Liquidation of a Large Block of Stock

Transcription

1 Liquidation of a Large Block of Stock M. Pemy Q. Zhang G. Yin September 21, 2006 Abstract In the financial engineering literature, stock-selling rules are mainly 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 position 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 a large block of stock by selling much smaller number of shares over a longer period of time. In particular, we treat the selling rule problem by using a fluid model in the sense that the number of shares are treated as fluid (continuous) and the corresponding liquidation is dictated by the rate of selling over time. The objective is to maximize the expected overall return. The underlying problem may 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 reported to illustrate the results. Keywords: Optimal control, state constraint, selling rule. This research was supported in part by the National Science Foundation. Department of Mathematics, Towson University, Towson, MD , mpemy@towson.edu Department of Mathematics, Boyd GSRC, The University of Georgia, Athens, GA , qingz@math.uga.edu Department of Mathematics, Wayne State University, Detroit, MI 48202, gyin@math.wayne.edu 1

2 1 Introduction Decision making in stock liquidation is crucial for successful trading and portfolio management. In [10], Øksendal formulated the optimal selling decision making as an optimal stopping problem and derived a closed-form solution. Using a smooth fit technique, the problem may be treated as a free boundary value problem and the idea may be traced back to McKean [9] in the 1960 s. Such results are extended to incorporate possible regime switching by Guo and Zhang [6] with the switching represented by a two-state Markov chain. In [6], using a smooth-fit technique, the optimal stopping problem was converted to a set of algebraic equations under certain smoothness conditions; closed-form solutions were obtained. Another way to formulate the problem is to consider a selling rule determined by two threshold levels, a target price and a stop-loss limit. One makes a selling decision whenever the price reaches either the target price or the stop-loss limit. The objective is to choose these threshold levels to maximize an expected return. In [18], Zhang obtained optimal threshold levels by solving a set of two-point boundary value problems. In particular, if the underlying Markov chain has only two states, the corresponding two-point boundary value problem has an analytic solution and the optimal solution can be obtained in a closed form. When the modulating Markov chain has a large number of states, a two-time-scale approach is used by Zhang, Yin, and Liu [19] for obtaining near-optimal selling strategies. Along another line, in Yin, Liu, and Zhang [16], an alternative approach using stochastic approximation methods was developed for a class of stock liquidation problems. The idea is to design a recursive algorithm to approximate the optimal threshold values. Convergence and rates of convergence of the algorithm were obtained. Simulation examples were presented and the computation results were compared to the analytic solutions when the Markov chain having two states. Moreover, such a method can also be extended to treat the case that the precise model is not available; NASDAQ real data were used for demonstration in [17]. Recently, Helmes [7] considered computational issues of the selling rule by using a linear programming approach. Pemy and Zhang [12] studied a selling rule in which the liquidation is constrained to be within a pre-specified time period. A common feature in the aforementioned papers is that the selling has to be done all at 2

3 once. This is feasible when a relative smaller number of shares of a stock is involved. Selling a large shares of a stock in a market place normally depresses the market resulting in poor filling prices, if it is sold in a short period of time. A typical strategy for selling stock of large size is to sell much smaller number of shares over a longer period of time. It is the purpose of this paper to consider large block stock liquidation. Here, we treat the selling rule problem using a fluid model in the sense that the liquidation is dictated by the rate of selling over time. The objective is to maximize the overall return, and the underlying control problem has state constraints, which makes the problem much more difficult to analyze. We use constrained viscosity solution techniques in Soner [14] 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 at time t. An easily implementable optimal selling strategy is obtained, which presents a threshold-like selling rule. Roughly, as demonstrated in the numerical experiments, the state-dependent threshold curve separates the entire region into two parts, namely, selling region and no selling region. To find numerical solutions, we use a finite difference method for solving the associated Hamilton-Jacobi-Bellman (HJB) equations and demonstrate its convergence. In this paper, we focus on the ideas, methods, and properties of optimal policies. All results presented can be rigorously justified. The detailed proofs are not included; they are available upon request from the authors, however. The rest of the paper is organized as follows. In the next section, we formulate the problem as a constrained stochastic control problem and provide needed assumptions. In Section 3, we study the properties of the value or the optimal reward function and establish that it is the only viscosity solution to an HJB equation. Also provided are the associated boundary conditions and optimal selling rate. In Section 4, we consider a finite difference method for solving the associated HJB equations and establish its convergence. In Section 5, we present numerical examples to demonstrate how to apply the method to find the optimal selling rule. Section 6 concludes the paper with further remarks. 3

4 2 Problem Formulation Use X(t) and u(t) to represent the stock price and the rate of selling, respectively. The stochastic differential equation dx(t) = X(t)(µdt + σdw) L(X(t), u(t))dt (1) states that the change in the price dx(t) has three components. The first two terms are Brownian motion with drift. The drift term is X(t)µdt with µ > 0. The diffusion term is X(t)σdW(t), where W(t) is a standard Brownian motion. The third term L(X(t), u(t))dt reflects the relative influence that sales u(t)dt have upon the change in the price. For an atomistic firm, this term is negligible, pure competition. When the firm s sales can affect the price, spoil the market, then it must be taken into account in deciding how much to sell at each time. There are three cases to consider. In Case (i), equation (1) becomes dx(t) = X(t)(µdt + σdw(t)) bu(t)dt, X(0) = x, (2) where µ, σ, and b are positive constants representing the stock return rate, volatility, and depreciation rate, respectively. The large firm effect L(X(t), u(t)) = bu(t), is proportional at rate b > 0 to the sales u(t). This case is the subject of the present paper. Case (ii) concerns the pure competition case L(X(t), u(t)) = 0. Case (iii) is L(X(t), u(t)) = X(t)φ(u(t)) where φ (x) > 0. The depressing effect of sales increases with the sales. For example, as the market realizes that the large firm is selling, other firms are more reluctant to buy and may also join in the selling. The solution for the optimal sales u(t) is different in each case. In a subsequent paper we shall discuss Case (iii). In our formulation, X(t) is one of the state variables and u(t) is the control action. At time t, the number of shares of a stock yet sold is denoted by Z(t). In our formulation, we use a fluid model. That is, we allow Z(t) to take values in the set of nonnegative real numbers. We let its rate of change to be driven by the selling rate given by the differential equation dz(t) = u(t)dt, Z(0) = z. (3) 4

5 Thus, the state at any time t consists of the pair (X(t), Z(t)), and the state space is S = [0, ) [0, N], where N < is the total number of the stock to be sold. Note that the control or the selling rate u(t) is allowed to take values in the set Γ = [0, 1]. Definition 2.1 We say that a control u( ) is admissible with respect to the initial values (x, z) S, if, (i) u( ) is an F t = σ{x(s) : s t} adapted; (ii) u(t) Γ for all t 0; (iii) the corresponding state process (X(t), Z(t)) S for all t 0. We use A = A(x, z) to denote the set of all admissible controls. The admissibility essentially requires the selling rate u(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)) being in S. The interest rate ρ > 0 reflects the opportunity cost of having capital tied up in stock inventory. Hence the higher the interest rate, the greater is the incentive to sell now rather than to hold. It will be clear that a crucial variable will be (ρ µ), the interest rate less the drift term µ. In our problem, a positive solution for sales u(t) > 0 will require that interest rate exceeds the drift term, (ρ µ) > 0. The expected present value of profits from sales of the stock is J(x, z, u( )) = E 0 e ρt X(t)u(t)dt. Our objective is to choose the rate of selling u( ) A so as to maximize the expected reward J(x, z, u( )). The optimal reward function is v(x, z) = sup J(x, z, u( )). (4) u( ) A Using stochastic control methods (see [4]), formally, we obtain a partial differential equations known as the Hamilton-Jacobi-Bellman equation (or HJB equation in short) satisfied by the optimal reward function: ρv(x, z) = max u Γ [ v(x, z) v(x, z) (µx bu) u + 1 x z 2 σ2 x 2 2 v(x, z) + xu x 2 ], 0 z N. One of the problems regarding the optimal reward function is that it may not be smooth as one would like to. We term a solution of the HJB equation a classical solution if it 5 (5)

6 is twice continuously differentiable with respect to x and continuously differentiable with respect to z. The non-smoothness poses problems: There may be no classical solution to the HJB equation. A convenient way of dealing with the non-smoothness of the optimal reward function is the method of viscosity solution, which provides a viable alternative. This, in turn, characterizes the optimal reward not as a classical solution of the associated partial differential equation but an appropriate weak solution. We first recall the following definition. It is easily seen that one overcomes the difficulties by shift the derivatives of the optimal reward function to some smooth test function. Such a method has become a standard technique for dealing with stochastic control problems. We refer the reader to [5] for further references. Definition 2.2 v(x, z) is a constrained viscosity solution of (5), if, (1) v(x, z) is a viscosity supersolution of (5) on (0, ) [0, N], i.e., ρv(x 0, z 0 ) max [(µx 0 bu) φ(x 0, z 0 ) u φ(x 0, z 0 ) + 1 u Γ x z 2 σ2 x 2 2 φ(x 0, z 0 ) 0 + x x 2 0 u], (6) whenever φ(x, z) is twice continuously differentiable with respect to x and continuously differentiable with respect to z such that v(x, z) φ(x, z) has a local minimum at (x 0, z 0 ) [0, ) [0, N]; (2) v(x, z) is a viscosity subsolution of (5) on (0, ) (0, N), i.e., ρv(x 0, z 0 ) max [(µx 0 bu) φ(x 0, z 0 ) u φ(x 0, z 0 ) + 1 u Γ x z 2 σ2 x 2 2 φ(x 0, z 0 ) 0 + x x 2 0 u], (7) whenever φ(x, z) twice continuously differentiable with respect to x and continuously differentiable with respect to z such that v(x, z) φ(x, z) has a local maximum at (x 0, z 0 ) (0, ) (0, N). 6

7 3 Properties of the Reward Function To reveal the features of the optimal selling strategy, it is necessary to scrutinize the optimal reward function. This section presents its properties. We first show that the optimal reward is Lipschitz continuous. Then we show that it is in fact the unique solution of the HJB equation in the sense of viscosity solution. Next, boundary conditions are provided. Finally, the optimal selling rule is specified. 3.1 Properties of the Reward Function The following lemma reveals that as long as ρ > µ, for a fixed z, v(, z) is nondecreasing; for each fixed x, v(x, ) is nonincreasing; it is Lipschitz in x and locally Lipschitz in z. Lemma 3.1. Assume ρ > µ. Then the following assertions hold. (a) For each z, v(x, z) is nondecreasing in x. (b) For each x, v(x, z) is nondecreasing in z. (c) v(x, z) is Lipschitz continuous in (x, z). More precisely, v(x 1, z 1 ) v(x 2, z 2 ) x 1 x 2 ρ µ + (x 2 + 1) z 1 z 2, for (x 1, z 1 ) and (x 2, z 2 ) in (0, ) [0, N]. Remark 3.2 The condition ρ > µ is used to guarantee the boundedness of the optimal reward function. It can be seen in the numerical examples that this condition can be replaced by a truncated reward function. With Lemma 3.1 at hand, we are able to demonstrate the uniqueness of v. This is stated as follows. Theorem 3.3 The optimal reward function v(x, z) is the unique constrained viscosity solution on S. 7

8 3.2 Boundary Conditions Let us consider the case that the optimal reward function is twice continuously differentiable with respect to x and continuously differentiable with respect to z. In this case, v(x, z) satisfies (7). Next, take φ(x, z) = v(x, z) + β(n z) with β > 0. Then, v(x, z) φ(x, z) reaches its minimum at (x 0, 0). Thus, (6) must be satisfied. Combining (6) and (7) yields the following boundary condition { max bu φ(x 0, 0) u φ(x } 0, 0) + x 0 u u Γ { x z = max bu v(x ( 0, 0) v(x0, 0) u u Γ { x z max bu v(x 0, 0) u v(x 0, 0) + x 0 u u Γ x z which leads to the boundary condition b v(x 0, 0) x v(x 0, 0) z + x 0 0. ) β } } + x 0 u There is no constraint at z = N because u(t) 0, which makes Z(t) N for all t. Similarly, taking φ(x, z) = v(x, z) βx, we obtain the boundary condition at x = 0 given by v(0, z) v(0, z) b x z 0, for all z [0, N]. These two boundary conditions are needed to guarantee the optimal feedback control u (x, z) to be feasible at the boundaries x = 0 and z = 0, i.e., u (0, z) = 0 and u (x, 0) = Optimal Selling Strategy In view of the standard verification theorem given in Fleming and Rishel [4] (see Soner [13] in connection of optimal control with natural resource consumption control and Zhou [20] for the corresponding version under the viscosity solution framework), the optimal selling rate (optimal control) should have the following form: v(x, z) v(x, z) 0, if x b < 0, u (x, z) = x z v(x, z) v(x, z) 1, if x b 0. x z 8, (8)

9 The boundary conditions at x = 0 and z = 0 demand that u (0, z) = 0 and u (x, 0) = 0, respectively, which force the corresponding state (X(t), Z(t)) stays in domain S. This optimal selling strategy is simple and easily implementable. It offers much insight for the trading practice. As shown in (8), the optimal selling strategy is to sale one unit of stock (the unit here could be 1,000 shares, for example) whenever v(x, z) v(x, z) x b x z 0, otherwise no sell action will be taken. Intuitively, also as shown in the numerical example to follow, for a given z, the optimal selling rule should be determined by a threshold level x so that one sells at full rate if x > x (when the price is attractive) and sells nothing if x < x. Similarly, for a given x, the optimal selling rule should be: There exists a z such that one sells at full rate if z > z (when more shares yet sold) and sells nothing when z < z. The simplicity of the selling rule should be particularly welcomed by the practitioners in financial market. 4 A Numerical Scheme In order to find an optimal selling rule u, one has to find the corresponding optimal reward function, which requires solving the associated HJB equation. In general, an analytical solution to the HJB equation is difficult to obtain. In this section we resort to numerical solutions. We consider an explicit finite difference scheme that converges to the unique viscosity solution of equation (5). For computational convenience, we focus on a truncated problem. Given a positive integer M, we consider the following optimal control problem with optimal reward v M v M (x, z) = sup E u A(x,z) 0 e ρt min(x(t), M)u(t)dt. It is clear that v M (x, z) v(x, z) as M. We can then write the corresponding HJB equation. Let B(R [0, N]) be the collection of bounded and continuous functions u(x, z) defined on R [0, N]. Let h satisfying 0 < h < 1 be the discretization stepsize for the variable x and k satisfying 0 < k < 1 be the stepsize for the variable z. We can construct {v k,h M (x, z)}, 9

10 a sequence of approximation of v M (x, z). It can be shown that as (h, k) 0, the sequence v k,h M converges locally uniformly on R [0, N] to the unique viscosity solution for v M. The derivation of the procedure is available upon request from the authors. 5 Numerical Examples In this section we provide several numerical experiments to demonstrate the numerical solutions of the optimal reward function and the corresponding optimal control policy. First, we solve the discrete version of the HJB equation using the successive approximations. Recall that the optimal selling strategy u (x, z) is determined in terms of the corresponding optimal reward function v(x, z). Both v(x, z) and the switching surface G(x, z) = x b v(x, z)/ x v(x, z)/ z are plotted in Figure 1 with µ = 0.3, σ = 0.1, ρ = 0.1, and b = 0.3. The optimal control u (x, z) is determined by the curve G(x, z) = 0. This example demonstrates the boundedness of the optimal reward function without the condition ρ > µ. 5.1 Dependence of u (x, z) on ρ Next, we consider the dependence of the optimal selling rate u (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. This says the optimal control u (x, z) should be increasing in ρ. In this section, we show the monotonicity with µ = 0.3, σ = 0.1, and b = 0.3, the corresponding u (x, z) increases in ρ = 0.1, 0.2, 0.3, 0.4. Each curve in Figure 2 given by x b v(x, z)/ x v(x, z)/ z = 0 separates the entire region into two parts. The corresponding u (x, z) = 1 for (x, z) is above the curve and u (x, z) = 0 is below the curve. In Figure 2, these curves are monotone decreasing in ρ. This implies the monotonicity of u (x, z) in ρ, namely, u (x, z) increases in ρ. 5.2 Dependence of u (x, z) on µ Here, we demonstrate the dependence of u (x, z) on µ. Intuitively, a larger µ would be more attractive to hold the stock, which leads to smaller u (x, z). To see this, we take ρ = 0.1, 10

11 stock liquidation value function z x optimal control policy 5 x v z bv x z x Figure 1: Graphs of v(x, z) and G(x, z) = x b v(x, z)/ x v(x, z)/ z with µ = 0.3, σ = 0.1, ρ = 0.1, and b = 0.3. σ = 0.3, b = 0.3, and µ = 0.3, 0.4, 0.5, 0.6. The curves plotted in Figure 3 confirmed the monotonicity property of u (x, z). 5.3 Dependence of u (x, z) on b Finally, we consider the dependence of u (x, z) on b. We plot the curves of x b v(x, z)/ x v(x, z)/ z = 0 in Figure 4 with µ = 0.1, ρ = 0.08, σ = 0.1, and b = 0.1, 0.2, 0.3, 0.4. It is clear from these pictures that the corresponding u (x, z) is monotone increasing in b. This suggests that selling activities put more downward pressure on the stock with larger b, which in turn urges more selling. In addition to the dependence on various parameters, these examples also indicate the 11

12 x x v z bv x x v z bv x >0 rho=0.1 rho=0.2 rho=0.3 rho= z Figure 2: Graphs of x b v(x, z)/ x v(x, z)/ z = 0 with µ = 0.3, σ = 0.1, ρ = 0.1, b = 0.3, and ρ =0.1, 0.2, 0.3, 0.4. monotonicity in x and z. That is, u (x, z) is increasing in x for each z; u (x, z) is increasing in z for each fixed x. 6 Conclusion In this paper, we considered a strategy of selling a large block of stock, inventory, where the seller can influence the price by varying the sales. Initially, we posed three cases where the sales affect the change in price, but we only described the solution in one case, namely, equation (2). There are several limiting features of this model that will be taken into account in subsequent work. First, the diffusion term, the risk plays no role in deriving the optimal rate of sales. This results from the fact that the agent is risk neutral and just looks at profits x(t)u(t) and not at some concave function of profits. Second, the solution is bangbang. Either the seller sells the maximum u = 1 or holds u = 0. In our subsequent paper, we consider Case (iii), where the depressing effect of sales upon the change in price L(X(t), u(t)) = X(t)φ(u(t)) where φ > 0. Then the control u(t) is not bang-bang, but 12

13 x v z bv x mu=0.3 mu=0.4 mu=0.5 mu= x x v z bv x > z Figure 3: Graphs of x b v(x, z)/ x v(x, z)/ z = 0 with ρ = 0.1, σ = 0.3, b = 0.3, and µ = 0.3, 0.4, 0.5, 0.6. varies inversely with φ > 0. However, in all of the cases, since there is risk neutrality, the volatility is not relevant in deriving the optimal rate of sales. Therefore the main task in subsequent research is to rectify these limitations. Acknowledgement We thank Professor Jerome Stein and Professor Wendell Fleming, who read several earlier versions of the paper and made many invaluable comments and detailed suggestions that helped us to substantially revise and reshape this paper. 13

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