## Abstract

The dynamic portfolio selection problem with bankruptcy and nonlinear transaction costs is studied. The portfolio consists of a risk-free asset, and a risky asset whose price dynamics is governed by geometric Brownian motion. The investor pays transaction costs as a (piecewise linear) function of the traded volume of the risky asset. The objective is to find the stochastic controls (amounts invested in the risky and risk-free assets) that maximize the expected value of the discounted utility of terminal wealth. The problem is formulated as a non-singular stochastic optimal control problem in the sense that the necessary condition for optimality leads to explicit relations between the controls and the value function. The formulation follows along the lines of Merton (Merton 1969 *Rev. Econ. Stat.* **51**, 247–257; Merton 1971 *J. Econ. Theory* **3**, 373–413) and Bensoussan & Julien (Bensoussan & Julien 2000 *Math. Finance* **10**, 89–108) in the sense that the controls are the amounts of the risky asset bought and sold, and they are bounded. It differs from the works of Davis & Norman (Davis & Norman 1990 *Math. Oper. Res.* **15**, 676–713), who use, in the presence of proportional transaction costs, a singular-control formulation in which the controls are rates of buying and selling of the risky asset, and they are unbounded. Numerical results are presented for buy/no transaction and sell/no transaction interfaces, which characterize the optimal policies of a constant relative risk aversion investor. The no transaction region, in the presence of nonlinear transaction costs, is not a cone. The Merton line, on which no transaction takes place in the limiting case of zero transaction costs, need not lie inside the no transaction region for all values of wealth.

## 1. Introduction

Merton (1969, 1971, 1996) shows how to construct and analyse optimal continuous-time allocation problems under uncertainty. Merton considers the model in which the prices of the risky assets are generated by correlated geometric Brownian motions, and assumes that the portfolio can be rebalanced instantly and free of cost. His objective is to maximize the net expected utility of consumption plus the expected utility of terminal wealth. For the utility functions in the constant relative risk aversion (CRRA) class, the optimal strategy consists of an infinite number of transactions in order to keep the proportions invested in the risky assets equal to a constant vector, and to consume at a rate proportional to the total wealth. It is well recognized (Atkins & Dyl 1997) that transaction costs affect the investor's holding period of a particular asset, and investors accommodate transaction costs by drastically reducing the frequency and volume of the trade. Also, the timings of the rebalancings are not predetermined but are dependent on the development of the economy, which is not certain from a time-zero perspective.

The Merton model has been extended by incorporating (i) proportional transaction costs (Magill & Constantinides 1976; Constantinides 1986; Davis & Norman 1990; Dumas & Luciano 1991; Shreve & Soner 1994; Tourin & Zariphopoulou 1994; Akian *et al*. 1996; Sulem 1997; Tourin & Zariphopoulou 1997; Leland 2000; Atkinson & Mokkhavesa 2003); (ii) fixed transaction costs (Eastham & Hastings 1992; Hastings 1992; Schroder 1995; Korn 1998); and (iii) fixed and proportional transaction costs (Chancelier *et al*. 2000; Oksendal & Sulem 2002; Chellathurai 2003). In the presence of proportional transaction costs, the problem is characterized by buy/no transaction (B–NT) and sell/no transaction (S–NT) interfaces in the portfolio space. The optimal transaction policy, in the presence of proportional transaction costs, is minimal trading to stay inside the wedge, preceded by an immediate transaction to the closest point in the wedge if the initial endowment is outside of it. It is found that the S–NT boundary is very insensitive to transaction costs while the B–NT boundary changes very quickly as the transaction costs increase. When there are strictly positive fixed and proportional transaction costs, the problem is characterized by B–NT and S–NT interfaces, and buy and sell targets in the portfolio space. If there are two portfolios that lie in the buying (selling) region so that their net values remain the same, then the risky asset is bought (sold) such that the two rebalancings result in the same buy target (sell target) portfolio (having the same net value) that lies within the no transaction region. In the presence of fixed (and proportional) transaction costs, the volume of the trade is large if it takes place.

Generally, as can be observed from the transaction fees charged by the New York Stock Exchange or any other financial institution, the incurred transaction costs depend upon the volume of the transactions in a nonlinear way (see for example Konno & Wijayanayake 2001). This is because the transaction cost rate is relatively large when the volume of the transaction is small, and it gradually decreases as the volume of the transaction increases. The researchers have approximated the transaction cost function to be linear (which corresponds to proportional transaction costs) or constant (which corresponds to a fixed cost for each non-zero transaction) or both so that the problem is amenable to analysis. Konno & Wijayanayake (2001) study the portfolio optimization problem under concave transaction costs in a single-period set-up. Konno & Wijayanayake (2001) propose an algorithm for calculating a globally optimal solution, and do not study the nature of optimal trading strategies and the no transaction region. There has been, to the best knowledge of the authors, no study on dynamic portfolio selection problems when the transaction cost function is nonlinear.

The existing continuous-time formulation for the dynamic portfolio selection problem with proportional transaction costs uses singular optimal control theory. In the singular control formulation, the controls are the rates of buying and selling of the risky asset (Davis & Norman 1990), and they appear only in the drift terms of the governing stochastic dynamical equations. Due to this, the maximization problem is linear, and the necessary condition for optimality fails to obtain feedback policies for controls in terms of the derivatives of the value function (Bell & Jacobson 1975; Stengel 1986, pp. 247–251; Fleming & Rishel 1975, p. 37). The optimal controls are bang–bang and they are either zero or unbounded, and the resulting Hamilton–Jacobi–Bellman (HJB) equation reduces to a set of partial differential inequalities (Krylov 1980, ch. 4; Davis & Norman 1990; Shreve & Soner 1994). It is extremely time-consuming to solve the system of time-dependent differential inequalities (e.g. Tourin & Zariphopoulou 1997). The complexities increase if the portfolio consists of more than one risky asset, and there is no corresponding formulation when the transaction costs are nonlinear.

In this paper, we study the portfolio selection problem with nonlinear transaction costs as a non-singular stochastic optimal control problem in the sense that the necessary condition for optimality leads to explicit relations between the controls and the derivatives of the value function. The controls are the amounts of the risky asset bought and sold, and they appear both in the drift and diffusion terms of the governing dynamical equations (Merton 1969, 1971; Bensoussan & Julien 2000); and the resultant maximization problem is quadratic. The controls are absolutely continuous and bounded. The rebalancing is instantaneous, and hence the rate of transaction is unbounded, and in this sense, the rate of transaction is singular. The resultant HJB equation leads to a set of inequalities involving the value function and its derivatives as optimality conditions, and these coincide with those of Davis & Norman (1990), in the case of proportional transaction costs (§3). However, it is not necessary to solve these inequalities. Instead, the HJB equation is solved iteratively due to the presence of quadratic terms for the controls (§4). The HJB equation resulting from the non-singular control formulation is a natural extension of Merton's HJB equation for the value function, in the sense that, by setting the transaction costs parameters to zero, the portfolio selection problem with no transaction costs can be analytically studied (§2). This is not the case with the formulation using singular optimal control theory.

The objective of this work is to formulate and analyse the portfolio selection problem with nonlinear transaction costs. We assume that the transaction cost function is piecewise linear. In §2, the dynamic problem is formulated when the portfolio consists of a risk-free asset, and a risky asset whose price dynamics is generated by geometric Brownian motion. In §3, transaction regions are identified and the optimal policies are characterized. At every point of the transaction regions, necessary conditions satisfied by the value function are derived. In §4, an algorthim is presented. At each time, the problem reduces to solving a constrained (static) maximization problem and a degenerate partial differential equation, sequentially. In §5, a monotone upwind finite difference scheme is developed to discretize the HJB equation so that the discrete system leads to an *M*-matrix that guarantees the discrete maximum principle. In §6, we present computational results for the optimal trading strategies when the utility function for terminal wealth belongs to the CRRA class of functions. Section 7 concludes the paper.

## 2. Formulation

The portfolio consists of a risk-free asset and a risky asset whose price is driven by geometric Brownian motion. The risk-free asset is characterized by(2.1)where *S*_{0}(*t*) denotes the price of one unit of the risk-free asset at time *t* and *r* is the instantaneous rate of return from the risk-free asset and it is assumed to be constant. The dynamics of the risky asset is given by(2.2)where *S*(*t*) denotes the price of one share of the risky asset at time *t* and d*B*_{t} is the increment of a Brownian motion that is normally distributed with mean zero and variance d*t*. In equation (2.2), *α* is the instantaneous conditional expected change in price and *σ*^{2} is the instantaneous conditional variance of the price of the risky asset. For small values of Δ*t*, the dynamical equations (2.1) and (2.2) can be approximated by(2.3)and(2.4)Let *x*(*t*) and *y*(*t*) denote the investor's holdings in the risky and the risk-free assets at time *t*, respectively. Unlike Merton (1969, 1971, 1996), we keep separate identities for holdings in the risky and risk-free assets rather than merging them into a single wealth process *W*=*x*+*y*. If there are no transactions, the asset holdings in (*t*,*t*+Δ*t*) evolve, because of equations (2.3) and (2.4), according to the dynamical equations(2.5)(2.6)Let 0≤*t*<*T*, and the interval [*t*,*T*) be uniformly discretized so that (*N*−1)Δ*t*=*T*−*t*, where *N* denotes the number of discrete time nodes in [*t*,*T*), *T*(year) being the planning horizon. On the discrete time nodes *t*=*t*_{1}, *t*_{2}, …, *t*_{N−1}=*T*−Δ*t* the controls *p*(*x*, *y*, *t*_{k}) and *q*(*x*, *y*, *t*_{k}), *k*=1, 2, …,(*N*−1), are defined for each state (*x*, *y*). Here, *p*(*x*, *y*, *t*_{k})≥0 is an amount from the risk-free asset used to buy the risky asset at time *t*=*t*_{k} in state (*x*, *y*); *q*(*x*, *y*, *t*_{k})≥0 is the amount of risky asset sold at time *t*=*t*_{k} in state (*x*, *y*).

The investor pays the broker *F*_{1}(*p*) or *F*_{2}(*q*) as transaction costs on purchase (*p*) or sale (*q*) of the risky asset, respectively. We assume the transaction cost functions are piecewise linear as shown in tables 1 and 2, where , and are proportional transaction cost rates in the buying intervals and the selling intervals , respectively. The transaction cost functions *F*_{1}(*p*) and *F*_{2}(*q*) are concave if and , respectively.

We assume that rebalancing takes place instantly at the beginning of the interval, and the investment holdings at a given time *t*_{k}, 1≤*k*≤(*N*−1) induce an income at the next time *t*_{k+1}=*t*_{k}+Δ*t*. Let *x*(*t*_{k}+) and *y*(*t*_{k}+) denote the investor's holdings in the risky and risk-free assets, after rebalancing takes place at time *t*=*t*_{k} in state (*x*, *y*). They are given by(2.7)(2.8)Since it is not optimal to buy and sell simultaneously, we have(2.9)Using equations (2.5) and (2.6), the holdings of the assets at time *t*_{k}+Δ*t* can be written as(2.10)(2.11)The investor's objective is to maximize the expected value of the discounted utility of terminal wealth *W*(*T*)(≥0), the amount of money in the risk-free asset resulting from the liquidation of the risky asset at terminal time *t*=*T*. The utility function for terminal wealth, *ϕ*[*W*(*T*)], is selected so that it is monotonically increasing and concave in its argument to represent a risk-averse investor. In this study we use a power-law utility function that belongs to the CRRA family.

Let(2.12)denote, for fixed *T*, the maximum expected value of the discounted utility of terminal wealth, starting at time *t* in state *x*(*t*)=*x*, *y*(*t*)=*y*, where *E*_{t} denotes the conditional expectation operator at time *t* and *ρ* denotes the subjective discount factor. The control set *Ω*_{1} is given byEquation (2.12) can be written as(2.13)whereandUsing Bellman's principle of optimality (Bellman & Dreyfus 1962), equation (2.13) can be written as(2.14)During rebalancing, the value of the value function is preserved (see for example Dumas & Luciano 1991; Eastham & Hastings 1992; Hastings 1992; Schroder 1995; Korn 1998; Chancelier *et al*. 2000; Oksendal & Sulem 2002), that is(2.15)Due to equation (2.15), we have(2.16)and hence(2.17)Using equations (2.10), (2.11) and (2.17), equation (2.14) can be written as(2.18)As Δ*t*→0, 0<*t*≤*s*<*T*, the dynamical equations (2.10) and (2.11) become(2.19)(2.20)(2.21)and equation (2.12) becomes(2.22)where *U* is the set of integrable non-anticipative processes (*p*(*x*, *y*, *s*), *q*(*x*, *y*, *s*)), *t*≤*s*<*T*, such that(2.23)In equation (2.23), *D* is the solvency region defined by , whereWhen and , the solvency region reduces to .

Expanding by Taylor's series and using the properties of Brownian motion, and taking the limit as Δ*t*→0, 0≤*t*<*T*, equation (2.18) becomes(2.24)where the arguments of *p*(*x*, *y*, *t*) and *q*(*x*, *y*, *t*), for brevity, are not shown. In the derivation of the above HJB equation, it is implicitly assumed that the value function *J*(*x*, *y*, *t*) is sufficiently smooth that the HJB equation (2.24) is valid in the solvency region *D*.

The HJB equation (2.24) needs to be solved backward in time *t*∈[0,*T*), in the solvency region *D*, subject to the terminal condition(2.25)In addition to the terminal condition (2.25), we need to specify a boundary condition for the value function at zero net wealth. The value function at zero wealth signifies the reward or penalty associated with bankruptcy. This value will have consequence on an investor's decisions. Karatzas *et al*. (1986) assign a utility value to bankruptcy and include this value as a parameter in their treatment of the consumption-investment problem with non-negative consumption requirement and bankruptcy. Following Merton (1996, ch. 6), to capture the requirement that the net wealth *W*(*t*) is positive, and that *W*(*t*)=0 is an absorbing state, we add the boundary condition(2.26)where ∂*D*_{0} is the part of the boundary of *D* on which net wealth is zero. The boundary condition (2.26) is basically a non-negative wealth constraint that rules out arbitrage opportunities (Cox & Huang 1989; Merton 1996, ch. 6).

The controls appear both in drift and diffusion terms of the HJB equation (2.24), and the governing dynamical equations (2.19) and (2.20). The HJB equation (2.24) is a natural extension of Merton's HJB equation for the value function. Defining the net wealth, *W*(*s*)=*x*(*s*)+*y*(*s*), as a new variable, in the absence of transaction costs, equations (2.19) and (2.20), *t*≤*s*<*T*, can be amalgamated into(2.27)Let(2.28)denote the proportional investment in the risky asset. With this notation, in the solvency region *W*>0, the dynamical equation (2.27) can be written as(2.29)which is exactly same as the eqn (4.12) in Merton (1996, ch. 4, p. 101) with zero consumption. Correspondingly, under the transformation *w*=*x*+*y*, *G*(*w*,*t*)=*J*(*x*,*y*,*t*), the HJB equation (2.24) reduces to(2.30)which is Merton's HJB eqn (4.17*a*) in Merton (1996, ch. 4, p. 102) with zero discount factor (*ρ*=0). This is in contrast to the formulation using singular stochastic optimal control theory where the maximization problem is linear (Davis & Norman 1990). Due to this linearity, optimal controls are bang–bang and the resulting HJB equation reduces to a variational inequality (Krylov 1980, ch. 4; Davis & Norman 1990; Shreve & Soner 1994). In contrast, the HJB equation (2.24) leads to a set of inequalities involving the value function and its derivatives as optimality conditions (§3), and it is not necessary to solve these inequalities. Instead, the HJB equation (2.24) is solved iteratively (§4) due to the presence of quadratic terms for the controls.

The HJB equation (2.24) can be written as(2.31)We have derived the HJB equation. It is now imperative to show that a sufficiently regular solution of the HJB equation satisfying the terminal condition (2.25) and the bankruptcy condition (2.26) is indeed correct, that is, it coincides with the value function of the stochastic optimal control problem, and the corresponding control strategy is optimal. The HJB equation acts as the necessary and sufficient condition for the optimal control problem. This is achieved by the following verification theorem which is a special case of theorem IV.3.1 in Fleming & Soner (1993, p. 163).

### (a) Verification theorem

Let be a solution of the HJB equation (2.29) satisfying the terminal condition (2.25) and the bankruptcy condition (2.26) with and continuous on the closure of the domain. Then

, .

If (

*p*^{*},*q*^{*}) is an admissible feedback control such that

(2.32)for all (*x*,*y*,*t*)∈*D*×[0,*T*), then , . Thus (*p*^{*},*q*^{*}) is optimal.

The verification theorem reduces the solution of the stochastic optimal control problem to two other problems. The first is to solve the nonlinear second-order partial differential equation with known controls (*p*,*q*), subject to terminal and bankruptcy conditions. The second is to find (*p*,*q*) by solving the static maximization problem for each (*x*,*y*,*t*). This is implemented in §4.

The above verification theorem has the inherent deficiency of having to assume the smoothness of the value function at all points of the solvency region *D*. However, the value function may not be differentiable in *y* at all points of the solvency region. This difficulty is overcome by the ‘vanishing viscosity’ method, whereby the HJB equation is modified by adding a small diffusion term through the upwind finite-difference method. Although ‘viscous’ effects may be negligible throughout most of the solvency region, the effect is strong where possible discontinuities in derivatives of the value function occur. Also, it picks out the physically correct unique ‘vanishing viscosity’ solution when no classical solution exists, as the added viscosity terms tend to zero (Krylov 1980, ch. 4, 1987, ch. 7; Fleming & Soner 1993; Bardi & Capuzzo-Dolcetta 1997, p. 20; Barles 1997; Kroner 1997, p. 22).

## 3. Characterization of optimal policies

In this section, we derive expressions for optimal transaction policies by solving the maximization problem associated with the HJB equation (2.24). The respective regions of buying and selling of the risky asset in the portfolio space are identified. We write ‘buying (selling) region’ to refer to the region where buying (selling) of the risky asset takes place; and write ‘no transaction region’ to refer to the region where neither buying nor selling of the risky asset takes place in the portfolio space. For brevity, we use *p*, *q* and *J* to refer to *p*(*x*, *y*, *t*), *q*(*x*, *y*, *t*) and *J*(*x*, *y*, *t*), respectively. The HJB equation (2.24) has the character of a two-dimensional, drift–diffusion equation with no diffusion along the *y*-direction when the optimal policies are known. The utility of the terminal wealth, which is investor dependent, may not be sufficiently smooth to guarantee that the value function *J* is twice differentiable in the solvency region, *t*∈[0,*T*). However, the analysis is based on the assumption that the value function *J* is twice differentiable in the solvency region. This analysis will help us to understand the optimal control problem when the original partial differential equation is replaced by an equivalent partial differential equation (which differs from the original partial differential equation by numerical truncation error which tends to zero as step sizes tend to zero, see §5, for details) satisfied by the (numerical) value function (which is twice differentiable) even when the terminal utility is not smooth (Hirsch 1988, ch. 7; see also Barles 1997). We also assume that the value function *J*(*x*, *y*, *t*) is monotonically increasing in its arguments *x* (holdings in the risky asset) and *y* (holdings in the risk-free asset). This assumption can be interpreted as a condition of no-arbitrage.

Let(3.1)and(3.2)The HJB equation (2.24) can be written as(3.3)whereSince buying and selling of the risky asset cannot take place simultaneously, the control set *Ω* can be split into *Ω*_{B} and *Ω*_{S} such that(3.4)and *Ω*=Ω_{B}∪*Ω*_{S}. The control sets *Ω*_{B} and *Ω*_{S} represent the states of no selling and no buying of the risky asset, respectively. The intersection of the two sets contains the set of all points in the solvency region where no transaction takes place.

If we define(3.5)and(3.6)the problem is reduced to solving the HJB equation(3.7)and to finding the optimal controls *p*_{opt} and *q*_{opt} at every point of the portfolio space.

### (a) Identification of buying region B_{1}

The maximization problem defined by equation (3.5) can be restated as the following constrained quadratic maximization problem(3.8)Any point in the buying region B_{1} of the portfolio space is characterized by(3.9)(3.10)(3.11)and(3.12)The equality sign in equation(3.12) corresponds to no transaction in the portfolio space. Equation (3.12) can further be simplified to(3.13)Inequalities (3.9) and (3.10) and equation (3.13) define the buying region *B*_{1}, and equation (3.11) defines the optimal buying strategy. Inequality (3.9) and the optimal buying policy given by equation (3.11) state that the optimal buying policy changes smoothly across the B–NT interface. In the neighbourhood of the boundary where the net wealth , equation (3.13) leads to the inequality(3.14)

### (b) Identification of buying region B_{2}

The maximization problem defined by equation (3.5) can be restated as the following constrained quadratic maximization problem(3.15)Any point in the buying region *B*_{2} of the portfolio space is characterized by(3.16)(3.17)(3.18)and(3.19)Inequalities (3.16) and (3.17) and equation (3.19) define the buying region *B*_{2}, and equation (3.18) defines the optimal buying strategy.

### (c) Identification of buying region B_{3}

The maximization problem defined by equation (3.5) can be restated as the following constrained quadratic maximization problem(3.20)Any point in the buying region B_{3} of the portfolio space is characterized by(3.21)(3.22)(3.23)and(3.24)Inequalities (3.21) and (3.22) and equation (3.24) define the buying region B_{3} and equation (3.23) defines the optimal buying strategy.

### (d) Identification of selling region S_{1}

The maximization problem defined by equation (3.6) can be restated as(3.25)Any point in the selling region *S*_{1} of the portfolio space is characterized by(3.26)(3.27)(3.28)(3.29)The equality sign in equation (3.29) corresponds to no transaction in the portfolio space. Equation (3.29) can further be simplified to(3.30)Inequalities (3.26) and (3.27) and equation (3.30) define the selling region, and equation (3.28) defines the optimal selling strategy. Inequality (3.26) and the optimal selling policy given by equation (3.28) state that the optimal selling policy changes smoothly across the S–NT interface. In the neighbourhood of the boundary where the net wealth , equation (3.30) leads to the inequality (3.14).

### (e) Identification of selling region S_{2}

The maximization problem defined by equation (3.6) can be restated as(3.31)Any point in the selling region S_{2} of the portfolio space is characterized by(3.32)(3.33)(3.34)(3.35)Inequalities (3.32) and (3.33) and equation (3.35) define the selling region S_{2}, and equation (3.34) defines the optimal selling strategy.

### (f) Identification of selling region S_{3}

The maximization problem defined by equation (3.6) can be restated as(3.36)Any point in the selling region S_{3} of the portfolio space is characterized by(3.37)(3.38)(3.39)(3.40)Inequalities (3.37) and (3.38) and equation (3.40) define the selling region S_{3}, and equation (3.39) defines the optimal selling strategy.

It may be noted that at points (*x*, *y*, *t*) where , and , , the value function *J*(*x*, *y*, *t*) may be convex in *x*.

### (g) Characterization of optimal policies

From equation (3.13), it can be verified thatwhere *G*_{1}, in the buying region B_{1}, satisfies(3.41)Similarly,in the buying region B_{2}, satisfies(3.42)andin the buying region B_{3}, satisfies(3.43)Similarly, from equation (3.30), it can be verified thatwhere *H*_{1}, in the selling region S_{1}, satisfies(3.44)andin the selling region S_{2}, satisfies(3.45)andin the selling region S_{3}, satisfies(3.46)Equations (3.41)–(3.46), suggest that the value function *J* is constant alongDue to this property of the value function *J*, the optimal buying strategy given by equations (3.11), (3.18) and (3.23) implies that, if (*x*_{E}, *y*_{E}) and (*x*_{F}, *y*_{F}) are two portfolios that lie in the buying region (B_{1} or B_{2} or B_{3}) then(3.47)If (*X*_{B–NT}, *Y*_{B–NT}) denotes any portfolio that lies on the B–NT interface, the characteristic line that passes through (*X*_{B–NT}, *Y*_{B–NT}), on which the value function is constant in the buying region B_{1}, is given by(3.48)That is,(3.49)which states that any portfolio (*x*, *y*) that lies on the characteristic line (3.48) in the buying region B_{1} buys the amount *X*_{B–NT}−*x* of the risky asset, after paying the transaction costs, , so that the rebalanced portfolio results in (*X*_{B–NT}, *Y*_{B–NT}).

The characteristic line that passes through , on which the value function is constant in the buying region B_{2}, , is given by(3.50)That is,(3.51)which states that any portfolio (*x*, *y*) that lies on the characteristic line (3.50) in the buying region B_{2} buys the amount *X*_{B–NT}−*x* of the risky asset so that the rebalanced portfolio results in (*X*_{B—NT},*Y*_{B–NT}). The transaction cost paid in this rebalancing is . The proportional transaction costs rate is up to and for the remaining amount.

The characteristic line that passes through on which the value function is constant in the buying region B_{3}, , is given by(3.52)That is,(3.53)which states that any portfolio (*x*, *y*) that lies on the characteristic line (3.52) in the buying region B_{3} buys the amount of (X_{B–NT}−*x*) of the risky asset so that rebalanced portfolio results in (X_{B–NT},*Y*_{B–NT}). The transaction cost paid in this rebalancing is . The proportional transaction costs rate is up to , in , and for the remaining amount.

If there are two portfolios that lie in the buying region so that their net values remain the same, then the risky asset is bought such that the two rebalancings result in the same portfolio (having the same net value) that lies on the B–NT interface. On each characteristic curve that lies within the buying region, the value of the value function remains the same, and the direction of buying is towards the B–NT interface portfolio that lies on that characteristic curve. This is schematically shown in the buying region of figure 1. The results reduce to those obtained by Davis & Norman (1990) when the transaction cost function *F*_{1}(*p*) is linear, that is, .

Similarly, any portfolio (*x*, *y*) that lies on the characteristic curvesells the amount *x*−*X*_{S–NT} of the risky asset so that the rebalanced portfolio results in (*X*_{S–NT},*Y*_{S–NT}), which lies on the S–NT interface. The rate of transaction costs paid depends upon the interval on which the amount of the risky asset sold lies.

If there are two portfolios that lie in the selling region so that their net values remain the same, then the risky asset is sold such that the two rebalancings result in the same portfolio (having the same net value) that lies on the S–NT interface. On each characteristic curve that lies within the selling region, the value of the value function remains the same, and the direction of selling is towards the S–NT interface portfolio that lies on that characteristic curve. This is schematically shown in the selling region of figure 1. Again, these results reduce to those obtained by Davis & Norman (1990) when the transaction cost function *F*_{2}(*q*) is linear, that is, .

## 4. Algorithm

There are no analytical solutions for the HJB equation (2.24) when there are transaction costs. In this section, we present an algorithm for solving the HJB equation (2.24), at a particular time *t*. The algorithm assumes the existence of the numerical approximation of the value function *J*(*x*, *y*, *t*); , and for its derivatives. At each iteration, the existence of , and is guaranteed by the monotone finite-difference scheme used to solve the HJB equation numerically in the ‘vanishing viscosity’ sense.

Guess a solution for . A good initial guess is the solution at

*t*+Δ*t*. At (*x*,*y*,*t*) we solve the static optimization problem (2.24). Since buying and selling of the risky asset cannot take place simultaneously, the new optimal controls*p*_{opt}and*q*_{opt}are calculated as follows.Let denote the control pair that maximizesand denote the control pair that maximizes

Using the calculated controls, solve the HJB equation (2.24) along with the terminal and boundary conditions to get new .

Repeat the process until convergence. The convergence criterion used is

where denotes the updated (new) value of the numerical value function at node *i* and denotes the value of the numerical value function at the previous iteration.

## 5. Numerical discretization

We develop an upwind finite difference scheme to discretize the linearized HJB equation given by(5.1)where(5.2)(5.3)(5.4)In equation (5.1), the control variables *p*(*x*, *y*, *t*) and *q*(*x*, *y*, *t*) are assumed to be known from the previous iteration. The value function at the previous time-step *t*+Δ*t*, *J*(*x*, *y*, *t*+Δ*t*) is known, where Δ*t* is the step size in time. At the first time-step, *t*=*T*−Δ*t*, *J*(*x*, *y*, *t*+Δ*t*) is the terminal wealth function *ϕ*(*x*(*T*), *y*(*T*), *T*).

The HJB equation (5.1) needs to be augmented with the physical boundary conditions to get the unique solution. The following Dirichlet boundary condition, for *t*∈[0,*T*), is prescribed(5.5)where *ϕ*(*x*(*T*), *y*(*T*), *T*) denotes the terminal wealth function defined at *t*=*T*. It may be noted that equation (5.5) is nothing but the bankruptcy boundary condition on the part of the boundary, where the net wealth is zero and equation (5.5) is consistent with the inequality (3.14).

The objective of the upwind discretization is that the resulting difference equations satisfy a modified partial differential equation (whose solution is twice differentiable in the solvency region), which differs from the original partial differential equation by truncation error. The determination of the modified partial differential equation, and in particular, the leading order truncation error provides essential information as to the behaviour of the numerical solution (Hirsch 1988, ch. 7; see also Barles 1997). Barles & Souganidis (1991) describe a general convergence result which applies to a wide range of numerical schemes for nonlinear, possibly degenerate, parabolic partial differential equations. They show that a consistent, stable and monotone discretization scheme converges to the unique viscosity solution of the partial differential equation. Krylov (2000), Barles & Jakobsen (2002, in press) and Jakobsen (2004) prove results on the rate of convergence of monotone approximation schemes for the HJB equations.

The HJB equation (5.1) is discretized on a rectangular grid of the solvency region *D*. The left and right neighbourhood nodes of any interior node (*i*, *j*) are indicated by the indices (*i*−1, *j*) and (*i*+1, *j*), respectively. The lower and upper neighbourhood nodes are identified by the indices (*i*, *j*−1) and (*i*, *j*+1), respectively. Let (*x*_{i}, *y*_{j}) denote the coordinates of the interior node (*i*, *j*), and letThe subscript denoting time is not shown and it is understood that *J*(*x*_{i}, *y*_{j}, *t*)=*J*_{i,j}, so that the numerical scheme is fully implicit. For brevity, we use *a*_{1}(*x*_{i}, *y*_{j}, *t*)=*a*_{1}, *a*_{2}(*x*_{i}, *y*_{j}, *t*)=*a*_{2} and *b*_{1}(*x*_{i}, *y*_{j}, *t*)=*b*_{1}. Discretizing the second-derivative term using the central difference scheme and the drift-derivative terms using the upwind formulae, at any interior node (*i*, *j*), we get (Chellathurai 2003)(5.6)where(5.7)(5.8)(5.9)(5.10)(5.11)At the boundary nodes, the Dirichlet boundary conditions (5.5) are prescribed by(5.12)Equations (5.6) and (5.12) constitute the required discrete equations at every node in the solvency region. In each row associated with a node, all off-diagonal entries *C*_{i−1,j}, *C*_{i+1,j}, *C*_{i,j−1} and *C*_{i,j+1} are non-positive and the diagonal entry is positive. Also, the sum of all the entries of each row corresponding to every interior node is positive. All these conditions satisfy the requirements for an *M*-matrix, ensuring that the discrete maximum principle applies. The numerical solution cannot have oscillation at any iteration (Strang 1986; Ciarlet 1970).

It is shown that (Chellathurai 2003) as *h*_{L}, *h*_{R}, *k*_{D}, *k*_{U} and Δ*t*→0, we get equation (5.1) defined on any interior node (*i*, *j*) at time *t*, and hence the scheme is consistent. If the order of , *h*_{L}*h*_{R}, , , *k*_{U}*k*_{D}, Δ*t*^{2} and other higher order terms are neglected, equation (5.6) reduces to(5.13)where(5.14)(5.15)From equation (5.13), we infer that if the linear terms involving *h*_{L}, *h*_{R}, *k*_{D} and *k*_{U} are included, we have some additional terms in the coefficients of and . Equation (5.13) says that the numerical scheme (5.6) introduces numerical viscosity (Hirsch 1988; see also Barles 1997), , along with the existing diffusion term, , in the direction of the *x*-axis; and in the direction of the *y*-axis. This is equivalent to taking the expected value of the (maximized) stochastic functionalsubject to the modified wealth dynamical equations(5.16)and(5.17)where , are viscous Brownian motions with zero correlation. The Brownian motion associated with the risky asset *x* is modified by the new diffusion due to the numerical scheme. Also, we have a new Brownian motion whose diffusion term is a function of step size, associated with the risk-free asset. These additional numerical diffusion terms tend to zero as the step sizes tend to zero. Equations (5.16) and (5.17) state that the numerical scheme adds artificial viscosity, ensuring that the value function (numerical) behaves smoothly at any point in the solvency region where the value function (exact) may not be smooth in the classical sense. That is, a possible discontinuity in derivatives of *J* is automatically smoothened by the numerical scheme. As the step sizes go to zero, the numerical solution tends to the classical solution *J*. It may be noted here that the approximation scheme will not pick out the physically correct unique ‘viscosity solution’ satisfying the entropy condition if we use central difference schemes for the first-order derivatives of the value function (Kroner 1997, p. 31).

The upwind discretization procedure is widely used to solve convection dominated problems in computational fluid dynamics (Hirsch 1988) to smoothen the flow variables whose gradients change abruptly. However, the classical upwind method diffuses in both *x*- and *y*-directions, and hence introduces excessive diffusion along the *y*-direction. Some minimum dissipation is needed to ensure that ‘vanishing viscosity’ solution is obtained. This may be achieved by employing streamline upwind methods, which take into account the local hyperbolic characteristics of the value function (Hirsch 1988; Kroner 1997; Remaki *et al*. 2003). As a future work, it is planned to implement this family of schemes to HJB equations.

## 6. Computational results

It has been shown that the portfolio selection problem is characterized by two time-dependent curves representing the B–NT and S–NT interfaces. Initially (*τ*_{1}=0) or at any other rebalancing time (*τ*_{k}, *k*>1) if the portfolio is outside the no transaction region, the portfolio is rebalanced instantly to a portfolio that lies either on the S–NT interface or on the B–NT interface. Once the portfolio is within the no transaction region, the holdings in the risky asset and risk-free asset, evolve according to (see equations (2.19) and (2.20))(6.1)and(6.2)in *t*∈(*τ*_{k},*τ*_{k+1}) where (*x*(*τ*_{k}),*y*(*τ*_{k})) is the rebalanced portfolio at time *t*=*τ*_{k}. Here, *τ*_{k+1} is the stochastic time *t*=*τ*_{k+1}>*τ*_{k}, at which the portfolio hits either the B–NT interface or the S–NT interface.

In this section, we present numerical results for the interfaces when the utility function for terminal wealth isIn the absence of transaction costs, the fraction of total wealth invested in the risky asset is given by (Merton 1969)(6.3)and, at any time *t*, the Merton line (*M*≠0, *p*(*x*, *y*, *t*)−*q*(*x*, *y*, *t*)=0)(6.4)divides the portfolio space into selling and buying regions of the risky asset. On the Merton line, defined by equation (6.4), no transaction takes place.

In the numerical calculations, the maximal holdings in the risky asset and risk-free asset are *x*_{max}=$4500, *y*_{max}=$4500, respectively. The planning horizon is *T*=0.5 (year), and the step size along the time axis is Δ*t*=0.01 (year). The number of discrete nodes on (*x*_{max}, 0<*y*<*y*_{max}) is 201, and the number of discrete nodes on (*x*_{max}, *y*<0) is 200. The total number of discrete nodes is equal to 125 751. The breakpoints that describe the nonlinear transaction cost functions are given by , . In all the numerical experiments reported here, the maximum number of nonlinear iterations required for convergence is three. The maximum time taken to solve the HJB equation at any particular time is less than 10 s in a SunOS 5.6 micro system.

As an illustrative example, results are given for a particular problem (*r*=0.05 (1/year), *α*=0.09 (1/year), , *ρ*=0.05 (1/year), *γ*=0.5, so that *M*=1/2). Figure 2 shows the B–NT and S–NT interfaces in the portfolio space at the beginning of the investment when there are proportional transaction costs (; table 3) only. As expected, the B–NT interface and the S–NT interface are straight-lines passing through the origin, and proportional transaction costs widen the region of no transaction in a skewed way. The B–NT interfaces are more sensitive to changes in proportional transaction costs than the S–NT interfaces. This is in agreement with Constantinides (1986), and Davis & Norman (1990).

Figure 3 shows the B–NT and S–NT interfaces in the portfolio space at the beginning of the investment when there are nonlinear transaction costs. The investor pays linear transaction costs if the traded amount is less than the exogeneously specified level , and she pays a fixed transaction cost of $20, if the traded amount exceeds $500 (table 4). That is, , . The no transaction region is not a cone, and the Merton line is not inside the no transaction region for values of wealth in (600, 1600). The Merton line is inside the no transaction region, when the investor's wealth is small (*W*≤600), and when it is large (*W*≥1600). The B–NT and S–NT interfaces are almost parallel to the *x*-axis (risky asset) in some interval. This implies that investment in the risky asset increases as wealth increases, and the investment in the risk-free asset is kept constant. This in turn leads to an increase in frequency of buying compared to the frequency of selling (see equations (6.1) and (6.2)).

Figure 4 shows the B–NT and S–NT interfaces in the portfolio space at the beginning of the investment when the transaction cost rate is 0.04 if the traded amount is in [0,500], 0.03 if the traded amount is in [500,1000]. The investor pays a fixed transaction cost of $35, if the traded amount exceeds $1000 (table 5). That is, , and . Again, the no transaction region is not a cone, and the Merton line is not inside the no transaction region for values of wealth in (1300,3000).

Figure 5 shows the B–NT and S–NT interfaces in the portfolio space at the beginning of the investment when the transaction cost rates are 0.04, 0.03, 0.02 in the intervals [0,500], [500,1000], [1000,∞), respectively (table 6). That is, , , and . In this case, the Merton line is inside the no transaction region.

Figure 6 shows the B–NT and S–NT interfaces in the portfolio space at the beginning of the investment when the transaction cost rates are 0.04, 0.05, 0.06 in the intervals [0,500], [500,1000], [1000,∞), respectively (table 7). That is, , , , and the transaction cost functions *F*_{1}(*p*) and *F*_{1}(*q*) are convex. The no transaction region is wider compared to the case with linear transaction costs (0.04), and the Merton line lies below the S–NT interface for some values of wealth.

The presence of transaction costs, for certain values of parameters of the problem, reduces the investor's desire for leverage, and the investor should trade to move from the Merton proportion to a less leveraged proportion. This behaviour depends on the wealth level and the structural nature of the transaction costs functions. This was first observed by Shreve & Soner (1994, remark 11.3, pp. 674–675), in the presence of proportional transaction costs, when the Merton proportion is greater than one. Akian *et al*. (1996, p. 360) and Oksendal & Sulem (2002, p. 1768) also report similar behaviour, in the presence of proportional transaction costs, when there is short selling of the risky asset.

The optimal policy, in the presence of nonlinear transaction costs, is infinitesimal trading to stay inside the no transaction region, which is defined by the B–NT and S–NT interfaces. The interfaces are, in turn, functions of the parameters and , that define the transaction cost functions, the market parameters *α*, *σ* and *r*, the risk aversion coefficient *γ* and the planning horizon *T*. The quantity of infinitesimal trading is not determined by these parameters, but the portfolio where the rebalancing is likely to take place is determined by these parameters. In all the test problems we have considered (see tables 3–7), the transaction costs rates in the first segments and are . And, the transaction costs rates do not vary above the wealth level . Consequently, the B–NT and S–NT interfaces corresponding to strictly nonlinear transaction costs tend to coincide with those corresponding to pure proportional transaction costs when the wealth level is less than , and much greater than (see figures 2–6). In other words, the B–NT and S–NT interfaces that characterize the optimal behaviour of investors with large wealth are determined by the proportional transaction costs, and the role of nonlinear transaction costs is minimal.

## 7. Conclusion

We have studied the portfolio selection problem in which the investor pays transaction costs as a (piecewise linear) function of the traded volume of the risky asset. Dynamic programming leads to an HJB equation for the value function. Various (in)equalities are derived in terms of the value function and its derivatives, as optimality conditions. In the presence of nonlinear transaction costs, either no transaction takes place or the random trading is infinitesimal (except possibly at time *t*=0). The problem is characterized by B–NT and S–NT interfaces in the portfolio space. If there are two portfolios that lie in the buying (selling) region so that their net values remain the same, then the risky asset is bought (sold) such that the two rebalancings result in the same portfolio that lies on the B–NT (S–NT) interface. The characteristic curves along which the value function is constant in the transaction regions, are piecewise linear. This is different from the case with proportional transaction costs, in which the characteristic curves are linear. The smoothness of the solution to the HJB equation is guaranteed by the numerical method which is used to solve the HJB equation. Optimal policy parameters that characterize the problem are calculated numerically when the utility function for the terminal wealth is a power-law function. The no transaction region, in the presence of nonlinear transaction costs, is not a cone, and the Merton line need not lie inside the no transaction region for all values of wealth. The strictly concave transaction costs imply more frequent trading (compared to base linear transaction costs case), and therefore they make the securities' market more liquid. The investor's demand for assets is sensitive to the variations in the transaction cost rates. The demand for the risk-free asset is more if the transaction cost rates increase as trading volume increases (compared to the decreasing transaction cost rates). The model still lacks closeness to reality in the sense that the trading strategies consist of infinitesimally small transactions. This is due to the absence of non-zero fixed transaction costs, and the assumption that the transaction cost functions are piecewise continuous. The transaction cost functions must behave like an impulse function in the neighbourhood of zero to incorporate fixed transaction costs in the model. We plan to pursue this research in the future.

## Acknowledgments

The authors thank the two referees for their constructive suggestions which greatly improved the paper. The authors dedicate this paper, with unbounded respect and gratitude to Prof. S. Jeyamony, M.Sc., M.Phil. and Mrs V. Sebastiyayee Jeyamony, B.Sc., M.Ed., Chempaud, Tamil Nadu, India, who provided everything when needed.

## Footnotes

↵† Present address: Decision Solutions, Retail Risk Management, Canadian Imperial Bank of Commerce, Toronto M5L 1A, Canada.

As this paper exceeds the maximum length normally permitted, the authors have agreed to contribute to production costs.

- Received March 15, 2004.
- Accepted May 18, 2005.

- © 2005 The Royal Society