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MATHEMATICS OF OPERATIONS RESEARCHVol. 15, No. 4, November 1990Prinled in U.S.A.

PORTFOLIO SELECTION WITH TRANSACTION COSTS"

M. H. A. DAVIS AND A. R. NORMAN^

In this paper, optimal consumption and investment decisions are studied for an investorwho has available a bank account paying a fixed rate of interested a stock whose price is alog-normal diffusion. This problem was solved by Merton and others when transactionsbetween bank and stock are costless. Here we suppose that there are charges on alltransactions equal to a fixed pwrcentage of the amount transacted. It is shown that theoptimal buying and selling policies are the local times of the two-dimensional process of bankand stock holdings at the boundaries of a wedge-shaped region which is determined by thesolution of a nonlinear free boundary problem. An algorithm for solving the free boundaryproblem is given.

1. Introduction. This paper concerns the optimal investment and consumptiondecisions of an individual who has available just two investment instruments: a bankaccount paying a fixed interest rate r, and a risky asset ("stock") whose price is ageometric Brownian motion with expected rate of return a and rate of returnvariation a^. Thus the stock grows at a mean rate a, with white noise fluctuations. Itis assumed that stock may be bought and sold in arbitrary amounts (not necessarilyintegral numbers of shares). The investor consumes at rate cit) from the bankaccount; all income is derived from capital gains and consumption is subject to theconstraint that the investor must be solvent, i.e. have nonnegative net worth, at alltimes. The investor's objective is to maximize the utility of consumption as measuredby the quantity

(1.1)

Here E denotes expectation and S > 0 is the interest rate for discounting. In thispaper the utility function uic) will always be equal to c'^/y for some y G T — {y e R:y < 1 and y # 0} or uic) = log c ("log" denotes the natural logarithm). Thesefunctions form a subset of the so-called HARA (hyperbolic absolute risk aversion)class.

In the absence of any transactions between stock and bank, the investor's holding*(,(/) and Slit) in bank and stock respectively, expressed in monetary terms, evolveaccording to the following equations, the second of which is an Ito stochastic

•Received July 27, 1987; revised April 24, 1989,AMS l%0 subject classification. Primary: 93E20, Secondaiy: 90A16, 60H10,lAOR 1973 subject clas^ication. Main: Stochastic control. Cross references: bivestment,OR/MS Index 1978 subject classification. Primary: MO Finance/portfolio. Secondary: 564 Probability/dif-fusion.iSey words. Portfcrfiio election, transaction costs, stochastic control, reflecting divisions, free boundaryproblem, local time.^Supported in part by an SERC Postgraduate Studentship.

676

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PORTFOLIO SELECTION WrTH TRANSACTION COSTS

differential equation driven by a standard Brownian motion zit).

(1.2) dsgit) = irsgit)-cit))dt.

677

(1.3) dt + as^(s)

The investor starts off with an initial endowment 5o(O) = x, s^iO) = y. To completethe specification of the problem we need to state how funds are transferred frombank to stock and vice versa. In the original paper of R. C. Merton [18], and in almostall subsequent work in this area, it assumed that such transfers can be made instantlyand costlessly. In this case we can re-parametrize the problem by introducing newvariables wit) = Sgit) + s^it) (the total wealth) and irit) = Siit)/wit) (the fraction oftotal wealth held in stock). Adding (1.2) and (1.3) and using these variables gives usthe basic wealth equation

(1.4) = [rw(t) + ia- r)TTit)wit) - c (0 ] dt + aTrit)wit

H'(O) =x + y.

Since transactions are free and instantaneous we can regard rrit)—as well asdt)—as a decision variable. We now have a completely formulated stochastic controlproblem: choose nonanticipative processes vit),cit) so as to maximize (1.1) subjectto (1.4) and the constraint wit) > 0 for all t.

It is a remarkable fact that this is one of the few nonlinear stochastic controlproblems that can be explicitly solved. It turns out, as we will show in §2 below, thatfor utility functions in the HARA class the optimal investment strategy is to keep aconstant fraction of total wealth in the risky asset, and to consume at a rate propor-tional to total wealth, i.e. the optimal rrit), dt) ate rrit) = v* and dt) = Cwit) forsome constants •IT*,C. This means that, optimally, the investor acts in such a way thatthe portfolio holdings are always on the line 5, = [7r*/(l — v*)]sg in the isg,s^)plane; we shall refer to this as the "Merton line" (Figure 1).

In a recent paper by Karatzas, Lehoczky, Sethi and Shreve [12], the constraintwit) > 0 is replaced by the stipulation that evolution of the wealth process terminateson bankruptcy (i.e. the first time at which wit) =^ 0), at which point the investor isretired on a lump-sum pension F. It turns out that if F is not large enough then it isoptimal to avoid bankruptcy altogether (this can always be done) and then the policydescribed above is optimal. All of these results hold under a basic well-posednesscondition, namely

(holdingsin stock)

Merton lineslope Tt7(1-Tf)

no- transaction region

(hoUngsinbank)

FIGURE 1. Spacx of Bank and Stock Holdings, Shewing "Merton Line" and No-TransactionWedge.

678 M. H. A. DAVIS & A. R. NORMAN

Condition A. 8 > y[r + ia - r)^/aKl - y)\

If this condition is violated then growth of discounted utility is possible and arbitrarilylarge utility may be obtained by policies of prolonged investment followed by massiveconsumption. The well-posedness condition for uic) = log c is 5 > 0, which is justCondition A with y = 0.

Any attempt to apply Merton's strategy in the face of transaction costs would resultin immediate penury, since incessant trading is necessary to hold the portfolio on theMerton line. There must in such a case be some "no-transaction" region in is^,s^)space inside which the portfolio is insufficiently far "out of line" to make tradingworthwhile. In this paper we shall consider the case of proportional transactioncosts:' the investor pays fractions A and /x of the amount transacted, on purchase andsale of stock respectively. All such charges are paid from the bank account. In thiscase the bank and stock holdings must retain their separate identities rather thanbeing merged into a single wealth process. The equations describing their evolutionare

ds^it) = [rsoit] - cit)] dt-il+X)dL, + ( l - fi) dU,,

(1.5)dsiit) = as^it) dt + 0-5,(0 dzit) + dL, - dU,,

where L,,U, represent cumulative purchase and sale of stock on the time interval[0, t] respectively. This allows for instantaneous purchase or sale of finite amounts ofstock as well as purchase and sale at a given rate and various other sorts of behaviour.One notices from (1.5) that purchase of dL units of stock requires a payment(1 + A) dL from the bank, while sale of dU units of stock realizes only (1 - fi) dU incash. Obviously, it will never be optimal to buy and sell at the same time. If ConditionA does not hold then arbitrarily high utility can be achieved as described above (the"prolonged investment" just has to be a little more prolonged). Under Condition A,we will show that the no-transaction region is a wedge containing the Merton line (Fig.1); equivalently, the proportion of total wealth held in stock should be maintainedbetween fractions irf and v}, which of course depend on A and M as well as theother constants in the problem. For example, when A = fi = 0.015 and r = 0.07,a = 0.12, o- = 0.4, S = 0.1, and the utility function is uic) = -1/c we find thatvf = 9.0%, TT$ = 19.8% whereas the Merton proportion is n* = 15.6% (see Figure3 below). There is no closed-form expression for 'n-,*,ir| but we state how tocompute them. The optimal transaction policy is minimal trading to stay inside thewedge, preceded by an immediate transaction to the closest point in the wedge if theinitial endowment is outside it. More technically, the optimally controlled process is areflecting diffusion inside the wedge and the buying and selling policies iL,,U,) arethe local times at the lower and upper boundaries respectively. Consumption takesplace at a finite rate in the interior of the wedge (in which the process lies almost all

of the time).Our interest in this problem was aroused by the stimulating paper of Magill and

Constantinides [17] on the same subject. This paper contains the fundamental insightthat the no-transaction region is a wedge, but the argument is heuristic at best and noclear prescription as to how to compute the location of the boundaries, or what thecontrolled process should do when it reaches them, is given. The paper was in factahead of its time, in that an essential ingredient of any rigorous formulation, namely

'The case of fixed costs, where the investor pays a flat transaction fee regardless of the amounttransacted, remains largely unexplored; this is a problem of imimlse control 13J. There is some work in thisdirection by Duflie and Sun [7).

PORTFOLIO SELECTION WITH TRANSACTION COSTS 679

the theory of local time and reflecting diffusion, was unavailable to the authors, beingat that time (1976) the exclusive property of a small band of pure mathematicalvotaries. Needless to say, Magill's and Ctonstantinides' paper is far more valuablethan many others of unimpeachable mathematical rectitude.

Stochastic control problems involving local time have received much attention inrecent years. Early pioneering work of Bather and Chernoff was followed by theappearance of papers by Benes, Shepp and Witsenhausen [2] and Harrison andTaylor [10] in which problems of "finite fuel" control and regulation of "Brownianstorage systems" were solved rigorously, taking advantage of developments in stochas-tic calculus in the 1970s which made it possible to handle local times and reflectingdiffusions in simple domains in a relatively straightforward way. The relevant theorycan now be found in very compact form in Harrison's book [9]. All of these works,and the present paper, essentially concern free boundary problems, and indeed manyof them are closely related to optimal stopping, as Karatzas and Shreve [14] haveshown. This paper differs from all others we are aware of, however, in that ourproblem involves "continuous control" (i.e. consumption) as well as "singular control"(transactions). This leads to a free boundary problem for a nonlinear partial differen-tial equation (PDE) as opposed to the linear PDEs which arise when singular controlis the only control. The problem is for this reason substantially more delicate.

Three papers directly related to the present work are Constantinides [4], Duffieand Sun [7] and Taksar, Klass and Assaf [20]. Constantinides considers essentially thesame problem as ours (or as the earlier paper of Magill and Constantinides [17]) andproposes an approximate solution based on making certain assumptions on theconsumption process. Some further remarks on his results will be found in §7 below.Duffie and Sun [7] consider the case of fixed plus proportional transaction charges.Their results are quite different in character from ours. Taksar, Klass and Assaf [20],using the model (1.5) with c = 0 (no consumption), study the problem of maximizingthe long-run growth rate

( | ( f ) + 5 , ( 0 )

In an ingenious analysis they reduce the problem to a 1-dimensional one and showthat a "two-sided regulator" is optimal, which means that the process is^it), s,(f)) is,as in our case, optimally kept inside a wedge by reflection at the boundaries. Thesolution thus looks very similar to ours, but the details of the problem and themethod of analysis are completely different. A study of the effects of transaction costson option pricing is given by Leiand [15].

The present paper is organized as follows. In §2 we give a self-contained treatmentof the Merton (no transaction cost) problem. This is included because later on weneed to use comparison arguments involving the Merton case, and also because nosimple complete treatment seems to be readily available.^ The transaction costsproblem is formulated in §3, where we give informal arguments which indicate whythe no-transaction region is wedge-shaped. We also show how the analytic problemmay be reduced to a one-dimensional free boundary problem. In §4 we prove"veriflcation theorems" which show that if the free boundary problem can be solvedthen a policy of minimal transaction to stay within the wedge defined by its solution isindeed optimal. Theorem 5.1 in §5 gives conditions under which the free boundaryproblem is solvable. In §6 we obtain a semimartingale representation of the evolutionof the "value process". Apart from having some intrinsic interest, this is needed to

^The extra generality of the treatment in [12] necessitates more complicated arguments.

6 8 0 M. H. A. DAVIS & A. R. NORMAN

complete some technical argument in §4, and can be used to show that under theoptimal policy the investor does not reach bankruptcy in finite time.

The main results of the paper are summarized in Theorem 7.1 in §7. This sectionalso contains an algorithm for solving the free boundary problem together withnumerical results, as well as concluding remarks. Finally, the Appendix contains atechnical analysis of some differential equations arising in the solution of the freeboundary problem.

A preliminary account of some of this work was given in Davis [5].

2. No transaction costs: the Merton problem. Throughout the paper (fi, g, P)will denote a fixed complete probability space and (^) ,>o ^ 8*^^" filtration, i.e. afamily of sub-tr-fields of g such that (i) ^ < ^ for .s < f and (ii) for each f > 0, ^contains all null sets of g. The stochastic process^ iz,),^g will be a standardBrownian motion with respect to ( ^ ) , i.e. (z,) has almost all sample paths continu-ous, is adapted to ( ^ ) , and for each s,t ^ 0 the increment z,+^ - z, is independentof S^, and is normally distributed with mean 0 and variance s.

In this section we study the "Merton problem" of choosing investment andconsumption policies (ir,, c,) so as to maximize utility when wealth evolves accordingto equation (1.4). Let U denote the set of policies. A policy is a pair (c,, ir,) ofJ^-adapted processes such that

(2.1) ( i) cit,(a)>0 and ('cis,o))ds K'x for a\\ it, o)),•'o

(ii) \'trit,o})\ < K for all it,(o), where K is a constant which may varyfrom policy to policy, and

(iii) wit,(») >0 for a\\ it, <a), where (w,) is the unique strong solution ofthe wealth equation

(2.2) dw^ = irw, + ia - r)rr,w, - c,) dt + w,ir,o-dz,,

Wg = W.

Here jv G R^ is the initial endowment and a,r,a are positive constants as describedin §1. The existence of such a solution follows from standard theorems, which showthat the map / -> w,ica) is continuous for almost all w G O. Let r •= inf r: w, = 0}.We note that (c, TT) G U only if c, = 0 a.s. for all t > r and hence w, = 0 for allt > r. The investment problem is to choose (c, TT) G U so as to maximize

(2.3) ^(c, 17) == Kfe'^'uic,) dt.•'o

THEOREM 2.1. (a) Suppose Condition A holds and that uic) = c^/y, y G T.Define

(2.4)

•'Throughout the paper we take the usual probabilist's license of denoting a random process such as (z,)interehangeably as z,, zit), z,im) or zit,u>). All ex<«enously defined processes are a^umed to bemeasurable.


where p •= ia — r)/a. Then swp^^ .^^^y^J^ic,!?) = viw) where

(2.5) viw) = \c^~'wy.

The optimal policy is

(2.6) C* = CH',, < = (i_^y)^-

(b) Suppose uic) = log c and that S > 0. Then

sup ^ ( c , 7 r ) ^^\'-+jP^-s]+ ^log8w(c,ir)eU " '• ••

and the optimal policy is c* — Sw,, TT* = jS/cr.

REMARK 2.2. As noted in §1, rr* is constant and c* is proportional to currentwealth. The condition 8 > 0 and the optimal policies (c*, v*) of case (b) are formallyobtained from case (a) by setting y = 0. We see from (2.6) that v* G (0,1) only whenr < a < r + il - y)a^. This is hedging: assets are split between stock and bank toreduce volatility. If a > r + (1 - y)o-^ then leverage is optimal: funds are borrowedfrom bank to invest in stock,"* while if a < r then the converse—i.e. shortselling—isoptimal. When a = r, TT* = 0; in this case viw) = US - yr)/il - y))^"'y"'»v^,which is just the utility of optimally consuming an initial endowment w in the bank.

PROOF. We will only prove part (a). Part (b) is proved by exactly the samearguments. The proof is an application of dynamic programming, cf. Chapter VI ofFleming and Rishel [8]. When c, rr are constants the wealth process w, given by (2.2)is a diffusion process with generator

A'-'^viw) = \it + (a - r)rT)w - c\v'iw) -f |W^77-V^P"(H')

acting on C^ functions v iv' = dv/dw). The so-called Bellman equation of dynamicprogramming for maximizing (2.2) over control policies (c,,-n-,), to be solved for afunction i), is

(2.7) max [A'^-^iJ + - c ^ - 8ij\ = 0.

The maxima are achieved at

so that (2.7) is equivalent to

(2.8) rwv' - ^ ^ + i ^ ( ^ ' ) - / " - ^ - 5^ = 0.

The justification for introducing this equation will be seen when we obtain thesemimartingale decomposition of the process M, defined below. (2.8) is satisfied by

*We are assuming that interest rates are the same on lending and borrowing.


V = V given by (2.5) and the maximizing v and c are equal to TT*, C* given by (2.6).Condition (2.3) ensures that,c*(O > 0.

Let (c, TT) e U be an arbitrary policy. The solution of (2.2) is given by

(2.9) w, = expirt + /"'((a - ) „ - jo-^vA du + crf'v^ DzA\ •'Q\ •^ / • •'O ;

xlw - fcêxp - fUa -r)ir« - iy(T^Tr^]du - a f ir^dzjds].I •'o L •'0 \ ^ I •'0 J ;

It follows from Holder's inequality and the fact that v is bounded that w, has finitemoments of all orders, and it is also clear that w, < wf where w° is the solution of(2.2) with the same v and c, = 0. Now define

(2.10) M, ••= - f'e-^'cj ds + e-^'v{w,),y •'o

where v is defined by (2.5). By the Ito formula,

M, - Mo = f'e-^'\A''''v + -c^ - 8v] ds + aC-^ ('e-^'TT.wJ dz,.' I y 1 '0

It follows from the above argument that the second term on the right is a martingale,while the first term is, in view of (2.7), a decreasing process which is equal to zerowhen (c, v) = (c*, ir*). Thus M, is a supermartingale, and is a martingale when(c, TT) = ic*, IT*), so that

(2.11) v(w) =Mo> KK = KIy

By using the Ito formula and the wealth equation (2.2) we find that

(2.12) e-*'< = w^G,expij'ais) ds\.j

where G, is the exponential martingale (it is a martingale since TT is bounded)

G, = exp<< f'yTTâdz^ ~ ^ f r^'^/<^^

ais) = y[r + (« - r)rT, - (c,M) - ^1 - r )^> ' ] - «•

When ic,TT) = ic*,TT*) we find that ais) = -C and hence from (2.12) thatEê'^'viw,) -> 0 as r -> 00. It now follows from (2.11) that viw) = Jîc*, TT*).

To complete the proof we have to consider the cases 0 < y < l , y < 0 separately.In the former,

(2.13) ais) <-il-y)C,

equality being achieved when Cj = 0 and v^ = TT*. This implies as above thatEê~^'viw,) -• 0 as / -* 00 and hence from (2.11) that, for any policy ic, TT), viw) >JJic, TT). Thus ic*, TT*) is optimal.


Now take y < 0. The preceding argument fails because there is no longer ana priori upper bound for ais). Instead, we proceed as follows. For e > 0 define

Then v'îw) = v'iw + e) etc. and we see from (2.8) that v^ satisfies

riw + .)v'Sw) - f M l ^ ^ lîvMr'-^' - ^Â-) = 0.

Since vîw) > 0 this shows that

I.e.

+ -cT - 8v\ < 0.

Now v^ is bounded on R+ with bounded first and second derivatives. We easilyconclude, by introducing the process M, as in (2.10) but with v^ replacing v, that forany ic,rr) G U and w > 0, ^^(H') > Jj,c,rr). Since vj.w)i viw) as e iO this showsthat viw) > sup(^^)eu^(c. '^)- But we know that viw) = Jjc*, TT*), SO (C*, 77*) is infact optimal, as claimed.

COROLLARY 2.3. Under the optimal policy ic*,v*) the wealth process wit) is givenby

(This formula also applies when uic) = log c, setting y = 0.) Using (2.5) we findthat the evolution of utility is as follows:

(2.14)

(cf. Remark 6.3 below).

3. Transaction costs: preliminaiy discussion. Let us now consider the situation,outlined in §1, in which transaction charges are imposed equal to a constant fractionof the amount transacted, the fractions being A and M on purchase and salerespectively. The investor's holdings in bank and stock at time t are denotedSgit), Slit) and these are constrained to lie in the closed solvency region

0 and x + il

We denote IQ' d*,d;^ the upper and lower boundaries of .y^^ respectively (see


Figure 2 below). It is clear that the investor's net worth is zero on 5^ U 5~. A policyfor investment and consumption is any triprfe (c,, L,, U,) of adapted processes suchthat (c,) satisfies (?.l) and iL,) and iU,) are right-continuous and nondecreasing withLQ = Uo = O.iL and U are the cumulative purchases and sales of stock respectively.)The investor's holdings isît), sît)) starting with an endowment ix, y) e ^ ^ evolvein the following way in response to a given policy (c, L, U):

(3.1) dSoit)=irSoit)-cit))dt-il+\)dL, + (i-tJi)dU,, SoiO)=x,

dsît) = asiit) dt + CT-s,(O dz, + dL, - dU,, 5,(0) = y.

It follows from Doleans-Dade [6] that equations (3.1), have a unique strong solutionat least up to the bankruptcy time T = inf{t > 0: isît), 5,(/)) ^ ^y^.J-

An admissible policy is a policy (c, L, U) for which T = oo a.s. or, equivalently, forwhich Hisoit), Slit)) e ^^^ for all f > 0] = 1. We denote by U the set of admissi-ble policies. This set is clearly nonempty; indeed let (c, L, U) be any policy such thatisît), 5,(0) does not jump out of ^ (i.e. it is never the case that isît'), 5,(r")) e^^^ but (5o(0, 5,(0) € -y^,^). Then an admissible policy (c, L,U) can be con-structed by terminating ic,L,U) at bankruptcy, i.e. setting (c(f)L(O, t/(0) =(c(0, L(0, t/(0) for r < T,

= 0

T) = 5 , ( T ) j

and L(/) = Lit), Uit) = {/(T), cit) = 0 for t > T. Then 5o(f) = 5,(0 = 0 forunder (c, L, U).

The investor's objective is to maximize over U the utility

Here E_, denotes the expectation given that the initial endowment is SgiO)=x,5,(0) = y. Define the value function v as:

(3.2) t'(;<:,y)= sup 7, ,,(c, L,{/).(c, Z.,

The following properties of v are easily established directly from the definition.

THEOREM 3.1. Suppose uic) = c^/y for y e T, or uic) = log c.Then(a) V is corwave.(b) V has the homothetic property: for p > 0

v(px,py) =-^\

PORTFOUO SELECTION WrTH TRANSACTION COSTS 685

PROOF, (a) This is easily established by considering convex combinations of initialstates and control process and using the linearity of equations (3.1) and concavity ofthe utility function. This idea appears in [13].

(b) Denote by Mix, y) the class of admissible policies starting at isgiO), s0)) =ix, y) e ^ ^ . Then it is easily checked from equations (3.1) that for any p > 0.

nipx,py) = [ipc,pL,pU): ic,L,U) e U(x,y)}.

Thus

vipx,py)= sup Ep;,,mpx,py)

= sup E J, fe-^'uipc,) dt =•• V.

When uic) = c'*/y we have uipc) = p^uic) so that v = p'^vix, y), whereas whenuic) = log c then uipc) = log p + uic) and v = (log p)/8 + vix, y). This completesthe proof.

In order to get some idea as to the nature of optimal policies, let us takeuic) = c'>'/y and consider a restricted class of policies in which L and U areconstrained to be absolutely continuous with bounded derivatives, i.e.

L, = fl, ds, U, = j'u, ds, 0 < l,,u, < K.•'o •'o

Equation (3.1) is then a vector SDE with controlled drift and the problem may beattacked in exactly the same way as in §2. The Bellman equation, to be solved for thevalue function v, is

max(^^•'•"5(^,3') + ^c^ -8iJix,y)\ = 0 ,c,i,u\ y I

where A"-'-" is the generator of (3.1) for fixed c, /, u. Written out in full this becomes

(3.3) msx.{^a^y^Vyy + rxv^ + ayv^ + -c^ - cv^c,l,u {•'• '

] ^ Q

where Vj, = dv/dx, Vy = dv/dy. Note that both of these derivatives must be positivesince extta wealth will provide increased utility. The maxima are achieved as follows:

K if Vy>i\-^

Q if f; < (1 +

O if Vy>i\-

K if t;, < (1 -


=0 )

FIGURE 2 Bank/Stock Space Showing Directions of Finite Transactions and Solvency Region

This indicates that the optimal transaction policies are bang-bang: buying and sellingeither take place at maximum rate or not all all, and the solvency region ^ ^ splitsinto three regions, "buy" iB), "sell" (5) and "no transactions iNT). At the boundarybetween the B and NT regions, v^ = il + k)v^ whereas at the boundary betweenNT and S, v^-îl - ti)v^. We now have to consider what shape these boundariesare, and here we use the homothetic property (b) of Theorem 3.1. This property doesnot hold for the restricted problem we are presently considering, but will hold in thelimit as K 00. Assuming that u is C' and homothetic, we find by direct calculationthat for p > 0

v,ipx,py) =p^-'vîx,y), Vyipx,py) = p^-'Vyix,y).

It follows that if vîx, y) = (1 + \)i)îx, y) or vîx, y) = (1 - ti)vj.x, y) for someix, y) then the same is true at all points along the ray through ix, y). This stronglysuggests that the boundaries between the transaction and no-transaction iNT)regions are straight lines through the origin. In the transaction regions, transactionstake place at maximum, i.e. infinite, speed, which implies that the investor will makean instantaneous finite transaction to the boundary of NT. These considerationssuggest the picture shown in Fig. 2: the no-transaction region NT is a wedge, theregions above and below it being the sell (5) and buy iB) regions respectively. Notethat a finite transaction in the 5 [B] region moves the portfolio down [up] a line ofslope - 1 / ( 1 - M) [ -1/ (1 + A)]. After the initial transaction, all further transac-tions must take place at the boundaries, and this suggests a "local time" type oftransaction policy. Meanwhile, consumption takes place at rate t;]/*'*"'^ In NT thevalue function vix, y) satisfies the Bellman equation (3.3) with / = M = 0:

max {T - c ^ - Sv\ = 0

I.e.

(3.4) +fj-y/i-y) ^ Sy = 0.

'The natural ranges of values for n and A are {0,1] and {0, ==] respectively, these values corre^jonding toslopes of the finite transaction lines in S and B between -45° and vertfcal (in 5 ) or horizontal (in B).

PORTFOLIO SELECTION Wmi TRANSACTION COSTS 687

To substantiate the conjectured solution just outlined we make essential use of thehomothetic property of Theorem 3.1(b), by which the nonlinear partial differentialequation (3.4) may be reduced to an equation in one variable. Indeed, defineiffix) — vix, 1). Then by the homothetic property vix, y) = y'îl/ix/y). If our conjec-tured optimal policy is correct then v is constant along lines of slope (1 - ^t)"' in Sand along lines of slope (1 + A)"' in B, and this implies by the homothetic propertythat

(3.5) îx) ^ Âix + 1 - fi)\ x^xo,

1 y

if/ix) =—Bix + 1 + \) , x>Xj,

for some constants A, B, where XQ and Xj are as shown in Figure 2. (The factor 1/yin these expressions turns out to be notationally convenient.) Using the homotheticproperty again we find that, with i/f' = di/z/dx.

(3.6) Vyix,l)

v^îx, 1) = -y(l - y)Hx) + 2(1 - y)xrix) + x^'Xx),

and hence equation (3.4) reduces to

(3.7) Psx^rix) + p2Xii>'ix) + PMx)

/ ' ' ) " ' ' ^* ' ' ' ' ' = 0, x&{x^,xr], where

(3.8) /3, = - io-2y(l - r ) + « r - ^ . /32=o-2(l - y ) + r - a , ^3 = 10-'-

The key to solving this problem is thus to find constants x^^, Xj, A, B and a globallyC^ function i/r such that (3.5), (3.7) hold. The definitions of /3,, /32, 3 in (3.8) will bemaintained throughout the paper.

4. A sufficiency theorem. In this section we will show that the existence of a C^function ^ satisfying (3.5) and (3.6) supplies a sufficient condition for optimality of apolicy ic,L,U) such that the corresponding process isît), 5,(f)) is a reflectingdiffusion in the wedge NT and L,,U, are the local times at the lower and upperboundaries respectively. We first verify in Theorem 4.1 that such reflecting diffusionsare well-defined in arbitrary wedges in the positive orthant of R . We then give thesufficiency theorems. Theorems 4.2 and 4.3. The existence of such a function i/r isdemonstrated in §5.

THEOREM 4.1. Take 0 <X(f < x^ and let NT be tiw closed wedge shown in Figure 2,with upper and lower boundaries dS, dB respectively. Let c: NT -* U^ be any Lipschitzcontinuous function and let ix, y) G NT. Then there exist unique processes SQ, S, andcontinuous increasing processes L,U such that for t < r = inf{t: isj^t), sît)) = 0}

(4.1) dsît) = [rsît) - cisît),s,it))] dt

dsît) = asît) dt + asît) dz, - dU,, s,(0) = y,

^> J hu,U)^sM))^'>B)dL^, U, = jJiu,î),st{O)ef>s

The process c, — ciSffit), s,(0) satisfks condition (2.lXi).


The proof will be omitted. Note that the directions of reflection are along thevectors ((1 - fi), - 1) and ( - (1 -t- A), 1) at the upper and lower boundaries respec-tively. These coindde" with the directions of finite transactions in 5 and B, as shownin Figure 2. The process is^, 5,) is a degenerate diffusion with coefficients which arenot bounded away from zero, and with oblique reflection at a nonsmooth boundary.Because of this combination of factors, standard results on existence and uniqueness(e.g. Stroock and Varadhan [19]) do not apply. However .since one is only interestedin the solution up to the first hitting time of the comer, the solution can beconstructed piecewise, using a sequence of stopping times as in Varadhan andWilliams [22], from diffusions reflecting off one or other of the two line boundaries.The technique also appears in Anderson and Orey [1]. The result may also be derivedfrom Tanaka's theory of reflecting diffusions in convex regions [21].

For the sufficiency theorems which follow, we will only consider policies that do notinvolve shortselling (although borrowing is allowed). This class of policies is definedformally as

(4.2) U' = { ( c , L , C / ) e U : ( 5 o ( f ) , * , ( O ) e ^ ' for alU > 0}

where ^ ' = [ix, y) e R : y > 0 and (x + (1 - /x)y) > 0}. The results can be ex-tended to cover policies which allow shortselling, but because of the way we haveparametrized the problem a separate argument has to be introduced to show thatshortselling is not optimal; it does not seem worth presenting this argument here. Forsome of the results it is necessary to restrict the admissible policies further to theslightly smaller class U" defined as follows. For M' > M let

') = [{c,L,U) e U: {s^it),s,it)) e yj for all t > O}.

Now define U" = U^.>^ U"(M')- Using a policy in U" means that the investor is alwaysable to absorb a slight increase in the transaction costs. It seems certain that theresults below remain true if U" is replaced by U', but our proof technique requiresthe smaller class.

Theorem 4.2, which we present next, covers the case uic) = c^/y for y e T. Thecorresponding results for uic) = log c are stated separately as Theorem 4.3.

THEOREM 4.2. Take y e T and assume that Condition A holds. Suppose there areconstants A, B, XQ, XJ and a function t/»: [-(1 - /u.), oo) -> R swc/i that

(4.3) 0 < Xo < J r < °°'

(4.4) 1 ix C^ and ^'ix) > 0 forallx,

(4.5) ^{x) = Aix + \ - fiY forx<Xo,

(4.6) Psx^rix) + fi2X^'{x) + pMx)

z J l j ) ] - ^ / " - ^ ^ = 0 forx e [x^,Xrl

(4.7) ^ix) = Bix + 1 + X)^ forx>xr.

Let NT denote the closed wedge {(jr,y) e R .: xf' < yx'^ <jro *} and kt B and S


denote the regions below and above NT as shown in Figure 2. For ix,y)& NT\ {(0,0)}define

(4.8) c*ix,y) =

Let cf = c*isgit),slt)) where isg,Si, L*,U*) is the solution of (4.1) with c ••= c*.Then the policy ic*it), L*it), U*it)) is optimal in the class W for any initial endowmentix, y) G NT when y G (0,1). When y < 0 this policy is optimal in the class U". Ineither case, if ix, y) ^ NT then an immediate transaction to the closest point in NTfollowed by application of this policy is optimal in U', U" respectively. The maximalexpected utility is

(4.9) vix,y)

For the proof, we require the following lemma.

LEMMA 4.3. Suppose v is defined as in (4.9). Thenii) V is concave.

(ii)

(iii) (1 - ij,)v^- VyÔ with equality in S,

- (1 4- A) u, + t'j, < 0 with equality in B.

(iv) Define

(4.10) Gv = \(T^yî\y + rxv^ + ayv^ - Sv + - y ^ ( I ' J

Then

(4.11) Gv=-max(â^y^V +irx-c)v^ + ayvy+-c''-8v^ and^Vyy +irx-c)v^ + ayvy+-

(4.12) Gv = 0 in NT ii.e. the Bellman equation is satisfied),

Gv <0 inSu B.

PROOF. These properties can be verified directly from the construction of v.

PROOF OF THEOREM 4.2. Let (c, L, U) be a policy in U' and let isgit), 5,(r)) be thecorresponding solution of (3.1) with initial point ix, y) G ' . For an arbitrary C^fimction x let Mf be the scalar process defined for F > 0 by

-^'hcHt) dt + e - ' ^ x i )


An application of the generalized Ito formula (Harrison [9, §4.7]) gives

(4.13) Mf - M§

==/, + I2 + I3 + I4 + 15-

In this equation, x, Xx etc are evaluated at isît), s^t)) unless noted otherwise.Suppose first that (c, L,U) = ic*, L*, U*) as defined in the theorem statement and

(jc^ y) e NT, and let v be given by (4.9). The value of c* is equal to [y^]"'/*'"^', themaximizing value obtained from the Bellman equation. Note from (4.8) thatc*ipx, py) = pc*ix, y) for p > 0, and hence that c*ipx, py) = c*ix, y), c*ipx, py)= c*ix, y). It follows that c* is Lipschitz continuous and that (4.1) has a uniquesolution isoit),sît),L*,U*) when c is equal to c*. Now consider (4.13) with(c, L, U) = ic*, L*, U*) and x = v. It follows from Lemma 4.3 that / , = /j = /3 = 0,and /4 vanishes since isît), 5,(0) is continuous. Hence

(4.14) vix,y)=MI, = fV^'lcyit) dt + e-'''vi'0 '

It is shown in Theorem 6.2 below that under ic*,L*,U*) the value processvisfjit), 5,(/)) can be represented in the form

(4.15) vis,it),

where Git) is the Girsanov exponential

Git) = expf/Jyo-(l - / ) dz - ^/Jr'^'(l - ff

and / and g are bounded functions in NT with g > 0. It is also shown in Lemma 6.1


that in NT

(4.16) yv,ix,y)=

Since / is bounded, E yG\t) < <x> and it follows from (4.15) that the last term in(4.14) is a martingale. Thus

V^'^cyit) dty

As r -> 00, the last term converges to zero in view of (4.15) and this shows that, forix, y) G NT,

(4.17) vix,y)=J,,yic*,L*,U*).

For ix,y)e^^'\NT it is clear that J^yic*,L*,U*)=J,.^y.ic*,L*,U*) where(x', y') is the point on the boundary of the wedge to which the initial transaction ismade. Since v is by construction constant on the line joining (JC, y) to ix', y'), itfollows that (4.17) holds throughout ^ ' .

We now show that vix, y) > / . / c , L, U) for arbitrary ic, L, U). We need separatearguments for the two cases 0 < y < 1 and y < 0.

Case (a): 0 < y < 1. Here we need the following lemma, whose proof is given later.

LEMMA 4.4. Let v be defined by (4.9) with y e (0,1). Then there is a constant K andfor each e > Q a constant K^ such that

(4.18) Q^vix,y) ^Kix+yY for all ix,y) ^ ^ ; ,

(4.19) 0 <yv{x,y) ^Kîl+x+y)

forallix,y) &Ulu{ix,y) e ^ ' : x + (1 - ti)y > e}.

Let ic, L, U) e U' be an arbitrary policy and isoit), Siit)) the corresponding solu-tion of (3.1). If we define w, := sît) + Siit) and TT, = Siit)/w, then (c,,ir,) is anadmissible policy for the Merton (no transaction costs) problem. From (3.1) we seethat w, satisfies

dw, = [(r + ( a - r)TT,)w, - c,] dt + (TW,TT, dz, - dA,

where A = XL + /JLU. By the Gronwall-Bellman lemma we conclude that w, < w, forall /, where w, is the solution of the Merton wealth equation (2.1) with WQ = x + y.Fix e > 0, define vîx, y) = vix + €, y), and consider equation (4.13) with x ~ f'-From (4.19) we see that for any T > 0,

dt < Klj\\ + e + wf dt

The last ejqjression is finite as was shown in the proof of Theorem 2.1. Thus, the lastterm /j in (4.13) is a martingale. From part (iv) of Lemma 4.3 and since vîx, y) =


vîx + e, y) etc. we find that

. Gvîx,y) = Gvix + e,y) - revîx + €,y).

In view of (4.6), (4.11) and (4.12) we see that for any c > 0 and ix, y) G ^ ' ,

- c ^ + irx - c)v^ + ayv'y + ja^y^v^y - Sv <0.

Hence /, is a decreasing process. It follows from Lemma 4.3(iii) that Ij and /j aredecreasing while Lemma 4.3(i) implies that I^ is decreasing. Thus Mf is a super-martingale, i.e.

(4.20) v'ix,y)=M^'>l,^

Now using (4.18) we have

for some K' > 0, y' G (y, 1). It follows from (2.12), (2.13) that

Thus taking the limit in (4.20) as 7 - ^ <» we obtain vîx,y) >J^yic,L,U). Nowv'ix, y)i vix, y) as e iO, so that vix, y) > J^^yic, L, U). Thus (c*, L*, U*) is optimalin the case y G (0,1).

Case (b): y < 0. The problem here is that the candidate value function vix, y) isunboundedly negative at low levels of wealth. We use an argument similar to theproof of Theorem 2.1, based on the following lemma, whose proof is again given atthe end of this section.

L E M M A 4 . 5 . Let v be the function defined by ( 4 .9 ) with y < 0, and for e , e > 0define v^-%x, y ) = v i x + e e , y + e). Fix fi' > fi. Then there exists 0 > O such that forall e > 0

Gv'%x, y) < 0 for all (x, y) G ^ , ' ,

where G is defined by (4.10).

We note from Lemma 4.3(iv) that this result implies that for all c > 0,

(4.21) â^y^v^f + irx- c)v^' + ayv; + ^c^ - dv^-' < 0, (x, y) G ^ , ' .

We also note that for any e,« > 0, t;'* is bounded on ^ . ' with bounded first andsecond partial derivatives. Now let ic,L,U) be an arbitrary policy in U"; then(c, L,U)e Wifi') for some fi' > ix. Consider the process Mf of (4.13) with x = v^'^-We conclude from (421) and Lenuna 43 as t«fore that Mf' is a supennartingale


and hence that

The second term on the right converges to 0 as T -> oo, while the bracket [... ] in thefirst term is monotone decreasing. It follows that

Now v^'îx, y)i vix, y) as e J.0 and hence

v{x,y)-^ sup J^yic,L,U).(c, Z,,t/)eU-

On the other hand we know that vix, y) = J^îc*, L*,U*) and it is clear that(c*, L*, U*) e U" because the solution process is trapped inside the wedge NT afterany initial transaction. Thus ic*, L*,U*) is optimal in U", as claimed.

PROOF OF LEMMA 4.4. Denote 5+= 5 n R^ and 5_= 5 \ 5+, with similar defi-nitions for B+, B_ (note that B^= B n ^ ' ) . Elementary geometry shows that

(4.22) y<jîx+y) for all (x, y) e ' .

Using this, (4.18) is readily verified in 5 U B+, while in NT,

where Mg = fi'"^ maxjg[^^^^^ji/f(^). Thus (4.18) holds throughout 5^. In view of (4.9),in ATT

(4.23) yv,ix,y) = yy^'^^f) -xy^-y(j) < yMîx + y)\

In J5+,

(4.24) yvîx, y)

and similarly in S+,

(4.25) yv,ix,y)

In 5_ we note from (4.22) that when ix + il - ii)y) > e.

(4.26) yvîx, y) Âyix -K (1 - /i))-)^"' < - 4 = T ( ^ + >•)fie

We can now choose K^ such that Kîl + x + y) majorizes the right-hand sides of(4.23)-<4.26X thus ^tablishing (4.19).


PROOF OF LEMMA 4.5. Take e,e > 0. We know from Lemma 4.3(iv) that

0 > Gvix + e-e,y + e)

= |o-^(y + e)^Vyyix + d€,y + e) + rix i )

aiy+ e)v^ix + 0e, y + e)

, y) + epie) where p(e) - \cr^i2y + e)y,V + ''^''x-' +

We need to show that pie) > 0 so that then Gv'-^ < 0. First, consider points ix, y)such that ix + 0e, y + e) e NT. Then, with i-^ ix + 0e)/(y + e), we have from(4.10)

v;''ix,y) = iy + ey

v;fix,y) = (y + €)''-\er + 2(1 -

and hence

(4.27) ( y + e)'~V(e) = rO^ + j<T^vi^r +

+ [ay

where 77 == (2y -h e)/iy + e). Now ^ e [XQ, X^] and 17 e [1,2], so that all the termson the right of (4.27) are bounded. Since ./r' > 0 it follows that pie) > 0 if 0 is chosensufficiently large. Note also that the minimum value of 6 required does not dependon e.

For ix, y) e 5 we have vix, y) = Aix + il - iJ.)y)''/y and hence

pie) =Aby

where b •= x + il - ti)y + ie + 1 - iM)e. In 5+ (i.e. when JC > 0) we have

2y + e , 2y 4- £ , 2

so pie) > 0 as long as

A similar calculation applies in B.^. The remaining case is 51= 5;, n {x < 0}. Here


we have x + il — fi')y > 0 for some fi' > fi, so that

2y + €

x + il-ti')y+ in'

2y + €

ifi' - iJ.)iy + e) + (1 - At')

Thus pie) > 0 if

the right-hand side being independent of e. Thus for 6 > 6Q, where 6^ is a constantnot depending on e, we have pie) > 0 throughout 5^. This completes the proof.

The following theorem covers the case uic) = log c. Its proof is similar to that ofTheorem 4.2 and is omitted.

THEOREM 4.3. Assume 8 > 0, let j8,, /Sj, P^ be defined as in (3.8) with y = 0, anddefine P^-=i— \(T^ + a — 8)/8. Suppose that there are constants XQ, XJ, A, B and afunction i / > : [ — ( 1 — / A ) , o o ) - > | R such that

0 < XQ < Xj- < 00,

il/ isC^ andtl/'ix) > 0 forallx,

^ix) '^ ^log[Aix + il - n))]

p^x^rix) + p^-log ilf'ix) =0 forxe[xo,Xr],

= -glog[Bix + I + X)]

Let regions NT, B,Sbe defined as before and for ix,y)eNT\ {(0,0)} define c*ix, y)= y[ili'ix/y)]~^. Then with the utility function «(c) = Iogc, the policy ic*it) •=c*isQit), Slit)), L*, U*), where (SQ, ^i, L*, U*) is the solution of (4.1) with c == c*, isoptimal in the class U" for any initial endowment ix, y) G NT. If ix, y) € NT then aninitial transaction to the nearest point in NT followed by application of this policy isoptimal. The maximum expected utility is

(4.28) vix,y)=\logy +

The existence of functions ^ satisfying the conditions of Theorems 4.2 and 4.3 isestablished in §5 below when a > r. When a = r, it is intuitive that we would investonly in the bank. Our final result in this section demonstrates that this is so.

THEOREM 4.4. Suppose a - rand 8 > yr. That the optimal strategy mU is to closeout any position m stock and to consume optimally from bank.


PROOF, [sketch] For y G T and (x, y) G ; define wîx, y) = Kix + (1 -ti)yy and Wjix, y) = Kix + il -^ A)y)^ where K ~ ii8 - yr) / ( l - y))^ 'y '. Inview of Remark 2.2, ^(x, y) ••= wîx, y) A W2ix, y) is the utility of the policy de-scribed in the theorem statement. Taking an arbitrary policy (c, L, t/) G U andapplying a supermartingale argument as in the proof of Theorem 4.2 successively tothe functions ^ , , ^ 2 , we find that J^yic, L,U) < wlix, y), i = 1,2 and hence that7 ic, L, U) < wix, y). This approach sidesteps the nonsmoothness of w at y = 0. Asimilar argument handles the case uic) = log c, using the functions w,(x, y) = K' +log[5(x -1- (1 - ti)y)]/8 and Wj = K' + Iorf5(x + (1 + X)y)]/8, where K' =ir - 8)/8\

5. Solution of the firee boundary problem. In this section we give our mainresult. Theorem 5.1, concerning the existence of a solution to the free boundaryproblem. To prove this, we have to establish certain properties of the system ofordinary differential equations (5.11) below; this is done in the Appendix. We findthat a technical condition, introduced in the Appendix as Condition B, is required.We are virtually certain that this condition is nugatory, i.e. is always satisfied, but wecan prove this only in some cases. Further comments are given in the Appendix.

THEOREM 5.1. Suppose that Conditions A and B hold, that fi G [0,1[, A G [0,OO[,

/t V A > 0, and that

(5.1) 0 < r < a < r + ( 1 - y ) o - ^

Then there is a C^-function i/»: ( - (1 - M)>'=°) -^ ^ and there are positive constantsXo, Xj., A, B with 0 < Xo < Xj < 00 satisfying:

(a) in case y G F:

(5.2) i^(x) = Âix + 1 - n.y, X < Xo,

X G lX(]

(5.4) t^(x) = -iS(x-I- 1 + A)^, x>Xr,

(5.5) yîx)ili"ix)+ il-y){^'ix)f <Q with equaUty when x <XQ or x

(b) in case y = 0:

(5.6) ^(x) = i l o g [ . 4 ( x + 1 - M ) ) ] .

(5.7) pMx) + PiX^X^) + j83xV(x) + ^

(5.8) i^(x) = |log[S(x +1+A)] ,

(5.9) l^^(^) ^'^'^C^) < 0 withe^mUtyforx

= 0,

PORTFOLIO SELECnON WITH TRANSACTION 697

REMARK 5.2. Properties (5.5), (5.9) will be needed to prove that the functions vconstructed in (4.9), (4.28) are concave. (5,1) is the same as the condition whichensures, in the Merton problem, that hedging, rather than leverage or shortselHng, isoptimal. See Remark 2.2.

Before proceeding to the proof, let us point out conceptually what is involved insolving the free boundary problem. At a given point XQ, (5.2) and (5.3) provide twodifferent expressions for if/". Equating these, we find that there is a unique value ofA, say Aix^^), such that if/" is continuous at x,,. Similarly, continuity of 4f" at Xj^ fixesBixj.). Now write ^ix) = iC\x), ^Hx)) = iipix), il/'ix)) and write the scalar second-order equation (5.3) as a 2-vector first-order equation in the form

(< in'i -—Kx) = fix Z'(jc))

This can be solved, at least locally, from any initial vector Co at .»: = x^^; denote thissolution ^(jc; ^,,, JC,,). NOW (5.2) implies that the initial vector ^Q is fixed once x,, isgiven; indeed

v v - i

Thus for given JC,, the solution of (5.3) is Cix\ C(^x^), x,,). Similarly we see that (5.4)specifies the value CT^XJ) of the terminal vector CT- Th"s the free boundary problemis solved if we can find x,,, Xj such that 7 ( :7-) = Cixj', Uix^^, x,,). Generically, thereis no reason why this equation should be solvable, but the proof below shows that infact such JC,,, Xj exist under the specified conditions. As will be seen, the key to theproof is the introduction of nonlinear coordinate changes under which (5.10) takes asimpler form.

PROOF OF THEOREM 5.1. (a) For / e [0,1] define quadratic functions Qif), Rif)by:

Rif) ••=

Notice that QiO) = MO), Qil) = Ril). Now consider the following one-parameterfamily of initial vale problems {(jc,,,/( ),M )); 0 < JC,, < (1 - ti)P2M - r)), whereJC,, is the starting value:

(5.11) - ~

h'ix) -Qifix))],


REMARK 5.3. The range of XQ implies that 0 < / „ < / „ = 1 - (a - r)/il - y)a^.Here /„ is the value of / at which Qif) achieves its minimum. Condition (5.1)ensures that /„ e (0; 1) and we find that

Qifm) = y - r - 2 ( 1 - y (

so that Condition A ensures that yQif^) > 0.It is shown in the Appendix that there exist XQ and Xj and a solution (/('),hi-)) to

(5.11) such that

(5.12) (i) 0 <JCo <Xj,

(iii) h'ix^) = h'ixr) = 0, M^o)

(iv) hi-),fi-) are increasing function on ix^, Xj^) with y/i > 0.(v) fix) e (0,1) for x e (;co, Xr).

It is also shown that /Q decreases and / ^ increases with respect to A for fixed ti.Next, define for x e [XQ, XJ]

' ^ ' ' " fix) •

In view of (5.12Xiv) above, pix) is decreasing on [xg, Xj-]. Elementary computationsshow that

From (ii) above, we have that

(5.13) qixo) = 1 - M, «(^r) = 1 + A.

Notice also that p'/p = - /« 'A = - 9 ' A + 95 this implies that q is increasing on[.>Co' ^T^- Now define

(5.14) Hx) ••= ^PnJf)[x + «(jc)]^ x>-il-ii).

For X e [xo, Xj-J we find, by using the alKJve formulas for p'(x) and q'ix) and the

PORTFOLIO SELECTION WrrH TRANSACTION COSTS 699

relationship x + qix) = x/fix), that

(5.15) nx) = \y'l>ix)fix)

(and hence that i^'(x) > 0 as (4.4) requires), and

X

Also,

Using these formulae it is easily verified that ^ satisfies (5.3). Putting A = pjix^),B =pyixT), we note that i ( ) has the required form outside [XQ, x^ ]. In view of(5.12Xiii), these continuations are in C , at least. Now, for x in [xo, Xj-]

- h - ^J^ ^,il - y)f']

with equality when x = Xo or x = Xr, by (5.12(iii)) and the fact that h is increasing in(XQ, XJ-). When - (1 - M) < ^ < ^0.

y,l,r + (1 - yW^ = yAHx + 1 - M)'^"' ^(y - D + (i - y)A\x + i -

and similarly vdien x > x^- TTien, since ^ x ) > 0 for all x > - ( 1 - 11), it followsthat ^"i-) h continuous. Hemx, the constructed function ^ is C^, and satisfies

7CK) M. H. A. DAVIS & A. R. NORMAN

(b) When y = 0, the argument is similar. The corresponding initial value problemsare:

(5.16) f'ix) = -^[

h ' i x ) ^ - ^ [ h - Q i i f ) ] , X > J : O > O . ' w h e r e

Q i i f ) = - S p , - 8 log S - ^

and 0 < XQ < (1 - ti)P2/i" - r). In this case we define

-hix)

Remarkably, the relation p' = -pq'/ix + q) still holds; indeed, we find that

As before, we have qixg) = 1 - fi, qixj) = 1 + k, and we define qix) = qixg) forX < Xg, qix) = qix-r) for x > Xj. Similarly, we define pix) =A= pixg) for x < Xg,pix) = pixj^) = B for x^ x-p. We can now verify in a similar manner to case (a) that

satisfies (5.6)-(5.11). This completes the proof.

6. A representation Oeorera fw the value process. From the function intrcKlucedin the proof of Theorem 5.1, we can obtain useful repre^ntations of yVyix, y) andvis^it), sfit}), evolving under the optimal policy (c*, L*,U*). These are needed tocomplete the proof of Theorem 4.2. TTiey also, sbcm that under the ^timal polio?,bankruptcy (toes not o<xur in finite time.

PORTFOLIO SELECTION WITH TRANSACTION COSTS 7 0 1

LEMMA 6.1.

(6.1) (i) yt;j,(j:, y) = y j l - / I — I jj;(j:, y) for ix, y) e NTwhen y G T

where f is given by (5.12)

(6.1') (ii) yvyix,y) = Ul-f(-]\ for ix,y) & NTwhen y = 0

where f is given by (5.16).

PROOF. For case (i), express Vy in terms of i^j using (3.6) and obtain / from (5.14),together with the relations (5.15) and (5.16). An analogous procedure gives (6.1').

THEOREM 6.2. (a) Let the conditions of Theorems 4.2 and 5.1 hold, let ic*, L*, U*)be the policy described by Theorem 4.2, let sît), s*it) be the solution of equation (4.1)with starting point ix, y), and define ^* = sît)/sît). Then, for uic) = c^/y, y e F,we have

(6.2) vis*iT),sriT)) = vi^

-fii,*))dzit) -

(b) When uic) = log c, the corresponding result is

(6.2') visgit),s

j{l-fii,*))dzit).

REMARK 6.3. (6.2) and (6.20 are the counterparts in the transactions costs case ofthe expression for the wealth process in the Merton problem obtained in Corollary2.3 above. Indeed, (6.2) reduces to (2.14) when A = / = 0, because then fH,*) = / „ ,fiii,*) = h^, where / „ and h^ = Qif„) are as given in Remark 5.3 above. Substitut-ing these values in i62), we obtain the Merton expression (2.14).

PROOF. First, observe that for any ix, y) e NT, we have:(a.i) When uic) = cVr , 7 ^ (0, D, then vix, y) > 0 and vix, y) ¥= 0 « (JC, y)^ #

(0,0)^.(a.ii) When uic) = c'^/y, r < 0, then vix, y) < 0 and vix, y) > -oo «=> (jc, y)^ ^

(0,0)^.(b) uic) = log c, then vix, y) takes both positive and negative values, and

v{x,y) > -00 «* ix, y)^ # (0,0)^.Soppose s%0} ••= isîO),sfiO)) = ix, y) e NT\{(^ (without loss of generality).

T ••= 'mSiti s*it) = 0}. Then T > 0 a.s. by rantinuity of the wealth process.

7 0 2 M. H. A. DAVJS & A. R. NORMAN

(a) Applying the Ito differentiation formula to lode-'^''^'^yvis*iT A T))] yields:

(6.3) log[e-'^''^^^yvis*iT A T))] - logri;(i*(0))

= r^^^lf-gy + rs^v.^ + asfVy + la^s^Vyy - c*vAdt•'o ^ L

r'-'iU-H + A)», + iv) dL^l) + ((1 - M)», - ".)•'o '^

f ""^VycrsJQ V

r^''-J—n-^,!*^,,^dtJo

since the boundary terms vanish, where

Gv = jja^y^Vyy + rxv^ + ayv^ - Sf + | —

First GfCs*) = 0. Next, recall from (4.9) that vix, y) = y^«/r(x/y). From (5.14), (5.15)

we find 'that iyv)-KvX'''''''^ P''''''''- ^ f " ^^^^ ' ^ ^'^Z'' =). Thus (6.3) simplifies to

2

Put /i = (d - y)/y)j9"'^/*''"^' and obtain the following, by exponentiation:

(6.4) yvis^iThr),sfiTAr))

= yvix,

Since f G (0,1), a y<ril -f)dz is a time-changed Brownian motion, finite for all talmost surely. Also yh is bounded (away from zero). Write the right-hand side of(6.4) as yViT A T). Suppose T < «=, on some set A of positive measure and dioose asample path in A.


If 0 < yvix, y) < 00, then 0 < lim, _^ yVit A r) < ec since Vit A T) is a continuousprocess. On the other hand by definition of r,

limyvisît A r),sît A r)) = 0 or «>l-»T

contradicting (6.4). Thus T = <», and 0 < yvisît), sfit)) < » for all f > 0 almostsurely. So (6.4) is equivalent to (6.2).

(b) In the log case,

(6.3)' vis*iT A r)) - u(s*(O)) = /"^^''[log t' + 8v]dt C'

Since v satisfies the Bellman equation, we find by introducing the functions p, q, f of§5 that the right-hand side of (6.3') is equal to

+ Qi^n^r)] dt + /J^ ' f (1 -

+

where h = —8 log p. So

visîT Ar),s*iT A r)) = vix,y)

Again, the integrands are bounded. So the assumption that T < oo leads to acontradiction, as before. Thus T = oo and visît), sfit)) > -oo for all t almost surelyif vix, y)> -00.

COROLLARY 6.4. The optimal strategies do not lead to bankruptcy in finite time.

7. Summaiy, numerical results and conclusions. Combining Theorems 4.2, 4.3and 5.1 we can now state the main result of this paper.

THEOREM 7.1. Stq/pose that /x G [0,1[, A G [0, oo[ a«d /a V A > 0.Case (a): utility function uic) = c^/y, y G T.Si^pose that Conditions A and B hold, and that

(7.1) r <a<r-t il-y)cr^.

Then there is a C^ soUttion ^ to the free boundary prt^jkm (5.2)-(5.4) with 0 <Xg<Xj. < op, mid the p^cy ic*, L*, U*) described in Theorem 4.2 is optimal in the class W(-y € (0,1)) or U" (y < 0). In partiadar, L*, U* are the local times cf the wealth


equation (4.1) at the boundaries of the no-transaction region NT. Thus the optimalstrategy is minimal trading to keep the proportion of wealth held in stock betweenTTf ••= (1 + Xj)-^'and T72* == (1 + Xo)'\

Case (b): utility function uic) = log c.Suppose that 8 > 0, that Condition B holds, and that r < a < r + a ii.e. il.l)

holds with y = 0). Then there is a C^ solution to the frte boundary problem (5.7)-(5.9)with 0 <Xo<Xj.<°°and the policy ic*, L*, U*) described in Theorem 4.3 is optimalin U".

If condition (7.1) is not met then the situation is similar to that described for theMerton problem in Remark 2.2: If a > r + (1 - y)<T^ then leverage is optimal andthe no-transaction region is a wedge in ^ ' n {JC < 0}, whereas if a < r shortsellingis optimal. If a = r then cashing out all stock holdings is optimal, as shown inTheorem 4.7. The case a = r + (1 - y)(T^ is unsolved, but we conjecture that itinvolves a wedge with vertical upper barrier, with a discontinuity in the secondderivative of the value function at this barrier. If Condition A does not hold thenarbitrarily high utility can be attained.

We now turn to computation of the optimal policy. This is best done in terms of thetransformed coordinates introduced in the proof of Theorem 5.1. Recall from (5.11),(5.12) that, for uic) = c^y, if tfiix), XQ, XJ is the solution of the free boundaryproblem (5.3)-(5.5) then there are functions fix), hix) which satisfy the system ofdifferential equations

(7.2)

(7.3) / ( ^ o ) = ^ ^ / i - ^ -/o^ hix,) = Qif,),

(7.4)

Now from (5.14) ^ is given by

where p, q are defined in terms of x,f,h by

We then see that

(7.5)

From (5.15) we have

(7.6)


while from (4.8) we know that the optimal consumption policy c*ix, y) is given by

In view of (7.5), (7.6) we can calculate c* directly in terms of the function /(.*:), hix):

c(x v) =^ '^^ il-y)fix/y)-

The problem therefore reduces to computing the solution to (7.2) with boundaryconditions (7.3), (7.4), which say that both at XQ and at Xj^ we have (/, h) e Q -{if,h): h = Qif)}- Figure 4 below shows some typical trajectories (one shouldappreciate that the "phase space" is actually 3-dimensional since (7.2) is nonau-tonomous). The minimum of Q occurs at / = /„ = 1 — (a - r)/io-^il - y)) e (0,1).

The algorithm for solving (7.2)-(7.4) is as follows.1. Choose any Xj such that fj G (/„, 1), where fj- ~ Xj/ixj + 1 -I- A). Define

2. Using numerical integration, solve (7.2) backwards (i.e. in the direction ofdecreasing JC) until the trajectory re-crosses Q. Let XQ be the value of x at which thishappens ixg = s\ip{x < Xj^: ifix), hix)) G Q } ) and let /„ == fixg).

3. Define

(7.7) m - x o - h l - ^ .Jo

We see from (7.7) that /Q = Xa/ixg -I- 1 - m), and hence, referring to (7.3), thatixg, Xj^) determined in this way solves (7.2)-(7.4) for the given value of A arui withfi = m. We can regard Steps 1-3 above as a function which maps x^- to m = mixj-).The solution of (7.2)-(7.4) is completed by embedding 1-3 in a one-dimensionalsearch procedure to find a value of Xj- such that mixj) = fi (the prescribed propor-tional cost for sales). The argument given in the Appendix shows that this search willalways be successful. The reason for integrating backwards rather than forwards inStep 2 is that this is the "stable" direction of (7.2): the solution integrated forwards isvery sensitive to the initial condition /(,, whereas the backwards solution is muchmore robust. An exactly similar algorithm, based on equations (5.16), solves theproblem for the utility function uic) = log c.

We plan to report more fully on the numerical results in a later publication, butFigure 3 shows some typical results. Our result says that the proportion of wealthheld in stock should be kept between 'n-f = 100(1 +Xr)'^% and i r | = 100(1 +XQ)~^%. ITie Merton proportion (no transaction costs) is ir* = 1(X) x 2ia - r)/a^.In the present case this is equal to 15.63%. We have taken fi — A (equal transactiona>sts on sale and purchase) and plotted ir,*, irf against A. The most noticeablefeature of these curves is that the upper (sell) barrier is very insensitive to A while thelower (buy) barrier decreases quite rapidly as A increases. This is probably due to theasymmetiy in the model: all consumption takes place from the bank, so stock must besold (and transaK fHi charges paid) before it can be realized for consumption. If thefiling diarge is high then this is unfortunate but unavoidable. On the other hand ifthe bulk of the invest(M°'s hcridtngs are in cash then the potential gains from investing


I T *

20

10

1

TtJ

x%FIGURE 3. Dependence of TT,*, TTJ on A (with /x = A). The Parameter Values are a = 0.12,

r = 0.07, a = 0.4, S = 0.1, -y = - 1 .

in stock and then reselling at some later date may not be worthwhile if the associatedcosts are too high. Hence the decreasing value for irf.

In [4], Constantinides considers exactly the same problem as in this paper andobtains approximate results by restricting the class of consumption policies dx, y) tothose satisfying

(7.8)cix,y)

= constant.

His results are qualitatively similar to those displayed in Figure 3. We can use ournumerical technique to check whether (7.8) is exactly or approximately satisfied fortruly optimal policies. For y =?*= 0 the optimal consumption c* is given by (4.9) and wesee that for ix, y) e NT

In NT, ^ ranges over the interval [XQ, Xjl Taking parameter values as above andA = ;.i = 0.01, we find that x^ = 0.406, Xj = 1.112, cix^^) = 0.188 and cixj) = 0.105.Thus the ratio in (7.8) actually varies over a range of nearly two to one. We have not,however, investigated how much utility is lost by imposing the restriction (7.8).

Finally, let us consider possible extensions of this work. Of couree, the mostinteresting extension would be to the case of m > 1 risky assets, but unfortunatelythis is essentially impossible except perhaps for m = 2 or 3. As Magill andConstantinides [17] point out, m risky assets imply 3" possible transaction regionsand, for example, 3'" = 60000. Althoi^h the coordinate transformation introduced in§5 generalizes to higher dimensions, it is not clear how to locate the boundaries evenwhen m = 2. TTie only readily solvable case is that in which transactions between riskyassets are costless. The lisky assets can then be combined via a mutud fund theoremtl7] and the problem reduces to the single risky asset case considered here. SeeMagill [16]. Another important question is nonconstant model parameters (interestrates, return on assets, volatility). It is unlikely that our problem could ever be solvedat the level of generality of, say, Karatzas et al. [11], w*ere these parameters aretaken as general stochastic pro(«sses. However, it niight be solvable if the random-ness of the parameters were nKxieUed in specific wa^, for eran^le as finite stateMarkov processes. This is an interesting area for fiuther research.

PORTFOLIO SELECTION WITH TRANSACTION COSTS

h

707

0 f»

FIGURE 4, (/, h)-Plane for y e (0,1), /„ < | ,

h=0

(i) (il) (iii)FIGURE 5, (/, h) plane for (i) y e (0,1), / „ > j , (ii) y < 0, / „ < ^, (iii) y < 0, / „ > | , When

y = 0 the pictures are similar, but parabola M becomes a straight line.

Acknowledgements. We would like to thank V. E. Benes, I. Karatzas, P. Varaiyaand R. J. Williams for helpful discussions and advice, and the two referees for theirdetailed attention and many valuable remarks. We would also like to thankD. Abeysekera for her unstinting help in the preparation of the manuscript.

Appendix. In this Appendix we examine the solution trajectories ix, fix), hix))e R^ of the nonautonomous first-order system defined by equation (5.11):

(5.11)

We show that for each /,i < 1 and A > -/x, these equations have a solution satisfying(5.12Xi)-(v).^ This will follow from the sequence of lemmas presented below. Theproofs of these lemmas are collected together at the end of the Appendix.

Denote by Cl and 31 respectively the graphs in R^ of the functions Q and R for/ e [0,1], and define Op == Kf, h):O<f< fj where / „ = = ! - ( « - r)/i\ - y)(r^is the value of / at which (2(/) attains its minimum. We also define i6 = R x D,£i,, = Ux€x, and SR = R X SR. £l and M enclose a region D in (JC, / , h) space andwe denote by D the projection of D onto the (/, A)-piane (see Figures 4 and 5). The

*We only need to prove existence because the uniqueness of the optimal payoff functions (3.2) will followimmediately.


only "stationary point" F ••= (1, QiD) = il, Ml)) = (l ,(S/y) - r) corresponds to astraight line trajectory F = (JC, 1, QiD) czQr^'Si.^

Fix fl < 1 and consider the solution of (5.11) starting at ix(^,fixo),hixo)) =ixo, /o. o) e Oo with JCo == XJ^fo) := (1 - /i)/o/(l " /Q)- We note that fix^) > 0and h'ixg) = 0, and that / ' , h' > 0 inside D. Thus, the trajectory moves monotom-cally in each variable across D and can only (I) exit through O, (II) exit through 91 or(III) tend to F. We call such trajectories types I, II, III respectively.

Suppose for a moment that (I) holds. Denote by x'j,fT,hj the exit values ofX, f, h, and define / = /(/o) == (^r / / r ) -Xj-l- Since QifJ = (5/y) - r - ia-rf/lil - y)(T^, Condition A implies that h > 0 ior all (/, h) & D when y > 0.When y < 0, at each point (/, h) on our trajectory we have h <hj< QiD = (5/y)- r < 0. Thus y/i > 0 in either case, and properties (5.13Xi)-(v) are all satisfied with\=l.We therefore need to show that for each k> -fi there is a type I trajectory with/ = A.

Consider first the extreme case fo = f^. We have the following result.LEMMA A . 1 . Fix fi < 1, as before. Set /o = / „ , hg = Qifo) and x^ = Xj^f^) > 0.

Then Wix^) = 0, /i'"(jto) < 0.This shows that the trajectory of (5.11) does not enter the interior of D, and hence

that XT = Xo, / r = / „ , ^r = Qifm) and / = -fi.Ît remains to show that /( /Q) ranges continuously up to infinity as we decrease the

starting value /o of the type I trajectory keeping (JÔ, fo, K) «" ô and maintainingthe relationship JCQ = Xj,fo). We shall discover the behaviour of /(/„) for fo < fn, byinvestigating the function qix) = Jt(l - fix))/fix) introduced in the proof of Theo-

i h ( ) 1 d i ) 1 + / We shall also usestigating the funct qi) ( f / f

rem 5.1, and noting that ^(JCO) = 1 - fi, and qixj) = 1 + /. We shall also use(without proof) the fact that solutions to (5.11) are line integrals which are continuousfunctions of the initial values. The following lemma is of crucial importance. It showsthat the projections onto D of trajectories with distinct starting points in DQ do notintersect.

LEMMA A . 2 . Let ix, fîx), /i,(jc)) and ix, f2ix), hîx)) be two solution trajectoriesof (5.11) starting at ifô, /J.o) ^ O Q , X.^ = ^^(/,oX i = 1,2. Define ®\ ={ifiix), hiix)): X > Jt:,o} and @, = ®,' (^D,i= 1,2. Suppose that fô > 0 andfô > 0,G(/io) < 2(1) and Qif^) < QiD, and hô > K^ isof^ < /,o)- Suppose if, A,) e ®,and (/,/i2) e ©^ /o r / e [0,1]. Then h^ > h^.

COROLLARY. Let ix, fîx), /I,(JC)) and ix, fîx), hîx)) be as above, and supposethat ix, fîx), hjix)) is a type I trajectory. Then ix, fîx), hîx)) is also of type I. Thusthe terminal value fr e (/„, 1) is a monotonic decreasing function offg.

Next, define / . == inf{/o: ixj, fixrX Kx^)) e &}. In view of the above results, alltrajectories with starting points /o such that / „ > /o > /» are of type I. We need thefollowing condition:

Cmdition B. / . > 0.

If y^ > i , 0 < y < 1, then Condition B automatically holds, since the trajectorywith Mxo) = QiD is necessarily of type II (see Figure 5(i) and (iii)). Numerical workconvinces us that /^ > 0 in the remaining cases, but we are unable at present to givea proof.

'The ii = \ case.*Whcn A - M = 0, this corresponds to the solution of the Merton prc*Jem, and indeed we find that

•cg = / ^ / ( l - / ^ X corresp<»diBg to the Merton pRqjortion v* of (Z6), namely it* l / d + *(a - r)/{\ - rV^.

PORTFOUO SELECTION WITH TRANSACTION COSTS 709

The next lemma completes the argument by establishing that for any specifiedA > -fl, there is a (type I) trajectory for which ^(A^) = 1 -f A.

LEMMA A.3 . Suppose Condition B holds. Then as /g decreases from / „ to /„,(i) qixj^ increases from 1 - fi to <x>, and(ii) Xj- increases from il — nXH - fm)/fn) '^ <».

PROOF OF LEMMA A.I. The two cases are:(a) y e r. Then

X-'X,,

Qif)] + j^h[h' - Q'f]

Also,

=h' + p.xfh"

But /(XQ) ^ 0, soNext,

= 0.

dx'= i83((2/' + xf")h' + 2if + xf')h"

Another expression for this derivative is

2h'[h' - Q'f] + h[h"

showing that h"'ixo) < 0.(b) y = 0. In this case.


So /I'XXQ) = 0. Further,

d^ ,.dx'

xf")h' + 2if + xf')h"

showing that h^ixg) < 0. This completes the proof.

PROOF OF LEMMA A.2. Consider two trajectories evaluated at common/. When y G r, we have

dh _ y h(h-Qif)\df - 1-y f\Rif)-hj-

Thus if /t,, /i2, lie in D (see Figure 5(i)), then

When y = 0, the corresponding expressions are:

dh_df

In either case, if /12 > h^, then dihz - h^)/df> 0. Since /12 increases in D, andh2o> hio then h2>hifoT all / , as claimed.

PROOF OF LEMMA A.3 . (i) Consider the case y G T first. We saw that qixj.) =I - fi when /o = /„• Take two trajectories with /J2 > Aj, as before. Now «2 increasessince q' = [h - Qifyi/Pif^ > 0. So, initiaUy ^2 > <?, = 1 - M- With a common / asin Lemma A.2, the corresponding values of x are x^, X2 given by:

r ^' - ^^^ + 9 ( » ^ i ) X2 + qix2)'


Then

h 2 ~ s 2 i f ) 1 .. i - - i i t v j / i i ^ Q

whenever ^2 > fl'i ('' l > 0)-In case y = 0, the expressions are the same, with Q, R replaced by (Qj, /?,. In both

cases, ^2 > ?i for all (common) / . Finally, since qj increases, we have ^2(^27) ^Qiixir), showing that qixj^) increases as fg decreases.

To see that qixj-) -• 00 as /Q decreases to f^, use the fact that 9 is a continuousfunction of fg and x and Lemma A.4 below which establishes that qix) -> 00 alongthe type III trajectory with fg = /„.

(ii) Since ^2/^1 ~ ^2/^1 > 1' we have X2 > Xi for all / , and since X2if) isincreasing, we have that X2r > x^f. That is, Xj. increases as fg decreases.When fg = / „ , we have fj = fg and x-r = Xg = il - /iXl - fj/f^, since / „ = /o =Xg/ixg + 1 — ix). Finally, since /^ = Xj-/ixj. + qixj^)) and qix^) is increasing andpositive, it follows Xj^ increases to infinity as ff increases to 1, proving (ii).

In the proof of Lemma 3 we needed the following fact concerning the behaviour ofthe special (type III) trajectory with fg = f^, / r = 1 (see Figures 4,5).^

LEMMA A.4. Along the type III iasymptotic) trajectory, qix) -* 00 as x -* 00.

PROOF, (a) For y G r .

q - x\R-Q

We know that h -> Qil), / -> 1 as x 00. By L'Hopital's rule,

h- - QTR' - Q')f

Write a=h-Q, b = R-h. Then

i im I '. F"v-,oc\ a + b

[0,1].

lim

r (^\ . ~^2 + 2/33(1 - y)lim(j) + ^^

'This corresponds to a no-transaction region in which the lower barrier lies on the jc-axis, i.e. the cast ofbtqnng into the stodc is prohibitive.


So,

n.K = 2/33(1 - y) - ^2 - r ^ • 2 ( i ) r ^ '

and /Ce [0,1], i.e.

'•= p,K' + (-P, - 213,(1 ~y)+p2- J ^ •'Qi^))^ + 2/83(1 - y ) -

= 0

' ' O-2 , , y

Since

Hi0)=a-r>0 and / / ( I ) = - y / ( l - y)(5/y - r) < 0,

it follows that the roots satisfy 0 < /C, < 1 < A 2- So

2-o- + a - r +

(b) When y = 0,

Rx-Qx

Writing

the corresponding quadratic is

\a^K^ + [-i tr^ - 5 - ( a - r)]K - a - r = 0,

So, 0 < f, < 1 < ^2 . and

. 2

(a - r) - y l^o-^ -h 6 ± (a - r)j - 2<r\a - r) .

Then, for large JC, 9 is asymptotic to jc* (respectively x'^). So, q -^ <^ as x -* «> alongthe special trajectory, as claimed.


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Reflecting Boundary. Nagoya Math J. 60 189-216.[2] Benes, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980). Some Solvable Stochastic Control

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DAVIS: DEPARTMENT OF ELECTRICAL ENGINEERING, IMPERIAL COLLEGE, LONDONSW7 2BT, UNITED KINGDOM

NORMAN: COUNTY NATWEST, LONDON, UNITED KINGDOM

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