϕ ) , we have the discretized approximation of the value function as follows: ϕ h ( y, t n ) := γ − (cid:16) N (cid:88) i =1 min {− µ i y i , λ i y i } (cid:17) γ for y ∈ G t n ∆ y , (4.10)and for y ∈ G t k ∆ y , k = 0 , · · · , n − ϕ h ( y, t k ) := E t k ,y (cid:104) ϕ h ( ˆ Y y,t k +1 h , t k +1 ) (cid:105) + h F ( y, t k , D ϕ h , D ϕ h , D ϕ h ) . (4.11)Once the linear operator L ˆ Y ϕ is chosen, the remaining nonlinear parts are handled by meansof classical centered difference approximation. Let e i be the unit vector in the y i direction andthen the first order term ∂ y i ϕ h ( y, t k ) is discretized by the centered difference approximation of thegradient, that is, ∂ y i ϕ h ( y, t k ) ≈ ϕ h ( y + ∆ y i e i , t k ) − ϕ h ( y − ∆ y i e i , t k )2∆ y i . (4.12) UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 12
Algorithm 1
Mixed Monte Carlo Simulation and Finite Difference Method Algorithm
Output:
The value function ϕ h ( y, t ) , and the optimal buying and selling boundaries Let h := T /n and t k = kh , k = 0 , , · · · , n be the time step Discretize the domain Θ N into uniform grid, denoted by G t k ∆ y , with the grid size ∆ y =(∆ y , ∆ y , · · · , ∆ y N ) (cid:48) in each state variable direction for each y ∈ G t n ∆ y do Set the value function ϕ h ( y, t n ) at time t n according to its terminal condition in (4.10) Evaluate ∂ y i ϕ h ( y, t n ) and ∂ y i y j ϕ h ( y, t n ) for i, j = 1 , , · · · , N in each state variable direc-tion by centered-difference approximation in (4.12) and (4.13) end for for (cid:96) = n − ; (cid:96) ≥ ; (cid:96) = (cid:96) − do for each y ∈ G t (cid:96) ∆ y do Generate M sample paths of ˆ Y y,t (cid:96) +1 h by the one-step-ahead Euler discretization in (4.6) Estimate the values ϕ h ( ˆ Y y,t (cid:96) +1 h , t (cid:96) +1 ) , ∂ y i ϕ h ( ˆ Y y,t (cid:96) +1 h , t (cid:96) +1 ) , and ∂ y i y j ϕ h ( ˆ Y y,t (cid:96) +1 h , t (cid:96) +1 ) bylinear interpolation if the simulated point ˆ Y y,t (cid:96) +1 h is not on the grid Approximate D κ ϕ h for κ = 0 , , in (4.9) by ˆ E M [ D κ ϕ h ( ˆ Y y,t (cid:96) +1 h , t (cid:96) +1 )] corresponding tothe sample size M Compute ϕ h ( y, t (cid:96) ) based on (4.11) end for for each y ∈ G t (cid:96) ∆ y do Find the grid point y ∗ where ˆ S i ϕ h ( y ∗ , t (cid:96) ) < and/or ˆ B i ϕ h ( y ∗ , t (cid:96) ) < for i = 1 , , · · · , N in (3.6) and (3.7), and then adjust the value ϕ h ( y ∗ , t (cid:96) ) by (4.2) such that the ˆ S i ϕ h ( y ∗ , t (cid:96) ) =0 and/or ˆ B i ϕ h ( y ∗ , t (cid:96) ) = 0 end for end for The cross derivative term ∂ y i y j ϕ h is discretized as follows ∂ y i y j ϕ h ( y, t k ) ≈ y i ∆ y j (cid:34) ϕ h ( y + ∆ y i e i + ∆ y j e j , t k ) + ϕ h ( y − ∆ y i e i − ∆ y j e j , t k ) − ϕ h ( y + ∆ y i e i − ∆ y j e j , t k ) − ϕ h ( y − ∆ y i e i + ∆ y j e j , t k ) (cid:35) . (4.13)Once we have the set of one-step-ahead random path simulations ˆ Y y,t k +1 h , the iteration com-putes the discrete solution ϕ h ( y, t k ) at time t k from ϕ h ( y, t k +1 ) by (4.9)-(4.11). Note that if thesimulated point ˆ Y y,t k +1 h is not on the grid G t k ∆ y , we will approximate the value ϕ h ( ˆ Y y,t k +1 h , t k +1 ) , ∂ y i ϕ h ( ˆ Y y,t k +1 h , t k +1 ) , and ∂ y i y j ϕ h ( ˆ Y y,t k +1 h , t k +1 ) by interpolation. The interpolated value at a querypoint is based on linear interpolation of the values at neighboring grid points in each respectivedimension.In view of the above interpretation associated with the value function, our numerical schemestudied in this paper can be expressed as a mixed Monte Carlo simulation and finite differencemethod. The Monte Carlo portion includes the choice of an underlying diffusion process while the UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 13 finite difference portion consists of the derivative approximation of the remaining nonlinearity. Wesummarize the two-step iterative procedure in Algorithm 1.
Remark . The numerical method in Algorithm 1 is inspired by Fahim et al. (2011) where theydeveloped a Monte Carlo scheme for fully nonlinear PDEs of the form (cid:40) L v + G ( y, t, v, Dv, D v ) v ( T, y ) = g ( y ) where L is a linear parabolic operator and G is a nonlinear parabolic operator. In the numericalscheme of Fahim et al. (2011), they use the linear parabolic operator L to generate sample pathsof the diffusion process. Therefore, one has some flexibility in choosing the underlying diffusionprocess of the samples paths; e.g. one can also choose a linear parabolic operator L to generatethe diffusion sample paths as long as the nonlinear term F ( y, t, ϕ, Dϕ, D ϕ ) := ( L − L ) ϕ + G ( y, t, ϕ, Dϕ, D ϕ ) , remains parabolic. G ( y, t, r, p, γ ) : R N × [0 , T ] × R × R n × M ( N, N ) → R is called parabolic if ∇ γ G is positive definite where ∇ denotes the vector differential operator.The numerical scheme in Algorithm 1 sets L ˆ Y ϕ := L ϕ = ∂ t ϕ + b · Dϕ + 12 diag ( η ) · D ϕ and leaves the off-diagonal terms N (cid:88) i,j =1 ,j (cid:54) = i η ij ∂ y i y j ϕ for the nonlinear part. It is simply because the diffusion process simulated by this parabolic op-erator is less complicated when we have only diagonal elements. On the other hand, inclusionof off-diagonal second order derivative terms does not affect the sufficient conditions in Fahimet al. (2011) for the convergence of the numerical scheme, i.e. consistency, stability and mono-tonicity. For instance, monotonicity in Fahim et al. (2011) is guaranteed by the assumption that Tr[ diag ( η ) − ∇ γ F ] ≤ ; that is, the diffusion coefficient in L dominates the derivative of thenonlinear operator F with respect to the component of D v . Since L has the diagonal ele-ments of the second order derivative and L − L has only the off-diagonal elements, we have Tr[ diag ( η ) − ∇ γ F ] = Tr[ diag ( η ) − ∇ γ G ] .It is worth mentioning that the adjustment in Step 15 of Algorithm 1 to handle the free boundarymakes it difficult to show the scheme is monotone. Therefore, we restrict our study to the numericalconvergence of the proposed scheme.5. N UMERICAL R ESULTS
The objectives in this section are: (1) to examine the performance of the mixed Monte Carlosimulation and finite difference method algorithm applying on the investment-consumption opti-mization problem; (2) to indicate the behaviors of optimal trading strategies.5.1.
Test 1.
In this first example, we consider the following set of financial parameters: N = 1 , r = 0 . , α = 0 . , σ = 0 . , β = 0 . , γ = 0 . , µ = λ = 0 . , and solve the problem for the time interval t ∈ [0 , . The theoretical properties of the solutionand free boundaries to the problem (3.4) for N = 1 case are presented in Dai et al. (2009). We UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 14 − . − . . . . . . . . F r a c t i o n o f w e a l t h i n s t o c k ˆ B t ˆ S t (a) α = 0 . − . − . . . . . . . . . . . . . . F r a c t i o n o f w e a l t h i n s t o c k α = 0 . α = 0 . α = 0 . (b) Varying α F IGURE
3. The estimated selling, buying, and no-trading regions for the N = 1 case. Test 1 parameters: r = 0 . , σ = 0 . , µ = λ = 0 . , γ = 0 . , β = 0 . will use the following proven statements to verify the numerical results obtained from our mixedMonte Carlo / finite difference method.Let τ = 1 α − r log (cid:16) λ − µ (cid:17) and ˜ y = − α − r − (1 − γ ) σ α − r . (5.1)According to Theorem 5.4 in Dai et al. (2009), the two free boundaries B t and S t for the N = 1 case should satisfy the following properties:(1) for t ∈ [0 , T ) , B t < S t , and S t ≥ S T − = 11 + (1 − µ )˜ y ; moreover, S t = 1 if α − r − (1 − γ ) σ = 0 ,S t > if α − r − (1 − γ ) σ > ,S t < if α − r − (1 − γ ) σ < . (2) for t ∈ [0 , T ) , B t ≤
11 + (1 + λ )˜ y , and B t = 0 if and only if t ∈ [ T − τ, T ) . Now we have the values α − r − (1 − γ ) σ ≈ − . < and τ ≈ . so that the selling andbuying boundaries should satisfy
11 + (1 − µ )˜ y ≈ . ≤ S t < for t ∈ [0 , T ); B t ≤
11 + (1 + λ )˜ y ≈ . for t ∈ [0 , T − τ ); and B t = 0 for t ∈ [ T − τ, T ) . UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 15
Concerning the numerical method, we use the time step h = 0 . and uniform grid with length ∆ y = 0 . . Another numerical parameter that we have used is the number of simulated samplepaths M = 10 . Figure 3(a) shows the numerical approximation of the optimal trading strategiesin the fraction of wealth in stock at each time step. The upper function is the selling boundarywhile the lower one is the buying boundary. Clearly, these two boundaries depend on time, andthe no-trading region is between these two boundaries. First, we have verified that the theoreticalproperties are satisfied for all the grid points at every discrete time step. Also, it shows that thevalue of the buying boundary tends to zero as the time is greater than T − τ ≈ which indicates thatit is suboptimal to buy a risky asset soon as the finite horizon is approaching. This phenomenon,known as “no-buying near maturity”, was first proved by Dai et al. (2009) with consumption andtransaction costs in finite time horizon. Furthermore, the optimal selling boundary is always greaterthan the buying one which mainly points out that a risk averse investor prefers to buy low and sellhigh.Figure 3(b) shows the optimal trading boundaries with varying α . We can observe that boththe buying and selling boundaries increase as the value of α increases, which indicates that theinvestor should hold a larger fraction of wealth in risky asset when the return of risky asset ishigher. If α < . , the selling boundary is less than one which means it is always suboptimalto leverage. However, leverage will be needed if α > . . The obtained numerical results areagain in full agreement with the theoretical properties stated in Dai et al. (2009) Theorem 5.4.5.2. Test 2.
In this second numerical test, the following financial parameter values have beenconsidered: N = 2 , r = 0 , β = 0 . , γ = 0 . , µ = λ = µ = λ = 0 . α = 0 . , α = 0 . , a = 0 . , a = 0 . , (a) positive correlated: a = a = 0 . , (b) negative correlated: a = a = − . , and the investment period is set to be one year ( T = 1 ). In this case, we investigate the optimaltrading strategy for a risk averse investor who can access two positively or negatively correlatedstocks as well as a risk-free asset. We use the time step h = 0 . , uniform grid with length ∆ y = (0 . , . in each dimension, and the number of simulated sample paths M = 10 .Figure 4 shows the decomposition of the domain Θ into selling ( S i , i = 1 , ), buying ( B i , i =1 , ), and no-trading ( N i , i = 1 , ) regions at time t = 0 . for the two positively correlated stockscase. It can be observed that the domain Θ is partitioned into nine different regions, with theno-trading region N ∩ N in the center surrounded by trading regions S ∩ S , S ∩ N , S ∩ B , N ∩ B , B ∩ B , B ∩ N , and B ∩ S in the clockwise order. In addition, the four intersections ∂S ∩ ∂S , ∂S ∩ ∂B , ∂B ∩ ∂S , and ∂B ∩ ∂B are a singleton. This means that if the initialportfolio position is in B ∩ S , for example, the investor should buy the first stock and sell thesecond one to reach the unique corner ∂B ∩ ∂S . The phenomena we observed are consistent withrigorous analysis results proved in Chen and Dai (2013).The numerical approximation of the no-trading region at different time steps is provided inFigure 5 for both the two positively and negatively correlated stocks cases. Here the expectedrate of return for the first stock α = 14% is more than that of the second stock α = 12% andtransaction costs for buying and selling stocks are kept equal. Since the first stock gives a higherrate of return, as expected the investor will not only put more fraction in the first stock but have a UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 16 − − − − − − − − y ) F r a c t i o n o f w e a l t h i n t h e s ec o nd s t o c k ( y ) (a) Θ region and trading strategy − − . . . − − . . . N ∩ N N ∩ S N ∩ B S ∩ S S ∩ N S ∩ B B ∩ B B ∩ N B ∩ S Fraction of wealth in the first stock ( y ) F r a c t i o n o f w e a l t h i n t h e s ec o nd s t o c k ( y ) (b) Enlarged no-trading region F IGURE
4. The estimated selling, buying, and no-trading regions for the N = 2 case at time t = 0 . . Test 2 parameters: r = 0 , α = 0 . , α = 0 . , a =0 . , a = 0 . , a = a = 0 . , µ = λ = µ = λ = 0 . , γ =0 . , β = 0 . larger inhibition to trade the first one. In Figure 5(a) since these two stocks are positive correlated,the region of inaction can only elongate along the main diagonal. An explanation of this behavioris that the investor does not loose much by having more fraction in one stock and less in theother because one partially hedges the other. On the other hand, the result for the two negativelycorrelated stocks case is displayed in Figure 5(b). As we can see from the figure, the no-tradingregion elongates along the anti-diagonal direction. This implies that when the price of one performsworse than usual, the other will likely do better than usual. The gain in one stock is therefore likelyto offset the loss in the other. Hence, the investor does not loose much by having more fraction inboth stocks. These observations are the same as the results obtained in Muthuraman and Kumar(2006) for the infinite time horizon problem. Moreover, we can observe that B = 0 and B = 0 when the time approaches to the maturity of the investment period, which confirms the “no-buyingnear maturity” phenomenon. Remark . Muthuraman and Kumar (2006) provide a computational method to solve the portfoliooptimization problem with infinite horizon. They use an iterative scheme that adapts the finiteelement method in order to capture the region of inaction (please see Muthuraman and Kumar(2006) for more detail). Their problem does not depend on time, so it can be focused on findingthe no-trading region only. However, if we consider the same optimization problem with finitetime horizon, different trading regions should be characterized in order to adjust the value functionbased on the different regions.We notice that if we follow the numerical scheme proposed by Muthuraman and Kumar (2006)and adapt the implicit finite difference method instead, we obtain the same no trading region.However, the estimated buying and selling regions are not acceptable. Take N = 2 for example.Figure 6 shows the comparison of results at time t = 0 . obtained by the mixed Monte Carlosimulation and finite difference method we proposed and the iterative scheme adapting the implicitfinite difference method using the same parameter settings in Test 2(a). Observe that when theiterative scheme adapting finite difference method is applied, the numerical result illustrates B (cid:54) = UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 17 − . . . . . . − . . . . . . t = 0 . t = 0 . t = 0 . y ) F r a c t i o n o f w e a l t h i n t h e s ec o nd s t o c k ( y ) (a) Positive correlated: a = a = 0 . − . . . . . . . − . . . . . . . t = 0 . t = 0 . t = 0 . y ) F r a c t i o n o f w e a l t h i n t h e s ec o nd s t o c k ( y ) (b) Negative correlated: a = a = − . F IGURE
5. The estimated no-trading region for the N = 2 case at different timesteps. Test 2 parameters: r = 0 , α = 0 . , α = 0 . , a = 0 . , a =0 . , µ = λ = µ = λ = 0 . , γ = 0 . , β = 0 . − − − − − − − − y ) F r a c t i o n o f w e a l t h i n t h e s ec o nd s t o c k ( y ) (a) Mixed MC and FD Algorithm − − − − − − − − y ) F r a c t i o n o f w e a l t h i n t h e s ec o nd s t o c k ( y ) (b) Implicit FD Method F IGURE
6. The estimated selling, buying, and no-trading regions for the N = 2 case at time t = 0 . . Test 2 parameters: r = 0 , α = 0 . , α = 0 . , a =0 . , a = 0 . , a = a = 0 . , µ = λ = µ = λ = 0 . , γ =0 . , β = 0 . and B (cid:54) = 0 as the time approaches maturity, which obviously violates the “no-buying nearmaturity” phenomenon. The implementation of the proposed Monte Carlo scheme could cure thisproblem, and therefore gives a compatible result with the theoretical analysis. UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 18 − . . . . − . . . . . y y y (a) Uncorrelated − . . . . − . . . . . y y y (b) Correlated F IGURE
7. The estimated no-trading region for the N = 3 case. Test 3 parameters: r = 0 . , β = 0 . , γ = 0 . , µ = λ = µ = λ = µ = λ = 0 . , α =0 . , α = 0 . , α = 0 . , a = 0 . , a = 0 . , a = 0 . . (a)Uncorrelated: a ij = 0 , i (cid:54) = j . (b) Correlated: a = a = 0 . , a = a =0 . , a = a = 0 . .5.3. Test 3.
In this numerical test, we consider both the correlated and uncorrelated stocks caseswith the following financial parameter values: N = 3 , r = 0 . , β = 0 . , γ = 0 . , µ = λ = µ = λ = µ = λ = 0 . ,α = 0 . , α = 0 . , α = 0 . , a = 0 . , a = 0 . , a = 0 . , (a) uncorrelated: a ij = 0 , for i (cid:54) = j, (b) correlated: a = a = 0 . , a = a = 0 . , a = a = 0 . , and the investment period is set to be one year ( T = 1 ). The numerical result is obtained by usingtime step h = 0 . , uniform grid with length ∆ y = (0 . , . , . in each dimension, and thenumber of simulated sample paths M = 10 . We test the computational method for three stockscase for two reasons. First, we would like to demonstrate that the proposed numerical method canbe applied to high-dimensional problem. Second, it allows us to see if the insights we have in thetwo stocks case carry over to higher dimensions.Figure 7 shows the approximated no-trading region for both the three independent stocks caseand the three correlated stocks case at time t = 0 . . First observe that the no-trading region is aclosed region set bounded by six surfaces in three dimensions. Also note that the no-trading regionof the independent stocks case in Figure 7(a) is close to a rectangular cubic while the no-tradingregion of the correlated stocks case in Figure 7(b) is askew, which are consistent with previousobservations in the two dimensional case. Since the rate of return for the first stock α = 0 . is greater than the other two, a large fraction of wealth in the first stock can be expected, and itis obvious that fewer transactions will be made for the first one. Finally, it also verifies the “no-buying near maturity” phenomenon as the time approaches the end of investment period for bothcases. UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 19 T ABLE
1. The estimated no-trading region for the N = 10 case at time t = 0 . .No-trading Region y y y y y y y y y y Lower Bound .
21 0 .
21 0 .
20 0 .
20 0 .
16 0 .
16 0 .
08 0 .
08 0 .
08 0 . Upper Bound .
22 0 .
22 0 .
21 0 .
21 0 .
17 0 .
17 0 .
08 0 .
08 0 .
08 0 . Merton Proportion . . . . . . .
08 0 .
08 0 .
08 0 . Test 4 parameters: N = 10 , r = 0 . , β = 0 . , γ = − , µ i = λ i = 10 − for i = 1 , · · · , ,α = (0 . , . , . , . , . , . , . , . , . , . (cid:48) ,a = diag (0 . , . , . , . , . , . , . , . , . , . ,a ij = 0 , for i (cid:54) = j. Test 4.
In this numerical test, we consider the following financial parameter values: N = 10 , r = 0 . , β = 0 . , γ = − , µ i = λ i = 10 − for i = 1 , · · · , ,α = (0 . , . , . , . , . , . , . , . , . , . (cid:48) ,a = diag (0 . , . , . , . , . , . , . , . , . , . ,a ij = 0 , for i (cid:54) = j. and the investment period is set to be one year ( T = 1 ). The numerical result is obtained by usingtime step h = 0 . , uniform grid with length ∆ y i = 0 . for i = 1 , · · · , in each dimension,and the number of simulated sample paths M = 10 . Since transaction costs are really small,this problem is approximately reduced to the Merton’s problem. As we can expect, the no-tradingregion under these parameter settings should be a bounded small region including the Mertonproportion at any time step. Denote the
Merton proportion with the power utility function by π ∗ ,and then we have π ∗ = 1(1 − γ ) a − ( α − re ) shown in Merton (1971). This solution gives a valuable comparison, and can be used as a bench-mark to test whether the algorithm we proposed could provide a qualitative result or not.Table 1 shows the approximated no-trading region for the ten independent stocks case at time t = 0 . . The lower and upper bounds mean boundaries of the no-trading region in each dimension.We can observe from the table that the no-trading region is such a small region that it is almost the Merton proportion point because small transaction costs are applied. This mainly indicates thatwhen the transaction costs are really small, the investor is willing to rebalance his portfolio positionso that the proportion of wealth in risky assets is nearly a constant. This result again demonstratesthat the proposed numerical method can be applied to high-dimensional problems.6. C
ONCLUSION
In this paper, we have proposed a mixed numerical method including Monte Carlo simulationand finite difference method to cope with the different difficulties associated with the optimal in-vestment and consumption problem in the presence of transaction costs during a finite investmentperiod, for which no analytical solution exists. The computed approximations satisfy all the qual-itative properties which have been theoretically proved for the one risky asset case. Furthermore,the numerical solutions provide the optimal approximated value function in the presence of trans-action costs and also determine the behaviors of optimal no-trading, selling, and buying regions.
UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 20
It is worthwhile to point out that we not only characterize boundaries of the optimal tradingpolicies but also provide admissible heuristics for a portfolio which includes many stocks. Webelieve the motivation behind this proposed numerical scheme in section 4 can be extended tovarious HJB models for singular control problems. For instance, it can be directly adapted to theoptimal investment problem with transaction costs.Indeed, this work carries out many directions of future research. Arguably we do not work ondifferent choices of diffusion coefficients in the Monte Carlo step. On the other hand, in order toobtain a more accurate approximation, we observe that the high-level refinement of the meshesis required as the dimension increases which leads to an increase in computational time. It isimportant to mention that it does not automatically give a better result if we only refine the gridsnear the no-trading region and regions in which we trade only one stock. We eventually hope toinclude theoretical analysis and improved algorithm for these parts.R
EFERENCES
Altarovici, A., Reppen, M., and Soner, H. M. (2016). Optimal consumption and investment withfixed and proportional transaction costs. arXiv preprint arXiv:1610.03958 .Arregui, I. and Vázquez, C. (2012). Numerical solution of an optimal investment problem with pro-portional transaction costs.
Journal of Computational and Applied Mathematics , 236(12):2923–2937.Bayraktar, E. and Fahim, A. (2014). A stochastic approximation for fully nonlinear free boundaryparabolic problems.
Numerical Methods for Partial Differential Equations , 30(3):902–929.Bichuch, M. (2012). Asymptotic analysis for optimal investment in finite time with transactioncosts.
SIAM Journal on Financial Mathematics , 3(1):433–458.Chen, X. and Dai, M. (2013). Characterization of optimal strategy for multiasset investment andconsumption with transaction costs.
SIAM Journal on Financial Mathematics , 4(1):857–883.Dai, M., Jiang, L., Li, P., and Yi, F. (2009). Finite horizon optimal investment and consumptionwith transaction costs.
SIAM Journal on Control and Optimization , 48(2):1134–1154.Dai, M. and Yi, F. (2009). Finite-horizon optimal investment with transaction costs: a parabolicdouble obstacle problem.
Journal of Differential Equations , 246(4):1445–1469.Dai, M. and Zhong, Y. (2008). Penalty methods for continuous-time portfolio selection with pro-portional transaction costs.
Available at SSRN 1210105 .Davis, M. H. and Norman, A. R. (1990). Portfolio selection with transaction costs.
Mathematicsof Operations Research , 15(4):676–713.Fahim, A., Touzi, N., and Warin, X. (2011). A probabilistic numerical method for fully nonlinearparabolic pdes.
The Annals of Applied Probability , pages 1322–1364.Fleming, W. H. and Soner, H. M. (2006).
Controlled Markov processes and viscosity solutions ,volume 25. Springer Science & Business Media.Goodman, J. and Ostrov, D. N. (2010). Balancing small transaction costs with loss of optimalallocation in dynamic stock trading strategies.
SIAM J. Appl. Math. , 70(6):1977–1998.Janeˇcek, K. and Shreve, S. E. (2004). Asymptotic analysis for optimal investment and consumptionwith transaction costs.
Finance Stoch. , 8(2):181–206.Kallsen, J., Muhle-Karbe, J., et al. (2010). On using shadow prices in portfolio optimization withtransaction costs.
The Annals of Applied Probability , 20(4):1341–1358.Liu, H. (2004). Optimal consumption and investment with transaction costs and multiple riskyassets.
The Journal of Finance , 59(1):289–338.
UMERICAL SCHEME FOR INVESTMENT-CONSUMPTION UNDER PROPORTIONAL TRANSACTION COST 21
Magill, M. J. and Constantinides, G. M. (1976). Portfolio selection with transactions costs.
Journalof Economic Theory , 13(2):245–263.Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model.
J.Econom. Theory , 3(4):373–413.Muthuraman, K. and Kumar, S. (2006). Multidimensional portfolio optimization with proportionaltransaction costs.
Mathematical Finance , 16(2):301–335.Possamaï, D., Soner, H. M., and Touzi, N. (2012). Homogenization and asymptotics for smalltransaction costs: the multidimensional case. arXiv preprint arXiv:1212.6275 .Shreve, S. E. and Soner, H. M. (1994). Optimal investment and consumption with transactioncosts.
Ann. Appl. Probab. , 4(3):609–692.Soner, H. M. and Touzi, N. (2013). Homogenization and asymptotics for small transaction costs.
Siam journal on control and optimization , 51(4):2893–2921.Tourin, A. and Zariphopoulou, T. (1994). Numerical schemes for investment models with singulartransactions.
Computational Economics , 7(4):287–307.Tourin, A. and Zariphopoulou, T. (1997). Viscosity solutions and numerical schemes for invest-ment/consumption models with transaction costs.
Numerical methods in finance , pages 245–269. W AN -Y U T SAI , A
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