Optimal Liquidation Problems in a Randomly-Terminated Horizon

# Optimal Liquidation Problems in a Randomly-Terminated Horizon

Qing-Qing Yang Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: kerryyang920910@gmail.com.    Wai-Ki Ching Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. Hughes Hall, Wollaston Road, Cambridge, U.K. School of Economics and Management, Beijing University of Chemical Technology, North Third Ring Road, Beijing, China. E-mail: wching@hku.hk.    Jia-Wen Gu Corresponding author. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China. E-mail: jwgu.hku@gmail.com.    Tak Kwong Wong Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. E-mail: takkwong@maths.hku.hk
July 12, 2019
###### Abstract

In this paper, we study optimal liquidation problems in a randomly-terminated horizon. We consider the liquidation of a large single-asset portfolio with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. Three different scenarios are analyzed under Almgren-Chriss’s market impact model to explore the relation between optimal liquidation strategies and potential inventory risk arising from the uncertainty of the liquidation horizon. For cases where no closed-form solutions can be obtained, we verify comparison principles for viscosity solutions and characterize the value function as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation.

Keywords: Dynamic Programming (DP) Principle; Hamilton-Jacobi-Bellman Equation; Randomly-terminated; Optimal Liquidation Strategies; Stochastic Control; Viscosity Solution.

AMS Subject Classifications: 35Q90, 49L20, 49L25, 91G80, 65M06.

## 1 Introduction

Understanding trade execution strategies is a key issue for financial market practitioners and has attracted growing attention from the academic researchers. An important problem faced by equity traders is how to liquidate large orders. Different from small orders, an immediate execution of large orders is often impossible or at a very high cost due to insufficient liquidity. A slow liquidation process, however, is often costly, since it may involve undesirable inventory risk. Almgren and Chriss  provided one of the early studies on the optimal execution strategy of large trades, taking into account the volatility risk and liquidation costs. In order to produce tractable and analytical results, they set the market impact cost per share to be linear in the rate of trading. Schied and Schoneborn  considered the infinite-horizon optimal portfolio liquidation problem for a von Neumann-Morgenstern investor under the liquidity model of Almgren , in which a power law cost function was introduced to determine optimal trading strategies. However, most of the literature on optimal liquidation strategies mainly considered a known pre-determined time horizon or infinite horizon. The case of unknown (or more precisely, randomly-terminated) time horizon is not fully addressed. In some situation, it is more realistic to assume that the liquidation horizon depends on some stochastic factors of the model. For example, some financial markets adopt the circuit-breaking mechanism, which makes the horizon of the investor subject to the stock price movement. Once the stock price hits the daily limits, all transactions of the stock will be suspended.

In this paper, we consider a randomly-terminated time horizon under three different scenarios that an agent might encounter in a financial market. Almgren-Chriss’s market impact model is employed to describe the underlying asset price:

 {dSt=f(θt)dt+σdWSt,˜St=g(θt)+St,

where the constant is the absolute volatility of the asset price , is an one-dimensional standard Brownian motion, is the actual transaction price, is an admissible control process, and represent, respectively, the permanent and temporary components of the market impact. We consider the liquidation problem of a large single-asset portfolio with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact.

We first consider the case with a pre-determined time horizon , which can be used as a benchmark for other cases with randomly-terminated time horizon. In general, it is required that a liquidation strategy should satisfy the hands-clean condition:

 XT=X0−∫T0θtdt=0,

where is the number of shares held by the trader at the time . We first work on a subclass of deterministic controls, which do not allow for inter-temporal updating, satisfying the hands-clean condition. Obviously, the deterministic strategy obtained in the subclass might be no longer optimal when taking into account the entire class of admissible controls. We then temporarily relax the hands-clean condition, and allow an immediate final liquidation (if necessary) so that the number of shares owned at the time is . We employ the dynamic programming (DP) approach to solve the stochastic control problems and prove that the optimal liquidation strategy actually converges to the deterministic strategy when the transaction cost involved by liquidating the outstanding position approaches to infinity.

We then move to analyze the randomly-terminated cases. Two different scenarios are analyzed to shed light on the relationship between liquidation strategies and potential position risk arising from the uncertainty of the time horizon. First, we consider the scenario where the liquidation process is terminated by an exogenous trigger event. We model the occurrence time of a trigger event to be random and its hazard rate process is given by . Once this event occurs, all liquidation processes will be forced to suspend. Compared with the case without trigger event, agents facing the scenario that an exogenous trigger event might occur during the trading horizon would like to accelerate the rate of liquidating to reduce their exposure to potential position risk and eventually in a smaller position when the trigger event occurs. Their strategy has a steeper gradient and is more “convex” when compared with those who are not threatened by this trigger event. Second, we consider the case when the liquidation process is subject to counterparty risk. Different from the exogenous trigger event setting, information set available to the counterparty risk modeler is more refined in terms of predictability. To model counterparty risk, we adopt the structural firm value approach, originated from Black and Scholes , and Merton , and let the firm’s asset value follow a geometric Brownian motion:

 dYtYt=βdt+ξdWYt.

The incorporation of counterparty risk into the study of optimal liquidation does not come without cost. In order to examine its impact on optimal trading strategies, we have to introduce and employ viscosity solutions. By verifying the comparison principles for viscosity solutions, we characterize the value function as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation. This equation can be numerically solved. We further analyze the effectiveness of the numerical method and illustrate that the computational error is sufficiently small.

The remainder of this paper is structured as follows. The background and basic models of an agent’s liquidation problem are introduced in Section 2. Section 3 discusses typical liquidating problems under the benchmark model. In Sections 4–5, we discuss two different scenarios with randomly-terminated time horizons. Viscosity solution approach is adopted in these sections to study in great generality stochastic control problems. By combining these results with comparison principles, we characterize the value function as the unique viscosity solution of the associated dynamic programming equation, and this can then be used to obtain further results. Finally, concluding remarks are given in Section 6. For the sake of self-containedness, we provide the technical proofs in the Appendix.

## 2 Problem Setup

In this section, we first describe the market environment of the agent. We then present a market impact model to discuss the optimal liquidating problem.

### 2.1 The Market Environment and Market Impact Model

The agent starts at and has to liquidate a large position in a risky asset by time . This terminal time can be either deterministic or random, depending on the scenario that the agent is facing. For simplicity, we assume that the agent withholds the liquidation proceeds. In other words, he/she does not deposit the liquidation proceeds in his/her money market account. At any time , we adopt the following notations for the agent’s portfolio:

(i)

;

(ii)

;

(iii)

;

(iv)

.

The initial conditions are , , and .

Suppose the risky asset can be continuously liquidated during the trading horizon, namely, there is always sufficient liquidity for their execution111For simplicity, the transaction fees will not be considered in this paper.. Let denote the liquidation process. The shares held by the trader at any time can be written as follows:

 Xt=Q−∫t0θudu.

We consider a probability space endowed with a filtration .

###### Definition 1

A stochastic process is called an admissible control process if all of the following conditions hold:

(i)

(ii)

(Non-negativity) , where is the set of nonnegative real values;

(iii)

(Consistency)

 ∫T0θtdt≤Q;
(iv)

(Square-integrability)

 E[∫T0|θt|2dt]<∞;
(v)

(-integrability)

 E[max0≤t≤T|θt|]<∞.

Furthermore, denote as the collection of admissible controls with respect to the initial time and as the collection of controls only satisfying condition (i), (iv) and (v).

We assume that the risky asset exhibits a price impact due to the feedback effects of the agent’s liquidation strategy. For any given admissible control , the market mid-price of the stock is assumed to follow the dynamics:

 dSt=f(θt)dt+σdWSt, (1)

where is a standard Brownian motion with filtration , the constant is the absolute volatility of the asset price, and is the permanent component of the market impact. For simplicity, we further assume that is time homogeneous, namely, is independent of .

Generally speaking, the actual transaction price is not always the same as the market mid-price , since the market is not perfectly liquid, see, for example, Almgren and Chriss . We assume and call the temporary price impact. Intuitively, the function captures quantitatively how the limit order books available in the market are eaten up at different levels of trading speeds.

Assumption 0. The price dynamics follow a simple Almgren-Chriss linear market impact model (see, Almgren and Chriss ):

 f(θ)=−η⋅θandg(θ)=−ν⋅θ,

where and are positive constants.

An agent who holds the stock receives the capital gain or loss due to stock price movements. Thus, if the agent’s position is marked to market using the book value, ignoring market impact that would be incurred in converting these shares into cash, at any time , the agent’s portfolio value satisfies

 {V0=Q⋅sdVt=(˜St−St)θtdt+XtdSt. (2)

At any time before the end of trading,

 Vt=V0+∫t0(˜Su−Su)θudu+∫t0XudSu=V0+∫t0[(˜Su−Su)θu+Xuf(θu)]du+∫t0σXudWSu.

### 2.2 Hands-clean Condition

Let us recall that our task is to liquidate a large-size position by the time . Generally speaking, it is required that the hands-clean condition should be satisfied:

 XT=X0−∫T0θtdt=0. (3)

This technical condition, however, introduces some unexpected properties to the stochastic control problem. To tackle this problem, we temporary relax the hands-clean condition and allow an inmmediate final liquidation (if necessary) so that the number of shares owned at equals zero. That is, given the state variables at the instant before the end of trading , if , then we will have an immediate final liquidation so that . The liquidation proceeds after this final trade is

 CT=CT−+XT−(ST−−Co(XT−)),

where , for some constant , is the cost involved from liquidating the outstanding position . Thus, we have

 VT=VT−−ϕX2T−.

The gain/loss from liquidating the outstanding position, , is given by

 RT=∫[0,T)[(˜St−St)θt+Xtf(θt)]dt+∫[0,T)σXtdWSt−ϕX2T−. (4)

### 2.3 Performance Criterion

Under the normal circumstance, investors are risk averse and demand a higher return for a riskier investment. The mean-variance criterion is popular for taking both return and risk into account. However, the mean-variance criterion may induce a potential problem of time-inconsistency, namely, planned and implemented policies are different. As mentioned in Rudloff et al. , a major reason for developing dynamic models instead of static ones is the fact that one can incorporate the flexibility of dynamic decisions to improve the objective function. Time-inconsistent criteria are generally not favorable to introduce in the study, since the associated policies are sub-optimal.

To take both return and risk into account, instead of adopting the mean-variance criterion, we are most interested in the mean-quadratic optimal agency execution strategies, as they are proved to be time-consistent in [2, 6, 22]. In this section, we will introduce the quadratic variation and the corresponding objective function as follows.

Formally, the quadratic variation of the portfolio value on is defined to be

 [V,V]([0,T))=∫[0,T)σ2X2tdt. (5)

From the interpretation of Eq. (5), minimizing quadratic variation corresponds to minimizing volatility in the portfolio value process.

#### 2.3.2 Objective Function

Let be a constant corresponding to the risk aversion. Then the agent’s objective is to find the optimal control for

 maxθ(⋅)∈Θ0E[RT−γ[V,V]([0,T))]=maxθ(⋅)∈Θ0E[∫[0,T)[(˜St−St)θt+Xtf(θt)−γσ2X2t]dt−ϕX2T−]. (6)

## 3 The Benchmark Model for Optimal Liquidation (Model 1)

###### Assumption 1

The liquidation horizon is a finite-valued, pre-determined, and positive constant.

In this section, we present our benchmark model under Assumption 0 for the optimal liquidation problem. We first work on a subclass of deterministic controls222Controls that do not allow for inter-temporal updating. satisfying the hands-clean condition (3), and then move to the dynamic programming (DP) approach considering over the entire class of admissible controls. We prove that when the transaction cost involved by liquidating the outstanding position approaches to infinity, the optimal liquidation strategy obtained from DP approach converges to the deterministic one.

### 3.1 Deterministic Control

Let us first consider the case in which ranges only over the sub-class of deterministic strategies in satisfying the hands-clean condition

 ∫T0θtdt=Q.

That is, , and the agent’s objective is to find the optimal strategy for

 maxθ(⋅)∈Θdet0E∫[0,T)[(˜St−St)θt+Xtf(θt)−γσ2X2t]dt=maxθ(⋅)∈Θdet0E∫[0,T)[g(θt)θt+Xtf(θt)−γσ2X2t]dt. (7)

The cost function of the deterministic control problem (7) is

 H(Xt,θt,Λt,t)≡g(θt)θt+f(θt)Xt−γσ2X2t−Λtθt,

where is the Lagrange multiplier (also called the adjoint state). The differential equation for the deterministic system is:

 dXtdt=−θtwithX0=Q.

We assume the Hamiltonian has continuous first-order derivatives in state, adjoint state, and control variables, namely, . Then the necessary conditions (also called Hamilton’s equation) for having an interior optimum of the Hamiltonian at , are given by

 (8)

It follows from the critical conditions in Eq. (8) and Assumption 0 that

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩¨Xdet,∗t=γσ2νXdet,∗t,Xdet,∗0=Q,Xdet,∗T=0. (9)

An explicit solution, which is unique according to Lasota and Opial , is given by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Xdet,∗t=Qsinh(√γσ2ν(T−t))sinh(√γσ2νT),\parθdet,∗t=Q√γσ2νcosh(√γσ2ν(T−t))sinh(√γσ2νT). (10)

There is a very interesting phenomenon in the deterministic control problem: the solution (10) has nothing to do with the permanent price impact . If a position of size units with initial market price is fully liquidated by time , i.e. , the expected value of the resulting cash becomes

 E[∫T0˜Stθtdt]=E[∫T0Stθtdt−ν∫T0θ2tdt]=Q⋅s+E[∫T0XtdSt−ν∫T0θ2tdt]=Q⋅s−E[ν∫T0θ2tdt](temporary impact cost)−12ηQ2.(permanent impact cost)

Clearly, the permanent impact cost is independent of the time taken or strategy used to execute the liquidation.

### 3.2 Dynamic Programming Approach

Obviously, if we are allowed to update dynamically, namely, replacing by the entire class of admissible strategies , then one will be able to further improve his/her performance. In this section, we consider a stochastic approach. We employ the DP method to solve the stochastic control problem (6). This approach yields a Hamilton-Jacobi-Bellman (HJB) equation. When this HJB equation can be solved by an explicit smooth solution, the verification theorem then validates the optimality of the candidate solution to the HJB equation. For more details about the verification theorem, we refer interested readers to Pham  (Chapter 3), Øksendal  (Chapter 11), and Øksendal and Sulem  (Chapter 3).

Let be the optimal value function beginning at a time with initial value , namely333It is worth noting that the value function does not depend explicitly on the stock price .,

 U(t,q)=maxθ(⋅)∈ΘtE[∫[t,T)[−νθ2u−ηXuθu−γσ2X2u]du−ϕX2T−∣∣Ft]. (11)

Temporarily assuming that 444 is the space of functions which is continuously differentiable in , and twice continuously differentiable in . From the DP principle, must satisfy the following HJB equation:

 ⎧⎪⎨⎪⎩∂tU−γσ2q2−minθt∈Θt{νθ2t+(ηq+∂qU)⋅θt}=0U(T−,q)=−ϕq2. (12)

We remark that the optimization problem included in Eq. (12) is a constrained optimization problem with constraints: (a1) ; and (a2) . Generally speaking, there is no straightforward method to solve this kind of problems. One simple way to handle this problem is to consider the corresponding unconstrained optimization problem:

 ⎧⎪⎨⎪⎩∂tU−γσ2q2−minθt∈ˆΘt{νθ2t+(ηq+∂qU)⋅θt}=0U(T−,q)=−ϕq2, (13)

and then verify that the obtained result indeed satisfies all the constraints. From the HJB equation, Eq. (13), the optimal trading strategy without constraints is given by

 θϕ,∗t=−12ν(∂qU+ηq).

Thus the value function solves the following Ordinary Differential Equation (ODE):

 ⎧⎪⎨⎪⎩∂tU−γσ2q2+14ν(∂qU+ηq)2=0U(T−,q)=−ϕq2. (14)
###### Theorem 1

There is at most one solution to Eq. (14).

Proof: Let and be two solutions to Eq. (14). Define . Then the new function satisfies the following Partial Differential Equation (PDE):

 ⎧⎪⎨⎪⎩∂t˜f+14ν[∂q(f1+f2)+2ηq]∂q˜f=0˜f(T−,q)=0.

Since the evolution equation for is linear and first-order, one can solve the above problem explicitly by the method of characteristics, and find that is the unique solution to this problem. As a result, .

To solve Eq. (14), we consider an ansatz that is quadratic in the variable :

 U(t,q)=a(t)+b(t)q+c(t)q2.

According to Theorem 1, if the above ansatz is a solution of Eq. (14), then it must be the unique solution. Under this setting, the optimal liquidating strategy takes the following form:

 θϕ,∗t=−12ν{b(t)+[2c(t)+η]q}.

A direct substitution yields that the coefficients , and must satisfy the following ODEs:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩˙c(t)=γσ2−14ν[2c+η]2˙b(t)=−12νb(t)[2c+η]˙a(t)=−14νb2 (15)

with terminal conditions: , and . Since System (15) is partially decoupled, we can find the exact solution via direct integrations. As a result, they are given by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩c(t)=12ξ[ζe−4γξσ2(T−t)−1ζe−4γξσ2(T−t)+1]−η2b(t)=0a(t)=0 (16)

where the constants and are given by

 ζ=1−ξ(2ϕ−η)1+ξ(2ϕ−η)andξ=12σ√γν.

It is worth noting that

 ˙Xϕ,∗t=−θϕ,∗t=12ν[2c(t)+η]Xϕ,∗t, (17)

and that . Therefore,

 Xϕ,∗t=Q⋅exp(12ν∫t0[2c(u)+η]du).

As to the results obtained in this section, we have the following proposition.

###### Proposition 1

It is assumed that model parameters satisfy the condition:

 2ϕ>η+2σ√γν. (18)

That is, market liquidity risk dominates the potential arbitrage opportunity introduced by permanent impact and potential position risk involved by price fluctuations. Then, is a strictly decreasing function in and for any . Furthermore, we have that

(b1)

, for any time ; and that

(b2)

.

The obtained optimal trading strategy (17) is also the optimal trading strategy for the constrained problem.

Proof: Notice that the graph of the function depends on the coefficient , with . Under Assumption (18), , and hence . Therefore,

 ∂c(t)∂t<0,

i.e., is a strictly decreasing function in , and always holds for any . Thus, we conclude that

 θϕ,∗t=−Q12ν[2c(t)+η]e12ν∫t0[2c(u)+η]du≥0,

for any time , and that

 ∫[0,T)θϕ,∗tdt=Q[1−e12ν∫t0[2c(u)+η]du]≤Q.

Let denote the value function of the optimization problem (11) with time horizon , then for any , we have

 UT1(t,q)>UT2(t,q), (19)

provided that the condition (18) holds. This is consistent with the fact that an investor’s ability to bear risk relates to his/her time horizon for investment555The ability to bear risk is measured mainly in terms of objective factors, such as time horizon, expected income, and the level of wealth relative to liability..

### 3.3 Relation between Deterministic and Stochastic Control

###### Theorem 2

When the transaction fees involved by liquidating the outstanding position approaches to infinity, the limit of the optimal stochastic control process satisfies the hands-clean condition and it converges (point-wise) to the optimal deterministic control process . Meanwhile, the optimal trajectory converges (point-wise) to the one determined in the deterministic system . That is, as , we have

1. ;

2.  point-wise;

3.  point-wise.

Proof: We complete the proof by the following two steps:

Step 1 (Hands-clean condition)

We first prove that, as , . We note that

 Xϕ,∗t=Q⋅exp(∫t012ν[2c(u)+η]du).

A simple calculation yields

 e∫tu12ν[2c(r)+η]dr=ζe−4γξσ2(T−t)+1ζe−4γξσ2(T−u)+1e−2γξσ2(t−u). (20)

As , , and hence

 Xϕ,∗T−=Q(ζ+1)ζe−2γξσ2T+e2γξσ2T→0.
Step 2 (Convergence)

We then prove that as ,

•  point-wise; and

•  point-wise.

First, we have

 limϕ→∞Xϕ,∗t=limϕ→∞Qe∫t012ν[2c(u)+η]du=Qe2γξσ2(T−t)−e−2γξσ2(T−t)e2γξσ2T−e−2γξσ2T=Xdet,∗t.

For any time ,

 limϕ→∞[2c(t)+η]=1ξe−4γξσ2(T−t)+1e−4γξσ2(T−t)−1.

Thus, we have

 limϕ→∞θϕ,∗t=limϕ→∞−12ν[2c(t)+η]Xϕ,∗t=Q2νξe−2γξσ2(T−t)+e2γξσ2(T−t)e2γξσ2T−e−2γξσ2T=θdet,∗t.

In Figure 1, we illustrate how the transaction fees involved by liquidating the outstanding position , , affects the agent’s liquidating speed. We chose the following values of the model parameters: , , , , and .

Figure 1 illustrates that the speed of liquidation which is free of hands-clean condition is always slower than that under the constraint of hands-clean condition. As the transaction fees involved by liquidating the outstanding position increases (namely, as increases), the agent’s liquidating speed increases, indicating that the optimal stochastic control moves closer to the optimal deterministic control. The embedded subfigures in Figure 1 show, respectively, the differences between the deterministic and stochastic liquidating strategies and the corresponding trajectories with . Both of them are of magnitude .

## 4 Optimal Liquidation Strategy Subject to an Exogenous Trigger Event (Model 2)

In this section, we extend our results to models with an exogenous event, which does not depend on the information structure .

###### Assumption 2

The liquidation process will be suspended, if an exogenous trigger event occurs.

We model the occurrence time of a trigger event, denoted by , to be random, and the hazard rate is given by . The survival probability at time is

 P(t)=exp(−∫t0l(u)du). (21)

The liquidation horizon is then defined by

 τ=min{T,κ}, (22)

where the constant is a pre-determined time horizon. A direct computation yields the following proposition:

###### Proposition 2

For , the density function of is

 fτ(t)=l(t)exp(−∫t0l(u)du).

The probability that takes the value of is .

Denote by the event . At any time , i.e., the trigger event has not occurred prior to time , the agent’s objective is to find the optimal control for

 maxθ(⋅)∈ΘtE[∫τ−tΠ(θu,Xu)du−ϕX2τ−∣∣Ft∨Gt] (23)

where

 Π(θt,Xt)=g(θt)θt+f(θt)Xt−γσ2X2t,

and is the information structure available to the agent up to and including time . If the trigger event occurs at time , all market transactions will be suspended at that time. The agent will end up with an outstanding position .

It is worth noting that

 E[∫τ−tΠ(θu,Xu)du∣∣Ft∨Gt]=E[∫[t,T)I{u<τ}Π(θu,Xu)du∣∣Ft∨Gt]=E[∫[t,T)P(τ>u|Gt)Π(θu,Xu)du∣∣Ft], (24)

and that

 P(τ>u|Gt)=P(τ>u|τ>t)=e−∫utl(r)dr.

Here, the indicator function takes the value 1 when its argument is true and the value , otherwise. The last equality in Eq. (24) follows from the assumption that the trigger event is exogenous and does not depend on information structure .

Therefore, we have

 maxθ(⋅)∈ΘtE[∫τ−tΠ(θu,Xu)du−ϕX2τ−∣∣Ft∨Gt]=maxθ(⋅)∈ΘtE[∫[t,T)e−∫utl(r)dr[Π(θu,Xu)−ϕ⋅l(u)X2u]du−ϕe−∫[t,T)l(r)drX2T−∣∣Ft]. (25)

That is, the optimal liquidating problem with a random horizon defined in Eq. (22) is equivalent to an optimal liquidating problem with a finite horizon , a consumption process , a discount process , and a terminal condition .

### 4.1 Deterministic Control

Let us first consider the case in which ranges only over the subclass of deterministic strategies in satisfying the hands-clean condition (3)666The hands-clean condition only makes sense for the equivalent problem (25). While considering the original optimization problem (23), where the terminal time is a stopping time, the hands-clean condition is no longer valid., namely, . Thus, the agent’s objective, before the trigger event occurs, is to find the optimal control for

 maxθ(⋅)∈Θdet0E∫[0,T)e−∫t0l(u)du[Π(θt,Xt)−ϕ⋅l(t)X2t]dt.

The cost function of the deterministic control problem is

 H(Xt,θt,Λt,t)≡P(t)[Π(θt,Xt)−ϕ⋅l(t)X2t]−Λtθt,

where is the Lagrange multiplier, and is the survival probability defined in Eq. (21). The differential equation for the deterministic system dynamics is

 dXtdt=−θtandX0=Q.

We assume that the Hamiltonian has continuous first-order derivatives in the state, adjoint state, and the control variable, namely, . Then the necessary conditions for having an interior point optimum of the Hamiltonian at are given by

 (26)

It follows from Eq. (26) that

 ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩¨Xdet,∗t=l(t)˙Xdet,∗t+γσ2+(ϕ−η2)⋅l(t)νXdet,∗tXdet,∗0=QXdet,∗T=0. (27)

Regarding this linear second-order boundary value problem (BVP), its existence and uniqueness are standard. Interested readers can refer to, for example Hwang , for more details.

Consider the case when , which corresponds to the case of constant hazard rate, an explicit solution is given by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Xdet,∗t=Qeλ2tsinh(α(T−t))sinh(αT),θdet,∗t=−Qeλ2t[λ2sinh(α(T−t))−αcosh(α(T−t))]sinh(αT),

where

 α=√λ24+γσ2+(ϕ−η2)λν.

It is worth noting that (i) when , the model degenerates to Model 1; (ii) as , and , for all ; and . That is, as the final liquidation fee, per share, approaches infinity, the trader would immediately complete the transaction at the beginning of the trading horizon.

### 4.2 Dynamic Programming Approach

Let us consider the case of allowing dynamic updating, i.e., replacing by the entire class of admissible strategies . Let denote the optimal value function of Eq. (25) at any time prior to the occurrence of the trigger event. Under appropriate regularity assumptions, satisfies the following HJB equation:

 l(t)F=∂tF−[γσ2+ϕ⋅l(t)]q2−minθt∈Θt{νθ2t+(∂qF+ηq)⋅θt} (28)

subject to the terminal condition: . Here, is the given hazard rate. Similarly, we consider relaxing the constraints associated with the HJB equation and solve the unconstrained optimization problem. We then prove that the obtained optimal control does satisfy all the constraints. The associated optimal trading strategy is

 θϕ,∗t=−12ν(∂qF+ηq),

and hence the value function satisfies

 ⎧⎨⎩∂tF−[γσ2+ϕ⋅l(t)]q2+14ν(∂qF+ηq)2−l(t)F=0F(T−,q)=−ϕq2. (29)

Regarding Eq. (29), we have the following theorem for the uniqueness of classical solutions.

###### Theorem 3

There is at most one solution to Eq. (29).

Proof: Suppose and are two solutions to Eq. (29). Define . Then the new function satisfies the following problem:

 ⎧⎪⎨⎪⎩∂t˜f+14ν[∂q(f1+f2)+2ηq]∂q˜f−l(t)˜f=0˜f(T−,q)=0.

Since the evolution equation for is linear and first-order, one can solve the above problem explicitly by the method of characteristics, and find that is the unique solution to this problem. As a result, .

Similar to Section 3.2, we consider an ansatz that is quadratic in the variable :

 F(t,q)=˜a(t)+˜b(t)q+˜c(t)q2.

Substituting the ansatz into Eq. (29), we know that the coefficients , and must satisfy the following partially decoupled system:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩˙˜c(t)=l(t)˜c(t)+γσ2+ϕ⋅l(t)−14ν[2˜c(t)+η]2˙˜b(t)=l(t)˜b(t)−12ν˜b(t)[2˜c(t)+η]˙˜a(t)=l(t)˜a(t)−14ν˜b2(t) (30)

with terminal conditions: , and .

It is straightforward to verify that

 ˜b(t)≡0and˜a(t)≡0.

However, the equation satisfied by is a Riccati equation, which can be reduced to a second-order linear ODE:

 u′′−l(t)u′−γσ2+(ϕ−η2)l(t)νu=0, (31)

where is defined implicitly via . For this second-order linear ODE, its existence and uniqueness are standard. Even though we know the existence and uniqueness of the solution, it is still difficult to solve it in a closed-form for a general hazard rate . The above second-order linear ODE can be easily solved in two cases: (i) its coefficients are constant; or (ii) its coefficients adopt particular forms.

If closed-form solutions cannot be obtained, finite difference method can be applied to solving the BVP numerically. For more details, see, for example, Hwang .

###### Theorem 4

(Constant hazard rate). When the hazard rate is a constant, i.e., , the unknown function can be explicitly solved. It is given by

 ˜c(t)=12^ξ