Optimal Decision for Selling an Illiquid Stock

Transcription

1 Optimal Decision for Selling an Illiquid Stock Baojun Bian Min Dai Lishang Jiang Qing Zhang Yifei Zhong February 8, 29 Abstract Most studies on stock-selling decision making are concerned with liquidation of the security within a short period of time. This is practically feasible only when a relative smaller number of shares of a stock is involved. Selling a large block of stock in a market place normally depresses the market if sold in a short period of time, which would result in poor filling prices. In this paper, we consider the liquidation strategy for selling an illiquid stock by combining selling with occasional buying over a period of time. The buying activities help to stabilize the stock price when heavy selling is in progress. We treat the problem by using a fluid model in the sense that the number of shares is treated as fluid (continuous) and the overall liquidation is dictated by the rates of selling and buying over time. The objective is to maximize the expected overall return. The underlying problem can be formulated as a stochastic control problem with state constraints. Method of constrained viscosity solution is used to characterize the dynamics governing the optimal reward function and the associated boundary conditions. Numerical examples are given to illustrate the results. Keywords: Optimal control, state constraint, selling rule. Bian and Jiang are from Dept of Math, Tongji University. Dai and Zhong are from Department of Mathematics, National University of Singapore (NUS), and Dai is also an affiliated member of Risk Management Institute, NUS. Zhang is from Department of Mathematics, The University of Georgia. We thank Changhao Zhang for assistance in some numerical implementation. Bian and Jiang are supported in part by NSFC (No ) and NBRPC (27CB81493), and Dai is supported by the Singapore MOE AcRF grant (No. R ) and the NUS RMI grant (No. R /646). 1

2 1 Introduction Equity markets are important components of the financial world. Recently, liquidation risk attracts increasing attention from both the industry and academia. This is especially the case following the recent financial crisis sparked by the subprime loan mess and the subsequent equity market meltdown. Liquidation of large block stocks in open market typically depresses the market to the extent that it is difficult to fill the sell orders at desirable prices. The size of positions is often so large that the execution of their order takes days to complete. Heavy selling from institution traders in the recent bear market demonstrates significant price impact. Managing liquidation of large block of stock during these turbulent times is crucial not only to the trading institutions (such as mutual funds and pension funds), but also to the general public at large because reckless of dumping large positions in open market exacerbates market downturn, shatters public confidence in the marketplace, and poses serious threat to the stability of the financial system. The purpose of this paper is to study optimal strategies on liquidation of large blocks of stock and design mathematically sound selling rules. There is an extensive literature for this fast evolving research field. Early theoretical studies can be traced back to Bertsimas and Lo [4] who studied an expected trading cost minimization in discrete-time. Under the assumption that the price is affected by each trade execution, they derived optimal trading strategies using a dynamic programming approach. Almgren and Chriss [2] considered a portfolio liquidation problem by virtue of a meanvariance criterion, where both the market impact risk (risk of rapid execution due to market impact) and volatility risk (risk of delayed execution) were incorporated into the modeling. They constructed an efficient frontier in the space of time-dependent liquidation strategies. As a subsequent work, Almgren [1] extended the results to incorporate nonlinear impact functions and trading-enhanced risk. Longstaff [15] studied the liquidation problem in which the investor is restricted to trading strategies with bounded variation and showed that the constrained investor has less control over managing his wealth distribution and acts as if facing borrowing and short selling con- 2

3 straints. Characterization of liquidity discount was considered by Subramanian and Jarrow [22] in connection with an optimal liquidation problem. Such characterization allows the modification of the standard value at risk computation to incorporate liquidity risk. Schied and Schöneborn [19] studied an infinite-horizon optimal liquidation problem for a von Neumann-Morgenstern investor who aims to maximize the expected utility of the proceeds of an asset sale. Using a stochastic control approach, they characterized the value function and the optimal strategy as classical solutions of nonlinear parabolic partial differential equations. In addition, they pointed out that the speed by which the remaining asset position is sold can be decreasing in the size of the position but increasing in the liquidity price impact. Schied and Schöneborn [2] further took the finite horizon problem into consideration. Pemy, Zhang and Yin [18] essentially followed a similar line of Schied and Schöneborn [19] except that the investor is risk-neutral. They formulated the problem using a fluid model in the sense that the liquidation is dictated by the rate of selling over time. The objective is to choose the selling rate to maximize the overall return. It is worthwhile pointing out that in the absence of illiquidity risk, the selling strategies under this framework have been extensively studied by Øksendal [16], Guo and Zhang [11], Zhang [25], Zhang, Yin, and Liu [26], Yin, Liu, and Zhang [23], Helmes [12], Pemy and Zhang [17], etc. This paper is motivated by the fact that in practice, an experienced trader normally carries out the liquidation task by heavy selling combined with occasional buying rather than straightforward unloading the entire position. The buying activities aim to stabilize the price to certain level and to prepare for additional selling. Similar as in Pemy, Zhang and Yin [18], we will assume that the stock price is affected by the rates of selling and buying over time and aim to maximize the overall return by choosing optimal rates. We use constrained viscosity solution techniques in Soner [21] to characterize the dynamics governing the value function and to treat boundary conditions. Instead of treating the stock price alone, we consider a pair of variables, namely, the stock price as well as the size of the stock yet sold at time t. An easily implementable optimal strategy is obtained, which presents a threshold-like control rule. By and large, as demonstrated by numerical experiments, the state-dependent threshold curves separate the whole region into three parts, 3

4 namely, selling region, buying region, and no action region. To find numerical solutions, we use a finite difference method for solving the associated Hamilton-Jacobi-Bellman (HJB) equation. This paper extends the results of Pemy, Zhang and Yin [18] in which they considered the problem with only selling involved. Moreover, the method used in the present paper is powerful and can be used to treat more general problems. The rest of the paper is organized as follows. In the next section, we formulate the infinite horizon problem as a constrained stochastic control problem and formally derive the HJB equation and associated boundary conditions that the optimal reward or the value function satisfies. A rigorous definition of viscosity solution to the HJB equation is placed in Appendix. In Section 3, we present an efficient finite difference method for solving the HJB equation with associated boundary conditions. In Section 4, we present numerical examples to demonstrate how to apply the method to find the optimal selling rule. In Section 5, we turn to the finite horizon model. Section 6 concludes the paper with further remarks. 2 Problem formulation Let X t denote the stock price at time t. Use u t and v t to represent the rate of selling and the rate of buying, respectively. The stochastic differential equation dx t = µx t dt + σx t db t a 1 u t dt + a 2 v t dt, (1) with positive a 1 and a 2, states that the change in the price dx(t) has four components. The first term is the drift µx t dt with µ > ; the second is the diffusion term σx t db t, where B t is a standard Brownian motion; the third and fourth terms reflect the relative influence that sales u t dt and purchases v t dt have upon the change in the price 1. Throughout this paper, we assume a 1 > a 2 >. In our formulation, X t is one of the state variables and (u t, v t ) are the control actions. At time t, the number of shares of a stock yet sold is denoted by Z t. In this paper, we use a fluid model, i.e., Z t takes values in the set of nonnegative real numbers and its rate of change is driven by the 1 In essence, it is assumed that the trading volume has a proportional influence on the change in stock price. If we instead assume that the trading amount has a proportional influence, that is, dx t = µx tdt + σx tdb t a 1u tx tdt + a 2v tx tdt, then it can be shown that the resulting value function is linear in stock price and the optimal strategy must be independent of stock price. 4

5 differential equation dz t = (v u) dt, Z = z. (2) Thus, the state at any time t consists of the pair (X t, Z t ), and the state space is S = [, ) [, N], where N < is the total number of the stock to be sold. We assume u t λ 1, v t λ 2. The control (u t, v t ) is allowed to take values in the set Γ = [, λ 1 ] [, λ 2 ]. Definition 2.1 We say that a control (u., v.) is admissible with respect to the initial values (x, z) S, if (i) (u., v.) is an F t = σ{x s : s t} adapted; (ii) (u t, v t ) Γ for all t ; (iii) the corresponding state process Z t N for all t. We use A = A(x, z) to denote the set of all admissible controls. The admissibility essentially requires the control (u t, v t ) not depending on future but only on the available information (namely, the stock price) up to time t, taking values in the control set, and the state the pair (X t, Z t ) satisfying Z t N. Given an initial position (X, Z ), we introduce a stopping time τ = inf{t > : Z t = or X t = } to reflect the first point in time that either all shares have been sold or the stock price becomes. The investor aims to sell out N shares and to maximize [ τ ] J (x, z; u., v.) = E e ρs (u s v s ) X s ds, where ρ > is the discount rate. In this paper, we assume µ < ρ. Remark 2.2 Our objective function focuses on money flow from a trader s point of view. It is different from the traditional wealth based reward function. In this paper, the discount rate ρ is treated as time scale factor rather than the traditional risk-free interest rate. It is used to determine the time horizon of the selling process, i.e., how long the position should be completely liquidated. For example, larger ρ encourages swift selling action. See Zhang [25] for further discussion in connection with a stock selling rule. Our objective is to choose (u., v.) A so as to maximize the expected reward J(x, z; u., v.). Define the value function as follows: φ(x, z) = sup J(x, z; u., v.). (3) (u.,v.) A 5

6 Using stochastic control methods (see [7]), formally, we obtain a partial differential equation known as the Hamilton-Jacobi-Bellman equation (or HJB equation in short) satisfied by the value function: or, where x (, ), z (, N), L φ + max u λ 1 (ul 1 φ) + max v λ 2 (vl 2 φ) =, L φ + λ 1 (L 1 φ) + + λ 2 (L 2 φ) + =, (4) L φ = 1 2 σ2 x 2 φ xx + µxφ x ρφ, L 1 φ = a 1 φ x φ z + x, L 2 φ = a 2 φ x + φ z x. At x = and z =, we naturally have the Dirichlet boundary value condition φ x= =, (5) φ z= =. (6) At z = N, we prescribe the following state constrained boundary condition L φ + λ 1 (L 1 φ) + z=n =, (7) which means no buying at z = N. In Appendix, we will provide a rigorous definition of constrained viscosity solution to the HJB equation and prove that the value function φ(x, z) is the unique state constrained viscosity solution of the equation (4) in (, ) (, N]. Furthermore, we will deduce the boundary condition (7) in the viscosity sense. Define SR {(x, z) : L 1 φ(x, z) > } = {(x, z) : φ z + x > a 1 φ x }, BR {(x, z) : L 2 φ(x, z) > } = {(x, z) : φ z + x < a 2 φ x }, NT = {(x, z) : a 2 φ x φ z + x a 1 φ x }, which respectively represent selling region (SR), buying region (BR) and no-transaction region (NT). We are interested in the properties of these regions which characterize the optimal policy. Due to lack of analytical solutions, we will employ numerical solutions to examine the optimal policy. 6

7 Remark 2.3 It is worth pointing out that (4) reduces to max {L φ, L 1 φ, L 2 φ} =, (8) as λ 1, λ Numerical scheme To numerically solve (4)-(6), we use the finite difference method. Consider a bounded domain [, M] [, N], where M > is big enough. We need to impose a Dirichlet boundary condition at x = M. Assume that at this level the optimal decision is to sell all shares at the maximum allowable rate, then all shares would be sold after a time of z/λ 1. This implies the following boundary condition: φ(m, z) = E X =M = z/λ1 z/λ1 λ 1 e ρs X s ds λ 1 e ρs [e µs M a 1λ 1 µ (eµs 1)]ds = λ 1(M a 1 λ 1 /µ) (e (µ ρ)z/λ 1 1) + a 1λ 2 1 µ ρ µρ (1 e ρz/λ 1 ). Let x and z be the step size in x- and z- direction, respectively, such that M = m x, N = n z. Let φ j,k denote the numerical approximation to φ(j x, k z). Let L h V j,k, L h 1 V j,k and L h 2 V j,k be the discretization of L V, L 1 V and L 2 V respectively at (j x, k z), where L h V j,k = 1 2 σ2 j 2 x 2 φ j+1,k + φ j 1,k 2φ j,k x 2 + µj x φ j+1,k φ j,k x = 1 2 σ2 j 2 [φ j+1,k + φ j 1,k 2φ j,k ] + µj [φ j+1,k φ j,k ] ρφ j,k, L h 1V j,k = a 1 φ j,k φ j 1,k x L h 2V j,k = a 2 φ j+1,k φ j,k x φ j,k φ j,k 1 z + φ j,k+1 φ j,k z + j x j x ρφ j,k Here we emphasize that the upwind technique has been used to discretize the first order terms φ x and φ z. Then the finite difference discretization for solving (4) is given as follows: L h V j,k + λ 1 ( L h 1V j,k ) + + λ2 (L h 2V j,k ) + =, for j = 1,..., m 1, and k = 1,..., n 1. 7

8 At z = (i.e., k = ) and x = (i.e., j = ), we are given V j, = for j = 1,..., m 1; V,k = for k = 1,..., n 1. Due to the use of the upwind technique, it is natural to discretize the boundary condition at z = N (i.e., k = n) : L h V j,n + λ 1 (L h 1V j,n ) + = for j = 1,..., m 1. It remains to deal with the nonlinear terms in the above discretization, which take the form of G +. Given an initial guess G, let G i be the value at i-th nonlinear iteration (i ). We then make use of the following nonsmooth Newton iteration: (G i+1) + = (G i) + + ( G i+1 G i) I {G i >} = G i+1 I {G i >}, where I is the indicator function. Essentially the above algorithm is similar to the penalty algorithm with λ 1 and λ 2 for equation (8) (see, [5], [6], or [1]). 4 Numerical results In this section we provide several numerical experiments to demonstrate the numerical solutions of the value function and the corresponding optimal policy. First, we solve the discrete version of the HJB equation using the successive approximations. We take a 1 =.3, a 2 =.15, µ =.1, ρ =.15, σ =.3, λ 1 = 1, λ 2 = 1. (9) SR 3 x NT BR z 8

9 Figure 1: NT, SR, and BR The optimal control (u (x, z), v (x, z)) is determined by the three regions: SR, NT, and BR in Figure 1, where the BR and SR are separated by the NT. The BR corresponds to low price region and SR the high price region. The simplicity of the strategy is particularly welcomed by the practitioners in financial market. 4.1 Dependence of (u (x, z), v (x, z)) on ρ Next, we consider the dependence of the optimal control (u (x, z), v (x, z)) on the discount factor ρ. Heuristically, the larger the ρ is, the higher discount into the future, which in turn, encourages sales of shares. We take ρ =.15 and.2 with all other parameters fixed as in (9). In Figure 2, it says that larger ρ leads to larger SR and smaller BR. There is a clear downwards shift of both buying and selling curves ρ=.15 Sell ρ=.15 Buy ρ=.2 Sell ρ=.2 Buy x z Figure 2: Trading strategies with varying ρ 4.2 Dependence of (u (x, z), v (x, z)) on µ In Figure 3, we demonstrate the dependence of (u (x, z), v (x, z)) on µ. Intuitively, larger µ would be more attractive to hold the stock, which leads to smaller SR and larger BR. To see this, we take µ =.9 and.1 with other parameters fixed. The curves plotted in Figure 3 confirm the shift of these regions when µ changes from.9 to.1. 9

10 µ=.1 Sell µ=.1 Buy µ=.9 Sell µ=.9 Buy x z Figure 3: Trading strategies with varying µ 4.3 Dependence of (u (x, z), v (x, z)) on a 1 We next consider the dependence of (u (x, z), v (x, z)) on a 1. We plot the regions in Figure 4 with a 1 =.3 and.4 and other fixed. Intuitively, a larger a 1 has greater impact of selling on stock price. It is clear from these pictures that large a 1 leads to both smaller SR and BR and larger NT a 1 =.3 Sell a 1 =.3 Buy a 2 =.4 Sell a 2 =.4 Buy 4 x z Figure 4: Trading strategies with varying a Dependence of (u (x, z), v (x, z)) on σ Finally, we examine the dependence of (u (x, z), v (x, z)) on σ. Intuitively, larger σ means larger volatility which would lead to more opportunity also more risk for future selling. In Figure 5, when σ increases from.2 to.5, there is a upwards shift of both SR and BR for small z and a slight downwards shift of both SR and BR for big z. So, the 1

11 impact of σ on trading strategies is non-monotonic σ=.5 Sell σ=.5 Buy σ=.2 Sell σ=.2 Buy x z 5 Finite horizon problem Figure 5: Trading strategies with varying σ. In this section, we consider the corresponding finite horizon problem in which the investor is facing a finite time horizon T. We can similarly define the set of all admissible controls, denoted by A t = A(x, z, t). To accommodate the finite time horizon, we have imposed a finite time proportional penalty 1 c on the value of remaining shares. Now, the investor aims to maximize [ ] T τ E e ρs (u s v s ) X s ds + ce ρ(t τ) Z T τ X T τ, where τ is the stopping time as defined in Section 2. Define the value function φ (x, z, t) = max E (u.,v.) A t which satisfies [ T τ Xt=x, Zt=z t t e ρ(s t) (u s v s ) X s ds + ce ρ(t τ t) Z T τ X T τ ], φ t + L φ + λ 1 (L 1 φ) + + λ 2 (L 2 φ) + =, (1) in x (, + ), z (, N), t [, T ). The final and boundary conditions are φ t=t = cxz, φ x= =, 11

12 φ z= =, φ t + L φ + λ 1 (L 1 φ) + z=n =. The problem can be solved by using an implicit difference scheme with the discretization and the non-smooth Newton iteration as presented in Section 3. We carry out numerical tests to examine the trading strategy with varying c. The default parameter values are the same as in the infinite horizon case. A time snapshot of the buy region, sell region and no-action region at T t =.5 is depicted in Figure 6 with c =.5 and c =.7, respectively. It can be seen that the shape of the no-action region resembles that of the infinite horizon case. There is a clear upwards shift of both SR and BR as c increases. This implies that the investor is encouraged to hold the shares for a bigger c due to a higher guaranteed terminal value of unsold shares c=.5 Sell c=.5 Buy c=.7 Sell c=.7 Buy x z 6 Conclusion Figure 6: A time snapshot of trading strategies with varying c. In this paper, we studied an optimal strategy of selling a large block of illiquid stock, where the seller can influence the price by varying the sales and occasional purchases. We established the mathematical framework for the optimal control and provided numerical examples that reveal dependence of the strategy on various parameters. These results provide useful insights into the nature of the problem and can be used as a guide to fund managers when unloading large blocks of illiquid stocks. 12

13 In this paper, our focus was to study a relatively simple model in order to examine the viability of our selling rule and its dependence on various parameters. It would be interesting to extend our results to more general models with nonlinear dynamics with various objective functions. In particular, it would be interesting to incorporate explicitly the risk factor into the objective function as in Almgren and Chriss [2]. In this connection, the robust control (minimax control and risk-sensitive control) idea appears to be relevant. Robust control emphasizes system stability (i.e., with less risk) rather than optimality (i.e., overall return). For detailed discussions along this line, see Fleming and Zhang [9] in connection with robust production planning of stochastic manufacturing systems. 7 Appendix In this appendix, we provide definitions of constrained viscosity solution, show that the value function is the unique constrained viscosity solution to the associated HJB equation and discuss boundary condition (7). Definition 7.1 φ(x, z) is a constrained viscosity solution of (4) in (, ) (, N], if (i) φ(x, z) is a viscosity supersolution of (4) in (, ) (, N), i.e., L ϕ(x, z) + max u λ 1 (ul 1 ϕ(x, z)) + max v λ 2 (vl 2 ϕ(x, z)) (x,z ), whenever ϕ(x, z) C 2,1 such that φ(x, z) ϕ(x, z) has a local minimum at (x, z ) (, ) (, N) and φ(x, z ) = ϕ(x, z ); (ii) φ(x, z) is a viscosity subsolution of (4) on (, ) (, N], i.e., L ϕ(x, z) + max u λ 1 (ul 1 ϕ(x, z)) + max v λ 2 (vl 2 ϕ(x, z)) (x,z ), whenever ϕ(x, z) C 2,1 such that φ(x, z) ϕ(x, z) has a local maximum at (x, z ) (, ) (, N] and φ(x, z ) = ϕ(x, z ). The constrained viscosity solutions for finite time horizon equation (1) can be defined similarly. By the standard method, it is easy to see that the value function φ(x, z) is continuous and grows at most linearly in x. 13

14 Theorem 7.2 The following assertions hold (i) The value function φ(x, z) is the unique state constrained viscosity solution of the equation (4) in (, ) (, N] that grows at most linearly in x. (ii) Furthermore, φ(x, z) satisfies the following equation at z = N in the viscosity sense L φ + max u λ 1 (ul 1 φ) =. (11) Proof. It is standard to prove that φ(x, z) is a state constrained viscosity solution of (4). The uniqueness can be obtained using the method of [13]. It remains to show (ii). First we show φ(x, N) is nondecreasing. Assume < x 1 < x 2. It is clear that A(x 1, N) = A(x 2, N). Let (u 1., v 1.) A(x 1, N), X 1 t be the solutions of (1) according to control (u 1., v 1.) with X 1 = x 1 and τ 1 = inf{t > : Z t = or X 1 t = }. Define (u 2., v 2.) such that (u 2 t, v 2 t ) = (u 1 t, v 1 t ) for t τ 1 and (u 2 t, v 2 t ) = (, ) for t > τ 1. Then (u 2., v 2.) A(x 2, N). Let X 2 t be the solutions of (1) according to control (u 2., v 2.) with X 2 = x 2 and τ 2 = inf{t > : Z t = or X 2 t = }. It is easily to see that τ 1 τ 2, a.s.. Using integration by parts, we get From (2) τ1 ( ) ( ) ( τ1 ) E e ρt u 2 t vt 2 Xt 1 Xt 2 dt = E e ρτ 1 (u 2 t vt 2 )dt (X ) τ 1 1 Xτ 2 1 τ1 +(ρ µ)e e ρt ( t ) ( (u s v s )ds Xt 1 Xt 2 t Z t = N + (vs 2 u 2 s)ds N. ) dt. This implies, for t τ 1 On the other hand t (u 2 s v 2 s)ds, a.s. for t τ 1. Therefore, we conclude that X 1 t X 2 t = (x 1 x 2 )e (µ 1 2 σ2 )t+σb t <. τ1 ( ) ( ) E e ρt u 2 t vt 2 Xt 1 Xt 2 dt. Since u 2 t v 2 t = u 1 t v 1 t for t τ 1 and u 2 t v 2 t = for τ 2 t > τ 1, J(x 1, N; u 1., v 1.) J(x 2, N; u 2., v 2.). 14

15 It follows that φ(x 1, N) φ(x 2, N). Now we show that φ is a viscosity subsolution of equation (11). For suppose not, φ is not a viscosity subsolution. Then exists x > and δ > such that L ϕ(x, N) + max u [,λ 1 ] (L 1ϕ(x, N)u) = 2δ, where test function ϕ C 2 ((, ) (, N]), such that φ ϕ attain its maximum at (x, N). We might as well assume φ(x, N) = ϕ(x, N). Then φ ϕ on S. Let us choose a neighborhood O(x, N) S such that L ϕ(x, z) L ϕ(x, N) + λ 1 L 1 ϕ(x, z) L 1 ϕ(x, N) + λ 2 L 2 ϕ(x, z) L 2 ϕ(x, N) δ, for all (x, z) O(x, N). Let K = sup ( L ϕ(x, z) + λ 1 L 1 ϕ(x, z) + λ 2 L 2 ϕ(x, z) ). O(x,N) Given (X, Z ) = (x, N) and (u., v. ) A(x, N). Let (X t, Z t ) be the solution of (1)-(2) and stopping time τ = τ h = min{h, inf{t > : (X t, Z t ) O(x, N) c }} for h >. By the dynamic programming principle, we have [ τ ] J (x, N; u., v.) E x,n e ρt (u t v t ) X t dt + e ρτ φ(xτ, Zτ) [ τ ] E x,n e ρt (u t v t ) X t dt + e ρτ ϕ(xτ, Zτ). Next we subtract φ(x, N) = ϕ(x, N) from both sides, use the Ito s formula E x,n E x,n τ [ τ E x,n( 1 2 Kτ 2 + δτ) + E x,n J (x, N; u., v.) φ(x, N) ] e ρt (u t v t ) X t dt + e ρτ ϕ(xτ, Zτ) ϕ(x, N) e ρt [L ϕ(x t, Z t ) + u t L 1 ϕ(x t, Z t ) + v t L 2 ϕ(x t, Z t )] dt τ [L ϕ(x, N) + u t L 1 ϕ(x, N) + v t L 2 ϕ(x, N)] dt 15

16 Since φ(x, z) is continuous, φ(x, N) is nondecreasing and (x, N) is a maximum point of φ ϕ, we conclude that ϕ x (x, N). This yields L 1 ϕ(x, N) τ u t dt + L 2 ϕ(x, N) Since (u., v. ) is admissible, we have from state equation (2) for all t [, τ]. Therefore, and Hence we have Z t = N + τ t τ τ v t dt L 1 ϕ(x, N) (u t v t )dt. (v t u t ) dt N, (u t v t )dt [, λ 1 τ], τ L 1 ϕ(x, N) (u t v t )dt τ max (L 1ϕ(x, N)u). u [,λ 1 ] J (x, N; u., v.) φ(x, N) E x,n( 1 2 Kτ 2 δτ). Taking the supremum over all admissible control and choosing h small enough, this yields a contradiction. Thus, φ is a viscosity subsolution of equation (11). Next let ϕ be a smooth test function such that (x, N) is a minimizer of φ ϕ and φ(x, N) = ϕ(x, N). Choose the constant control (u., v. ) = (u, ) with u [, λ 1 ]. It can be deduced by the standard arguments L ϕ(x, N) + max u [,λ 1 ] (L 1ϕ(x, N)u). This implies that φ is a viscosity supersolution of equation (11). This completes the proof. Let u (x, z) and v (x, z) be the optimal selling and buying rate, respectively. From (ii) in Theorem 7.2, we deduce v (x, N) = for all x (, ), which interpret the boundary condition (7). Then, the selling region, buying region, no-transaction region defined in Section 2 can be rewritten as follows: SR = {(x, z) : u (x, z) >, v (x, z) = }, BR = {(x, z) : v (x, z) >, u (x, z) = }, NT = {(x, z) : u (x, z) =, v (x, z) = }. 16

17 We next give a verification theorem in terms of the value function along the line of Fleming and Rishel [7]. Theorem 7.3 Let w(, ) C 2,1 b (a) w(x, z) J(x, z, u., v.) for all admissible (u., v.) be a solution to the HJB equation (4). Then, (b) Define the selling rate and the buying rate as follows: { u, if L1 w(x, z), (x, z) = λ 1, if L 1 w(x, z) >, { v, if L2 w(x, z), (x, z) = λ 2, if L 2 w(x, z) >. (12) Then w(x, z) = φ(x, z) = J(x, z, u, v ). That is, (u, v ) is optimal. Proof. The proof is standard. We sketch the main steps below. First using Dynkin s formula and the HJB equation (4), we have Sending T, we have T Ee ρt w(x T, Z T ) w(x, z) E e ρs (u s v s )X s ds. w(x, z) E e ρs (u s v s )X s ds. The equality holds if u = u and v = v given in (12). References [1] R.F. Almgren, Optimal execution with nonlinear impact functions and trading-enhanced risk, Applied Math Finance, 1 (23), pp [2] R. Almgren and N. Chriss, Optimal execution of portfolio transactions, Journal of Risk, 3 (2), pp [3] O. Alvarez, J.M. Lasry and P.L. Lions, Convexity viscosity solutions and state constraints, J. Math. Pures Appl., 76 (1997),

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