por folio op tim is at i on under transaction costs

8/8/2019 Por Folio Op Tim is at i on Under Transaction Costs

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Mathematical Finance, Vol. 18, No. 1 (January 2008), 115134

SIMULATION-BASEDPORTFOLIOOPTIMIZATION FORLARGE

PORTFOLIOSWITHTRANSACTION COSTS

KUMAR MUTHURAMAN

McCombs School of Business, University of Texas at Austin

HAINING ZHA

School of Industrial Engineering, Purdue University

We consider a portfolio optimization problem where the investors objective is to

maximize the long-term expected growth rate, in the presence of proportional transac-

tion costs. This problem belongs to the class ofstochastic control problems with singular

controls, which are usually solved by computing solutions to related partial differential

equations called the free-boundary HamiltonJacobiBellman (HJB) equations. The

dimensionality of the HJB equals the number of stocks in the portfolio. The runtime

of existing solution methods grow super-exponentially with dimension, making them

unsuitable to compute optimal solutions to portfolio optimization problems with even

four stocks. In this work we first present a boundary update procedure that converts

the free boundary problem into a sequence of fixed boundary problems. Then by com-

bining simulation with the boundary update procedure, we provide a computational

scheme whose runtime, as shown by the numerical tests, scales polynomially in dimen-

sion. The results are compared and corroborated against existing methods that scale

super-exponentially in dimension. The method presented herein enables the first ever

computational solution to free-boundary problems in dimensions greater than three.

KEY WORDS: portfolio optimization, simulation, transaction costs, stochastic control, Hamilton

JacobiBellman equation, free boundary problem

1. INTRODUCTION

We consider the continuous time portfolio optimization problem with proportional trans-

action costs. Such portfolio optimization problems are usually formulated as stochastic

control problems with controls that are termed as singular control. These singular con-

trols (e.g., transactions) can bring about an instantaneous change in the state variable

(e.g., fraction invested in an asset) rather than just a change in the rate of change of state.Solutions to such singular control problems are sought by first arguing that it is equivalent

to solving a related partial differential equation known as the HamiltonJacobiBellman

(HJB) equation. The arising HJB equation is of the free boundary type, that is, the bound-

aries of the region in which the HJB is to be solved are not pre-specified and have to be

We thank S. Kumar, D. Duffie, B. Schmeiser, H. Feng, and A. Chockalingam for their comments andfeedback. We are also thankful to the anonymous referees and the editors for their valuable comments andsuggestions.

Manuscript received November 2005; final revision received October 2006.Address correspondence to Kumar Muthuraman, McCombs School of Business, University of Texas at

Austin; e-mail: [email protected].

C 2008 The Authors. Journal compilation C 2008 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148,


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116 K. MUTHURAMAN AND H. ZHA

solved as a part of the solution itself. Moreover, the dimensionality of the HJB equa-

tion equals the number of stocks in the portfolio. Hence solving a portfolio optimization

problem with a large number of stocks essentially reduces to solving a free-boundary

problem in large dimensions. Runtimes of existing solution methods grow super expo-

nentially with dimensionmaking them unsuitable for problems with more than even

three stocks (Muthuraman and Kumar 2006). The primary objective of this paper is todevelop a computational scheme that scales well (polynomially) with dimension so one

can potentially solve large portfolio optimization problems.

We specifically consider the objective of maximizing long-term growth rate of the

portfolio in a market that contains one risk-free asset (bank) and multiple risky assets

(stocks). Denoting the portfolios total wealth over time as a stochastic process W(t), the

objective is to maximize

lim inft

E

log W(t)

t

.(1.1)

Price processes of stocks aremodeled as a multi-dimensional geometric Brownian motion.We will also allow for correlation between prices of various stocks. The investor is given

an initial position in various assets. In time, he can choose to either buy stocks with money

in the bank or add money to the bank by selling stocks. Transacting, that is, buying or

selling stocks, incurs proportional transaction costs. The investor pays a proportion of

the value transacted to a third party that enables the transaction. This proportion may

depend on the particular stock being transacted as well as on whether the transaction is a

purchase or a sale. The investor is allowed to trade in continuous time and in infinitesimal

quantities.

For the optimization problem stated above, it can be argued that the optimal policy

is specified by a no-transaction region. When the proportions of the investors wealthinvested in each of the stocks lie within this region, the investor does not make transac-

tions. When fluctuations in the price processes drive the proportions of wealth invested

in the stocks out of the boundary of the no-transaction region, the investor transacts

the minimal amount required to keep the proportions in the region. The key difficulty

in obtaining the solution, either analytically or computationally, is that the domain over

which the HJB equation must be solved is not pre-specified. In such free-boundary prob-

lems the boundary is a part of the solution and needs to be computed. Obtaining the no

transactions region in one dimension is a search for two scalar boundary points. But as

dimensionality increases the problem size grows quickly, for example when three stocks

are considered the optimization is a search for six surfaces in three-dimensional space. In

general for the N stock case the search is for 2N hyper surfaces in N dimensional space.

1.1. Placing the Work in Context

Continuous time portfolio optimization models can be broadly classified into two

classes based on the objectives that are considered. The first set of models consider

another decision variable, consumption, and maximize a function (usually discounted

utility) of consumption. The second set of models do not consider consumption and

maximize a function of wealth in the portfolio directly. Our objective falls within the

second set of models since we maximize the long-term expected growth of the portfolio

wealth.

Portfolio optimization problems in continuous time was firstintroduced and considered


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PORTFOLIO OPTIMIZATION PROBLEM 117

this setting, the optimal policy obtained by Merton continuously transacts to hold fixed

fractions of total wealth in various stocks and consumes a (different) fixed fraction of

wealth. Mertons policy requires that an infinite number of transactions be made in any

finite time interval. This suggests that in the presence of even very small transaction costs,

Mertons policy would no longer be optimal. With transaction costs, the investor would

want to make fewer transactions. In particular, transactions would be necessary only ifthe fraction of stock holding is sufficiently far away from Mertons optimal fraction

to warrant the transaction. Magill and Constantinides (1976) first considered one-stock

portfolio optimization problem with proportional transaction cost and conjectured that

the optimal policy would be characterized by a no-transactions interval, such that the

optimal policy would not transact when the fraction of wealth in stock lies in this interval.

When the fraction lies outside the interval, the optimal policy would be to buy or sell just

enough to bring the fraction into the interval.

Taksar et al. (1988) were the first to recognize that the portfolio optimization problem

with proportional transactions costs can be analyzed under the stochastic singular control

framework. They restricted their analysis to the one-stock case and maximized the long-term expected growth (1.1), which we consider in this paper. They reduce the portfolio

optimization problem to a one-dimensional PDE and show that a bang-bang type

policy is optimal (as in our case too). Davis and Norman (1990) considered proportional

transactions costs in Mertons setting, again restricting their analysis to the one-stock

case. They provided detailed characterization of the optimal policy and conditions under

which the HJB equation has a smooth solution. A comprehensive review of portfolio

optimization with transaction costs for the one stock case can be found in Zariphopolou

(1999). A representative list of other papers that consider the one stock case include

Constantinides (1979, 1986), Duffie and Sun (1990), Shreve and Soner (1994), Tourin

and Zariphopolou (1994), Korn (1998), Weiner (2000), Janecek and Shreve (2004), andMuthuraman (2006).

Relatively, the number of papers that treat the multiple stock case with transaction

costs are much lower. This is specifically due to the curse of dimensionality inherent in

the multiple stock problem. Akian et al. (2001) considered the same objective we consider

in this paper. They approximate the problem by a discounted control problem, show the

existence of a viscosity solution to the variational inequality (HJB) and the uniqueness of

the long-term expected growth rate. Numerically they solve a two-dimensional example

using policy iteration and full-multigrid-Howard algorithm. Muthuraman and Kumar

(2006) consider the multi-dimensional problem in Mertons setting with transaction costs

and maximize the discounted utility of consumption. They transform the arising free-

boundary problem into a sequence of fixed boundary problems that are solved using

a variant of the finite element method. Both computational schemes can theoretically

handle portfolios of any size, but their runtimes grow super exponentially with dimension

making them inadequate for even solving problems with four stocks. Other papers that

consider the multi-stock portfolio optimization problems under various other model

settings include Akian et al. (1996), Atkinson et al. (1997), Bielecki and Pliska (2000),

Leland (2000), Liu (2004), Lynch and Tan (2002), Morton and Pliska (1995), and Pliska

and Selby (1994).

1.2. Contribution and OutlineThe primary contribution of this paper lies in developing a computational scheme


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inherently multi-dimensional. We use the method developed here to compute results for

the portfolio optimization problems for up to seven stocks, that is, in seven dimensions

and show that the runtime grows close to cubically with dimension. To our knowledge this

is the first ever computational solution to a free boundary problem in a dimension more

than three. When we say this, of course we are excluding the class of multi-dimensional

free-boundary problems that can be reduced analytically to one-dimensional problems. Anice example of a multi-dimensional portfolio optimization problem that can be reduced

to a set of one-dimensional problems can be found in Liu (2004). Though the scheme

that we develop in this paper is focused on the particular portfolio optimization prob-

lem, we believe that the sprit behind the scheme can easily be adopted to solving other

free-boundary problems as well.

In Section 2, we describe the model and the free-boundary HJB equation. Section 3

argues and shows that the free-boundary problem can be transformed to a sequence

of fixed boundary problems. Theorem 3.1 provides the theoretical guarantees for this

transformation, in the one-stock case. In Section 4 we introduce a simulation based

procedure that takes advantage of the boundary update procedure and finds the optimalsolution. Finally, in Section 5 we provide measures of performance of our scheme and

discuss some results using numerical examples to help the reader enhance intuition for

these optimal policies.

2. MODEL FORMULATION AND THE HJB EQUATIONS

Consider a market consisting of one risk-free (bank) and N risky assets (stocks). Let

S0 R denote the wealth invested in the risk-free asset and S RN denote the vector

whose i-th component represents the wealth invested in stock i. We take as our source

of uncertainty the N dimensional Brownian motion B = {B(t) : t 0} on its standardfiltered probability space (, F,P), where F= {F(t) : t 0} is a right continuous filtra-tion of-algebras on this space that represents the information revealed by the Brownian

motion. The price process of stocks are then modeled as a geometric Brownian motion

with RN denoting the local mean rates of return of stock and RNN denoting a

positive definite symmetric matrix that represents the covariance structure. The investor

is given an initial position of y0 dollars invested in the bank and y = [y1,y2, . . . ,yN]T

dollars invested in N stocks, that is, S0(0) = y0 and S(0

) = y.

Two Ft-adapted processes L(t) and U(t) which are right continuous with left limitsare used to model transactions. L(t) (U(t)) is a N-vector whose i-th element represents

the cumulative amount of money spent to buy (obtained by selling) stock i. Thus, L(t)and U(t) are non-negative and non-decreasing processes. The reader can note that L(t)

and U(t) completely specify the trading policy we use. Let = [1, 2, . . . , N]T 0 and

= [1, 2, . . . , N]T 0 be vectors representing the transaction costs for buying and

selling, respectively(the inequalities hold for each component). To be more precise, buying

a unit worth of stock i will cost (1 + i) in cash from the bank and selling a unit worth

of stock i will result in (1 i) in cash added to the bank. In order to avoid the trivial

case, we will assume that

i(i + i) > 0.

With transactions, the controlled evolution of S0 and Scan be described by the equa-

tions

dS0 = r S0 dt (e + ) dL + (e ) dU,(2.1)


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For the sake of readability, unless necessary, we will suppress the dependence on time t

when denoting the processes B(t), S0(t), S(t), L(t), U(t). Here denotes the standard

dot product and e denotes the N-vector of ones. At time t = 0, instantaneous transactions

can be made if chosen to do so by adjusting (L, U). Therefore,

S0(0) = y0 (e + ) dL(0) + (e ) dU(0),(2.3)

S(0) = y + dL(0) dU(0).(2.4)

We define a solvency region by

, =

(y0, y) (R,R

N) : y0 +N

i=1

min((1 + i)yi, (1 i)yi) 0

.

This is the set of portfolio weights from which the investor can conduct transactions

to move to a point of non-negative value in all assets. The initial portfolio (y0, y) and

its future evolution are restricted to lie in ,. We assume that the initial endowment

(y0, y) is in ,. For illustrative purposes the solvency region in the one-stock case is

shown in Figure 2.1.

A trading policy (L, U) is called admissible if S0 and S given by equations (2.1) and

(2.2) lie in , for all t 0. Therefore, an admissible policy is one that ensures that

bankruptcy does not occur in finite time. We will use U to denote the set of all admissiblepolicies. U is clearly non-empty, since given an initial endowment (y,y0) , we canalways move all wealth to the bank at time 0 and thereby construct an admissible policy.

The investors objective is to choose a (L, U) U so as to maximize (1.1), where W Rdenotes the total wealth in the portfolio, that is, S0 + Ni=1 Si.

F G 2 1 S l i f th t k


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Introducing a change of state variable would help make the problem more tractable.

The new state variable X is a vector whose i-th component denotes the fraction of wealth

in stock i, that is, S/W. Further we define processes L and U) by the equations,

dL = diag(S) dL,

dU = diag(S) dU.(2.5)

The processes L and Ucan be interpreted as the cumulative percentage of stocks bought

and sold, respectively. Now first expressing d(log (W)) in terms ofX, dL and dUby using

the Itos formula and then considering the expectation of its integral, we have

1

tE{log W(t)} =

1

tlog W(0) + r

1

tE

t0

h(X) ds +

t0

gl(X) dL(s) +

t0

gu (X) dU(s)

,

(2.6)

where

h(X) =1

2XT TX ( r )TX,(2.7)

gl(X) = Tdiag(X) and(2.8)

gu (X) = Tdiag(X).(2.9)

The dynamics ofX can be obtained by applying Itos formula to X= SW

,

dX = diag(X)(I e XT)( r TX) dt

+ diag(X)(I e XT)dB+ (I + XT) diag(X) dL

+ (I + XT) diag(X) dU.

(2.10)

In the above Iindicates the identity matrix of appropriate dimension. Now the problem

is to minimize the following average expected cost up to time t:

lim supt

E1

t

t0

h(X) ds +

t0

gl(X) dL(s) +

t0

gu (X) dU(s)

.(2.11)

with the dynamics ofX given by (2.10). Letting x =y/W(0) we have X(0) = x.

Note that when there are no transaction costs the second and third terms vanish. Thus

the minima of h(X) at every point in time gives us, X, the optimal weight allocation

fraction for the no transactions cost case. First order maximization condition on h(X)

directly yields, X = ( T)1( r). The above is the same as the optimal portfolio

weights in the classical Merton problem with no transaction costs and consumption

Merton (1969).

Now suppose that the optimal policy was found and the optimal value of (2.11) is d.

Then using a standard representation (Bather 1968; Taksar et al. 1988) we can represent

the cumulative expected cost when the process X starts from x as t d + V(x). V(x) is

then called the differential cost of starting at x or the differential cost function. Then by

using dynamic programming arguments and Itos formula as in Taksar et al. (1988), we


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min{V(x) + h(x) d, (BiV(x) | i = 1, . . . , N), (SiV(x) | i = 1, . . . , N)} = 0(2.12)

were the i-th component of the vectors BV(x) and SV(x) are,

i

N

j=1

xjVj + Vi + i and i

N

j=1

xjVj Vi + i, respectively, and(2.13)

V(x) = V [diag(x)(I exT)( r e Tx)]

+1

2tr {D2Vdiag(x)(I exT) T(I xeT) diag(x)}.

(2.14)

The notations V, D2V and tr{} denote the gradient of V, the Hessian of V and thetrace of a matrix, respectively. The above equation is often called the HJB equation.

For notational simplicity we will write (2.12) as

min{LV(x),BiV(x),SiV(x)} = 0,(2.15)

whereLV(x) V(x) + h(x) d.Atleastoneofthe2N+ 1 terms in the above equationneed to be tight and the tight term dictates the optimal transaction that needs to be carried

out. If the LVterm is tight no transactions are to be carried. Else, if for example terms S1and B4 were tight, then the optimal transaction would be to sell stock 1 and buy stock4, instantaneously. Therefore the state space of the variable x can be viewed as a union

of 2N + 1 regions: one no-transaction region ( LV(x) = 0), N sell regions ( SiV(x) = 0)and N buy regions ( BiV(x) = 0). As noted earlier the sell and buy regions for differentstocks need not be disjoint. Figure 2.2 shows the no-transaction and the buy/sell regions


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for the two stock case. We will represent the sell (buy) region and its boundary by si(bi)

and si(bi), respectively. The no transaction region is represented by .

The problem then becomes a free-boundary problem, since all that needs to be found

are the boundaries of the regions such that the respective equations hold within the

regions and equation (2.15) holds in the entire state space. Notice that if V(x) solves

(2.15) then V(x) + K0 also solves (2.15) for any constant K0. Hence for uniqueness werestrict V(0) = 0. The existence and uniqueness of d as well as the characterization of

the differential cost function, V, as the solution (in a viscosity sense) to the HJB can be

found in Akian et al. (2001).

3. MOVING BOUNDARY APPROACH

Before we describe the computational scheme, we need to establish a boundary update

procedure that transforms the free boundary problem into a sequence of fixed boundary

problems. Let n be an arbitrary no-transaction region and (Ln , Un ) be the control

processes that keeps X(t) in n forever. Suppose that (2.11) takes the value dn when the

transaction policy is to keep X in n, then let tdn + Vn(x) denote the cumulative cost

incurred till time t using the policy (Ln , Un ) and starting from x.

From the arguments in the previous section we know that,

LVn (x) Vn (x) + h(x) d = 0 in n(3.1)

with boundary conditions BiVn = 0 and SiVn = 0. Moreover,

min{LVn(x),BiVn (x),SiVn (x)} 0.

Now suppose that for n the solution pair (Vn

, dn

) is known. If we can create a boundary

update sequence that could give us an n+1 from {n, (Vn, dn)} such that dn+1 < dn and

also the assurance that the sequence ofs constructed by the procedure converges, then

we have effectively converted the free boundary problem into a converging sequence of

fixed boundary problems.

Such an update procedure is described by the following equations for n+1bi (ith stock

buy boundary) and n+1si (ith stock sell boundary),

n+1bi = inf > nbi | is the hyper surface formed by the local minimizers ofBiVn

(3.2)

n+1si = sup

< nsi | is the hyper surface formed by the local minimizers ofSiVn(3.3)

This is equivalent to moving the boundary nbi(nsi) towards the interior to the first set of

points where BiVn (SiVn) is minimized.Notice that the boundary update procedure shown above moves the boundaries in a

monotonic fashion. Hence the generated sequence of s are nested, that is, n+1 n.

This makes it obvious that for the boundary update procedure to work we require that

our initial guess, 0, contains the optimal no-transactions region . For any given n

and Vn, the following condition assures that n,

BiVn |nbi > BiVn |nbi+eii and(3.4)


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for some > 0 and all (0, ). Here ei is a N-dimensional vector with i-th element

equals to one and all other elements equal to zero. The above conditions simply say that

it is necessary that the derivative ofBiVn (SiVn ) along the xi axis is negative (positive).Therefore if 0 and V0 satisfy the above condition, it guarantees that the arbitrarily

chosen 0 was large enough. If either of the above conditions fail, then it indicates that

the arbitrarily chosen 0 was not large enough. A restart of the procedure with a larger0 is required. A good way to choose a larger 0 in such cases is to move each boundary

half way between the old position and the boundary of the solvency region and check

(3.4) and (3.5) again. Once 1 0, subsequent s will be nested, that is, n+1 n.

For the one-stock case (N = 1), the fixed boundary PDE (3.1) can be simplified as

2(x)V

n + 1(x)V

n + h(x) dn = 0 in chosen n = (nb ,

ns )(3.6)

with boundary condition

BVn = V

n j(x) = 0 at nb ,(3.7)

SVn = Vn + k(x) = 0 at ns ,(3.8)

where

2(x) =1

22x2(1 x)2,(3.9)

1(x) = 2

x

r

2

x(x 1),(3.10)

h(x) =1

22x2 ( r )x,(3.11)

j(x) =

1 + x ,(3.12)

k(x) =

1 x.(3.13)

The primes denote differential with respect to x.

Consider the Vn, dn that solves equation (3.6) with boundary conditions (3.7) and

(3.8) in n. Suppose that n+1 is the new no-transaction region that is obtained from

n and Vn using the policy update procedure (3.2) and (3.3). One of the main results

established in Theorem 3.1 is that dn+1 < dn, that is, long-term growth rate increases by

moving to n+1. Further it also shows that n+1, which is equivalent to showing

that conditions (3.4) and (3.5) hold for n+1

, Vn+1.

THEOREM 3.1. Consider the differential equation (3.6) with boundary conditions (3.7)

and (3.8). Assume that X is less than 1, that is, r2

< 1. Say, Vn C2(n) solves (3.6)

(3.8) in n (nb , ns ) and also that (BVn )

|nb < 0 and (SVn )|ns > 0.

Define n+1 (n+1b , n+1b ) as,

n+1b = min

x > nb |(BVn )|x = 0

and(3.14)

n+1s = max

x < ns |(SVn )|x = 0

.(3.15)

If Vn+1 C2(n+1) is such that it solves (3.6)(3.8) in n+1 (n+1

b

, n+1

s

), then

1. n+1b , n+1s exist,


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3. (BVn+1)|n+1b

< 0 and (SVn+1)|n+1s > 0,

4. nb b and

ns

s .

Proof of Theorem 3.1 is provided in the Appendix.

4. THE COMPUTATIONAL SCHEME

The boundary update procedure described in Section 3 essentially transforms the free

boundary problem into a sequence of fixed boundary problems. Provided we know how

to solve the fixed boundary problem, it would be sufficient to build a computational

method that can find the optimal . Powerful PDE methods, like the finite element

method, can be invoked to solve the fixed boundary problem (Muthuraman and Kumar

2006), but the runtimes/complexity of PDE solutions techniques tend to grow super-

exponentially in dimension. Our primary goal in this paper is to be able to solve portfolio

optimization problems in large dimensions. We build a computational scheme in this

section that still uses the boundary update equations (3.2) and (3.3) and conditions (3.4)

and (3.5). But we avoid using numerical methods to solve the PDEs. Instead, for any

fixed region n (that is, for any given transaction policy), we use simulation to estimate

the differential cost function V for a set of points in n, but still rely on the boundary

update equations obtained by the PDE-based arguments to improve policies.

First we define the notations we need. Mn is an arbitrary increasing sequences of

positive integers such that Mn . A discretization of n will be represented by a set

n. Obviously n is countably finite such that x n implies x n. An estimate ofV

that uses M + n sample paths will be denoted by VMnn .

In step 1, we start the computation with a guess 0 and n = 0. For a given n, in step 2,

using Mn sample paths we obtain an estimate ofV(x) (that is VMn

n ) for each x n. We

use standard simulation techniques for the estimations. Since Vis only an estimate ofV,

we are not guaranteed (as in Theorem 3.1) that the ns obtained from update conditions

(3.2) and (3.3) always contains . We need to check at each step if n and Vn satisfy

(3.4) and (3.5), which we do in step 3.

If conditions (3.4) and (3.5) hold, we update the boundaries using (3.2) and (3.3). At

this stage, after an inward movement of the boundaries, we define two sets of variables

nbi and nsi by

nbi = a

nbi n1bi

and(4.1)

n

si = a

n

si

n1

si

(4.2)for some a (0, 1). Both nbi and

nsi are recalculated only when a boundary update is

made using (3.2) and (3.3). Hence they can be interpreted as a fraction a of the difference

between n and n1 during the last inward movement of the boundaries. The conver-

gence of is checked after each inward movement and the iteration is continued if not

converged. Convergence can be checked by either testing the convergence of d or n.

If any of conditions (3.4) and (3.5) fail, then it indicates that (under the accuracy

permitted by Mn sample paths) our n has overshot inwards due to estimation error and

a backing out is necessary. We back out by redefining n. To this extent we first define

new by

newbi = nbi

nbi(4.3)


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FIGURE 4.1. The computational scheme flow.

Upon consecutive backing out it is possible (though rarely) that new (from (4.3)(4.4))

is no longer a subset of the initial guess 0, in which case we set n = 0 otherwise we

set n = new. Note that, by our assumption, is a subset of 0. Figure 4.1 shows a

chart that summarizes the computational scheme.

The idea behind using an increasing number of sample paths to estimate Vis to improve

on the computational efficiency. Since during the early stages of the iteration n

tends tobe relatively further away from , the chances of over-shooting due to a cruder estimate

ofVtends to be lower. Moreover, as Mn the scheme itself converges to the boundary

update procedure with estimate Vconverging to V. Hence we can get arbitrarily close to

the optimal and also for the one-stock case as Mn , Theorem 3.1 is sufficient to

establish the convergence of this simulation based scheme.

In the next subsection we discuss the policy space approximation we use and how

this approximation helps make the scheme scale polynomially in dimension. Section 4.2

contains some further remarks.

4.1. Policy Space Approximation

We turn to the problem of choosing an appropriate discretization set in this subsec-

tion. The simplest discretization scheme would be to discretize each dimension ofn into

Pdiscrete points. This would result in a set of size PN, that is, the number of elements

in grows exponentially. Moreover, the 2N boundaries that completely represent the

optimal policy for the N-stock case are each hyper-surfaces. Even the data structure that

is required for the representation of a general hyper surface grows exponentially with di-

mension. Thus, there would be little hope of being able to construct a scheme that scales

polynomially with dimension with this discretization.

By restricting the no-transaction regions to a hyper polygons we can build a that

grows polynomially in dimension. For a portfolio optimization problem with a dis-

counted utility objective, based on the results from a number of experimental runs, in


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that the difference in the value function with and without the approximation is only of

order 102 %. We will use this approximation to describe the construction of the set

in this section and also for results in Section 5. However, later in this section, we make

a remark on how this assumption can be relaxed without sacrificing the polynomial

growth.

With the hyper-polygonal approximation we can represent the no-transactions regionby

AX B,(4.5)

where

A =

1 ab12 ab1N

......

......

abN1 abN2 1

1 a

s

12 a

s

1N...

......

...

asN1 asN2 1

and B =

bb1...

bbN

b

s

1...

bsN

(4.6)

The elements ofA and B describe the buy and sell boundaries. The ith stocks buy and

sell boundaries are given by

xi bbi

j=1...N,j=i

abi jxj and(4.7)

xi bsi

j=1...N,j=i

asi jxj,(4.8)

respectively. Thus our search for the optimal policy becomes a search for matrix A and

vector B.

We first consider the two stock case (Figure 4.2) for the sake of easier description and

visualization. To construct n+1 from n, we require the new boundaries n+1b1 , n+1b2 ,

n+1s1

and n+1s2 . To move nb1 to

n+1b1 using the boundary update equation (3.2), we seek the local

minimizer ofBVn . Since the boundary b1 is approximated by a straight line we wouldonly need two points where BVn is minimized in order to determine

n+1b1 . By discretizing

the boundaries ns2 and nb2 by P points each, we can obtain the estimate Vn on the 2P

points. However since BVn depends on the gradient, we would need to estimate Von twomore lines parallel to ns2 and

nb2. Then we can use simple finite differences for gradient

estimation. Therefore to update boundary n+1b1 and n+1s1 for the first stock, we only need

to estimate the differential cost function Vn for discretized points on four lines (4 P points).

We use the estimates on these points to calculate the gradient on two lines (2 P points).

We will call the lines on which we seek estimates of Vn as well as its gradient as main

lines and the lines on which we seek estimates of Vn only to facilitate the calculation of

the gradients on the main line as auxiliary lines. Note that main lines and accompanying

auxiliary lines are parallel. These lines are also shown in Figure 4.2 for the 2 stock case.

Now consider the general N-dimensional case. For a particular stock i, the update of

the buy i boundary and the sell i boundary requires the estimation of Vn on N2 lines.

These N2 lines are in N groups ofN lines each. Each group has one main line and N 1

auxiliary lines. The gradient is estimated on the N main lines using the estimate ofV on2


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FIGURE 4.2. Shape of the no-transaction region in two-stock case.

the boundary update conditions (3.2) and (3.3) give 1 buy point and 1 sell point. Fitting

a N-dimensional hyper-plane for the new buy(sell) boundary using the N new buy(sell)

points is straight forward. For each of the Nstocks we do the same. This implies that the

estimation of V needs to be done on N2 N = N3 lines. The discretization of these N3lines comprises our discretization set . If each line is discretized by P points then the

size ofn is P N3, growing polynomially in N.

4.2. Further Remarks

With the assumption of hyper-polygonal no-transaction region , the number of pa-

rameters ((4.5)(4.6)) that are needed to represent any boundary for the N stock case is

N. Now consider relaxing the hyper-polygon assumption on . The key insight that can

be obtained from the previous discussion is that as long as the number of parameters

needed to describe any boundary grows polynomially with dimension, then we can ob-

tain a that grows only polynomially in dimension. In other words one can relax the

hyper-polygon assumption as long as we choose any parameterized hyper-suface such

that the number of parameters required to describe the hyper-surface has polynomial

growth. Of course such a relaxation is accompanied by an increase in the complexity and

runtime. The number of elements in would be larger than order N3 and more than N2

lines would have to be discretized for each boundary update.

The choice of sequences Mn and the backing out fraction a have not been discussed

yet. Though the computational scheme would work and converge for any increasing

sequences M and any a (0, 1), the runtimes of the scheme can greatly be reduced

by prudently choosing these, like in many other simulation based schemes like simulated

annealing and retrospective approximations. We have found that the following parameter

choices perform well and we also use these for all our computational results in Section1


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5. RESULTS AND DISCUSSIONS

The computational scheme that has been proposed uses simulation to estimate Vn (the

differential cost function) for a given policy n, during each boundary iteration. It is

also possible to use a PDE solution technique to obtain Vn during each iteration. The

primary reason we rely on simulation is that by using simulation and a careful choice of

discretization we hope to obtain a scheme that scales polynomially. In this section we first

demonstrate that the proposed scheme indeed scales polynomially with dimension and

compare it to the runtimes of the scheme that uses the Finite element method (FEM) to

solve for each Vn (this is similar to the method used in Muthuraman and Kumar (2006).

Then we illustrate the sequence of boundary movements during various iterations for the

one-stock case and a two-stock example with correlation between the two stocks.

The code is implemented in Matlab and the runtimes are based on execution by a

single processor Pentium IV machine running at 3 Gz with 1GB RAM. We consider

a sequence of problems of increasing dimension. The first problem considers only one

stock, the second problem considers two stock and so on. For the sake of comparison in

we always consider independent stock with i = 0.14, i = 0.3, i = i = 5%, and r =

10%. Table 5.1 records the runtimes for each of these problems for both the PDE based

scheme and the proposed simulation based scheme. The NA (for Not Available)

indicates cases where very large runtimes made computations infeasible.

Figure 5.1 plots the logarithm of the runtimes against logarithm of dimension, showing

the nature of runtime scaling. Suppose the runtime tr is a polynomial function of dimen-

sion N: tr(N) = CNK, where Cand Kare constants and lower order terms are ignored for

estimation purposes. By taking logarithm on both sides, we have log tr = log C+ Klog N.

Thus log tr and log Nhaving a linear relation, confirms the polynomial scaling. Moreover

the slope of the line, K, is the order of the polynomial. Table 5.2 shows K = 3.4455.

Figure 5.1 shows the movement of the boundaries with each iteration of the boundaryupdate procedure. The PDE based scheme converges in six iterations. The number of

iterations for convergence of the simulation based scheme obviously depends heavily on

the sequence Mn (we use Mn = 200n12 ). Also a sequence that starts with a very large M0

will assure convergence in lesser number of iterations but a much larger runtime, hence a

bad choice. Figure 5.1 is shown to illustrate the nature on convergence and not the rate

of convergence. While the convergence of the PDE-based scheme is strictly monotonic

the convergence of the simulation based scheme is not, due to the backing out step that

accounts for estimation errors. In Figure 5.1 iteration 7 is a backing out and moves the

boundary outward. Figure 5.2 also shows the convergence of dn for both cases.

TABLE 5.1

Runtime Data under Two Schemes

N PDE Simulation

1 34 sec 4 mins

2 20 mins 58 mins

3 45 hrs 3.4 hrs

4 NA 8.6 hrs

5 NA 18.7 hrs6 NA 36.6 hrs

7 NA 62 3 hrs


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1 2 3 4 5 6 7

30 secs

1 min

10 mins

30 mins

1 hr

5 hrs

10 hrs

50 hrs

Dimension (log scale)

Runtime(logscale)

Simulation runtime

Simulation trend

FEM runtime

FIGURE 5.1. Computational runtime against dimension under two schemes.

To illustrate the convergence of boundaries in the two stock case we consider two

stocks that are correlated with one another. The specific parameters we choose are, =

(0.3 0.10.1 0.3), 1 = 2 = 0.141 = 1 = 2 = 2 = 5%, and r = 10%. Figure 5.3 shows the

boundaries obtained with each iteration. For this particular parameter set convergence is

achieved in 13 iterations. Iterations 4, 7, 9, and 11 are outward movements of the bound-

aries and are shown as dotted lines. As can be noticed, when the stocks are correlated, the

region of inaction is compressed along the main diagonal (1, 1) and elongated along the

off-diagonal (1, 1). A heuristic reasoning for such a change in structure of the optimal

policy is provided in Muthuraman and Kumar (2006) and is as follows. As the correlation

between stocks become larger, it becomes less likely that an increase in the value of stock

1 is accompanied by a decrease in the value of stock 2. That is, in the region of inaction,

it is less likely that sample paths of the value processes will turn away from the main

diagonal. Hence transactions are more likely to be inevitable along the main diagonal.

Given this inevitability, one does not save much on transaction costs by giving the sam-

ple paths room to turn away from the boundary along the main diagonal. Therefore, a

new region of inaction that is compressed along the main diagonal will provide a better

value function because it does not let the value function deteriorate as much before it

intervenes. Of course, one can only shrink the region so much along the main diagonal


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

8

9

x

Iterationnumber

Simulation Bds

PDE Bds

FIGURE 5.2. Convergence of boundariesone stock case.

8 7.5 7 6.5 6 5.5 5

x 103

0

1

2

3

4

5

6

7

8

9

d

Iterationnumber

Simulation

PDE


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x2

FIGURE 5.4. Convergence of boundariestwo stock case.

6. CONCLUSION

Several examples of stochastic singular control models can be found in finance. Being able

to solve such problems in large dimensions is of natural interest since the dimensionality of

such problems usually represents the total number of sources of randomness. By focusing

on a particular singular control problem arising in portfolio optimization with transac-

tion costs, we have been able to construct a computational scheme and demonstrate nu-

merically that the computational scheme scales polynomially with dimensionsthereby

being capable of tracting problems of large dimensions. Though the scheme presented in

this paper has been specifically constructed for the portfolio optimization problem the

methodology, we believe, is much more general. We hope that other researchers will be

able to adapt the scheme for similar large dimensional singular control problems.

Using a typical single processor machine available today, our implementation on Mat-

lab can solve as much as a seven dimensional problem in reasonable time. The scheme

described here directly lends itself to distributed computing. With an implementation of

the scheme in a complied language (like C++) on a distributed computing infrastructure

can make problems of much higher dimensions tractable.

APPENDIX

Proof of Theorem 3.1. Without loss of generality let n = 1.

(1) Since SV1|1s = 0,


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At 1s ,

BV1|1s = V

1(1s ) +

1 + 1s(A.2)

=

1 1s+

1 + 1s> 0.(A.3)

At 1b we have BV1|1b = 0 and (BV1)|1b

< 0. Since BV1|1s > 0, .BV1|1b = 0

and (BV1)|1b < 0, there exists at least a point x0 (

1b ,

1s ) such that (BV1)

|x0 =

0. Since 2b is defined as the infimum of all such x0s, 2b exits. The existence of

2s can be argued similarly.

(2) Let f(x) = V2(x) V1(x). Now, from the above (BV1)

< 0 in [1b , 2b )

with BV1 = 0 at 1b . Hence BV1(

2b ) < 0, which implies

V1(2b ) < j(

2b )(A.4)

and from boundary conditions we have

V2(2b ) = j(

2b ).(A.5)

Therefore f(2b ) > 0. Similarly we can argue that f(2s ) < 0. Because f(

2b ) > 0

andf(2s ) < 0, there exists a point x0 in (2b ,

2s ), such thatf(x0) = 0, f

(x0) < 0.

In 2 (2b , 2s ), V1 and V2 satisfy

2(x)V

1 + 1(x)V

1 + h(x) d1 = 0,(A.6)

2(x)V

2 + 1(x)V

2 + h(x) d2 = 0,(A.7)

respectively. Defining d = d2 d1, subtracting one from the other and evaluat-ing at x0,

d = 2(x0) f(x0) + 1(x0) f(x0) = 2(x0) f

(x0).(A.8)

Since 2(x) =12

2x2(1 x)2 > 0 for x (0, 1) we have d < 0, that is, d2 < d1.

Note that d is a constant in 2. This shows that the expected cost per unit time

decreases as we update the boundary using (3.14) and (3.15), which is a policy

improvement.

(3) Let (BV2)|2b =

K and we know that (BV1)|2b = 0, that is,

(V

1 j(x))

|2b = 0 and(A.9)

(V2 j(x))|2b =

K(A.10)

subtracting (A.9) from (A.10),

K = f

2b

=d 1

2b

f

2b

2

2b = d

2

2b 1

2b

f

2b

2

2b .(A.11)

Now since

d < 0,(A.12)

2

2b

=1

22x2(1 x)2

> 0(A.13)


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1

2b

= 2

x r

2

x(x 1)

2b

> 0 and(A.14)

f2b > 0,(A.15)

we have K < 0, that is, (BV2)|2b < 0. Similar arguments yield (SV2)

|2s > 0.

(4) The sequence {nb } is clearly increasing and bounded above and hence converges.

While {ns } is a decreasing sequence bounded below and hence also converges.

The optimal b and s are the limits of the sequences {

nb } and {

ns }, respectively.

REFERENCES

AKIAN, M., J. L. MENALDI, and A. SULEM (1996): On an Investment-Consumption Model with

Transaction Costs, SIAM J. Cont. Opt. 34(1), 329364.

AKIAN, M., A. SULEM, and M. TAKSAR (2001): Dynamic Optimization of Long-Term Growth

Rate for a Portfolio with Transaction Costs and Logarithmic Utility, Math. Financ. 11(2),

153188.

ATKINSON, C., S. R. PLISKA, and P. WILMOTT (1997): Portfolio Management with Transaction

Costs, Proc. R. Soc. Lond. A. 453, 551562.

BATHER, J. A. (1968): A Diffusion Model for the Control of a Dam, J. Appl. Prob. 5, 5571.

BIELECKI,T.R.,andS.PLISKA (2000): Risk Sensitive Asset Management with Transaction Costs,

Financ. Stoc. 4, 133.

CONSTANTINIDES, G. M. (1979): Multiperiod Consumption and Investment Behavior with Con-

vex Transaction Costs, Manag. Sci. 25, 11271137.CONSTANTINIDES, G. M. (1986): Capital Market Equilibrium with Transaction Costs, J. Polit.

Econ. 94(4), 842862.

DAVIS, M., and A. NORMAN (1990): Portfolio Selection with Transaction Costs, Math. Operat.

Res. 15, 676713.

DUFFIE, D.,and T. SUN (1990): Transaction Costs and Portfolio Choice in a Discrete-Continuous

Time Setting, J. Econ. Dyn. Contl. 14, 3551.

JANECEK, K., and S. E. SHREVE (2004): Asymptotic Analysis for Optimal Investment and Con-

sumption with Transaction Costs, Financ. Stochast. 8(2), 181206.

KORN, R. (1998): Portfolio Optimization with Strictly Positive Transaction Costs and Impulse

Control, Financ. Stochast. 2, 85114.LELAND, H. E. (2000): Optimal Portfolio Management with Transaction Costs and CapitalGains

Taxes, Haas School of Business Technical Report.

LIU, H. (2004): Optimal Consumption and Investment with Transaction Costs and Multiple

Risky Assets, J. Financ. 59, 289338.

LYNCH, A. W., and S. TAN (2002): Multiple Risky Assets, Transaction Costs and Return Pre-

dictability: Implications for Portfolio Choice, Working paper.

MAGILL, M.J. P.,and G. M.CONSTANTINIDES (1976): Portfolio Selection with Transaction Costs,

J. Econ. Theor. 13, 245263.

MERTON, R. C. (1969): Lifetime Portfolio Selection under Uncertainty: The Continuous Time

Case, Rev. Econ. Stat. 51, 247257.

MORTON,A.J.,andS.R.PLISKA (1995): Optimal Portfolio Management with Fixed Transaction


20/21


MUTHURAMAN, K. (2006): A Computational Scheme for Optimal InvestmentConsumption

with Proportional Transaction Costs, J. Econ. Dyn. Contl. 31, 11321159.

MUTHURAMAN, K., and S. KUMAR (2006): Multi-Dimensional Portfolio Optimization with Pro-

portional Transaction Costs, Math. Financ. 16(2), 301335.

PLISKA, S. R., and M. SELBY (1994): On a Free Boundary Problem That Arises in Portfolio

Management, Phil. Trans. R. Soc. Lond. A. 347, 447598.

SHREVE,S.E.,andH.M.SONER (1994): Optimal Investment and Consumption with Transaction

Costs, Annal. Appl. Probab. 4(3), 609692.

TAKSAR, M., M. J. KLASS, and D. ASSAF (1988): A Diffusion Model for Optimal Portfolio

Selection in the Presence of Brokerage Fees, Math. Operat. Res. 13, 277294.

TOURIN, A., and T. ZARIPHOPOLOU (1994): Numerical Schemes for Investment Models with

Singular Transactions, Comput. Econ. 7, 287307.

WEINER, S. M. (2000): The Effect of Stochastic Volatility on Portfolio Optimization with Trans-

action Costs, Ph.D. thesis, University of Oxford.

ZARIPHOPOLOU, T. (1999): Transaction Costs in Portfolio Management and Derivative Pricing,

Proc. Symp. Appl. Math. 57, 101163.


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