# A Class of Optimal Portfolio Liquidation Problems with a Linear Decreasing Impact

Authors: Jiangming Ma; Zheng Yin; Hongjing Chen
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A problem of an optimal liquidation is investigated by using the Almgren-Chriss market impact model on the background that then agents liquidate assets completely. The impact of market is divided into three components: unaffected price process, permanent impact, and temporary impact. The key element is that the variable temporary market impact is analyzed. When the temporary market impact is decreasing linearly, the optimal problem is described by a Nash equilibrium in finite time horizon. The stochastic component of the price process is eliminated from the mean-variance. Mathematically, the Nash equilibrium is considered as the second-order linear differential equation with variable coefficients. We prove the existence and uniqueness of solutions for the differential equation with two boundaries and find the closed-form solutions in special situations. The numerical examples and properties of the solution are given. The corresponding finance phenomenon is interpreted.


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## Body

## 1. Introduction

The investor often considers optimal execution in the market of transactions within a fixed time period, optimizing some trade-off between risk and reward. Slow trading reduces transaction costs and fast liquidation decreases the volatility risk. The investors must choose an optimal strategy which could bring them the maximization of expected revenues and minimization of the variance of revenues. The mean-variance model is given by Markowitz [1, 2] and is a pretty important position in modern portfolio theory. The model gives an optimal investment portfolio method when investors want to get the highest return specifying their acceptable risk level. Samuelson [3], Hakansson [4], and Pliska [5] extend the model from the single-period case to the multiperiod discrete-time case. Many scholars continue to make further researches in the recent decades, and the continuous-time case is studied by Cox and Huang [6], Duffie and Richardson [7], Karatzas et al. [8], Schweizer [9], and so forth. The mean-variance optimal problem becomes how to maximize the expected return of time t and minimize the variance of the return of time t at the same time. In the model of Almgren and Chriss [10], the market impact is divided into a linear permanent impact and temporary impact. The temporary impact only affects the trade when the trade happens but the permanent impact affects all trading process. Almgren [11] uses the mean-variance and studies the optimal execution with a nonlinear temporary market impact.Since the model in Almgren and Chriss [10] is proposed, some scholars have investigated the optimal investment problem based on the model. Huberman and Stanzl [12] show that permanent market impact must be linear in the trade quantity and symmetry between buyers and sellers in order to rule out any price manipulation strategies. Bouchaud et al. [13] considered that market impact must be temporary and decay as a power law. However, Attari et al. [14] obtain a conclusion that the market price impact model only has temporary impact and show that the elastic market has the profitable liquidity provision. Schoeneborn and Schied [15] acquire that the illiquid market has multiplayers. They consider a case where the market has predatory trading. Carlin et al. [16] present an episodic illiquidity model when the cooperation breaks up in financial markets. Schied and Schoeneborn [17] find the dynamic optimal liquidation strategies in the illiquid market by the stochastic control approach. Carmona and Joseph Yang [18] change the permanent component of market price impact to solve the equilibrium of the predatory trading game and comparative statics from the HJB equation. Besides the continuous time, many scholars discuss the discrete time and the optimal strategies from the utility function maximization. Schied et al. [19] show that the utility function maximization is equal to the mean-variance maximization and the utility function maximization is simplified to maximize mean-variance. They prove that the results not only meet the continuous-time model but also satisfy the discrete-time model. Lorenz and Almgren [20] find the mean-variance optimization strategies by the means of a dynamic programming principle. At the same time, several scholars investigate the situation in which the market impact function has the decay factor. Gatheral [21] studies the properties of the market impact function with the exponential decay, power-law decay, and other situations. Except the above two decay functions, Schied [22] further discusses the capped linear decay and Gaussian decay and gives some useful properties.Schied and Zhang [23] study n risk-averse agents who trade a finance product and get the equilibrium strategies in both the finite and infinite time horizons. They give the market price process in which the coefficients of permanent and temporary price impact components are constant and prove the existence and uniqueness of Nash equilibria.Based on the models of Almgren and Chriss [10] and Schied and Zhang [23], we discuss a case where the coefficient of temporary market impact decreases linearly in a short time. This case is different from those presented in [10, 23]. There exists a unique Nash equilibrium in the market. The existence and uniqueness of the Nash equilibria are equivalent to the existence and uniqueness of solutions for the second differential equation with variable coefficients. The concrete form of optimal strategy in special cases is given. We discuss the various strategies in which the volatility and temporary impact of market change and get some interesting phenomena.The paper is organized as follows. In Section2, we state the optimal investment in the Almgren-Chriss framework and drive the equation of market impact price, the return function, and the optimal objective function. In Section 3, we give the system of the Nash equilibrium and prove the existence and the uniqueness. When the market has two agents, the optimal solution is given. In Section 4, the numerical examples and the corresponding financial interpretations are illustrated.


## 2. Statement of Background

### 2.1. Introduction of Models

In this paper, we use the model which is a continuous-time framework of Almgren and Chriss [10]. In this framework, every investor is active in a fixed time period [0,T]. An investor in this framework is required to hold x shares at the initial position and completely trade at the terminal time T. Investors know all market information. The information flow is considered as a filtration (Ft)t>0 on a given probability space (Ω,F,P). The investor’s trading strategy is defined by X=(X(t))t∈[0,T]. The trading strategy satisfies the following properties:(i)
X is absolutely continuous and bounded with derivative X˙(t) and satisfies the liquidation constraint X(T)=0.(ii)
X is adapted to the filtration (Ft)t>0. There is a progressively measurable process X˙(t) which satisfies ∫0T(X˙(t,ω))2dt<∞. X satisfies(1)Xt,ω=X0,ω+∫0tX˙s,ωds,t∈0,T.The class of all strategies is denoted byX(x,T). X(x,T) satisfies X(0)=x for given x∈R. The subclass of all strategies is denoted by Xdet(x,T). It does not depend on ω and is in X(x,T). S0 is defined by the unaffected price process. It describes the fluctuations of asset prices which are perceived by an investor in the symmetric information market during the time interval [0,T]. In the Almgren-Chriss model, it is considered as the price of the risky asset which follows the Bachelier model [24]. Further, an extra drift in addition to the current model is based on the Bachelier model [24]. Therefore, we have (2)S0t=S0+σWt+∫0tbsds,where S0 is the initial price, W(t) is the standard Brownian motion, the positive constant σ is the volatility of unaffected price process, and b(t) is an extra drift which is deterministic and continuous.The investor chooses a trading strategy that is described byXt which is the number of shares at time t. The market impact is the adverse feedback effect about the quoted price of a stock caused by the investor’s own trading and it is divided into three components: unaffected price process and permanent and temporary impact components. Firstly, the temporary impact shows the temporary imbalances between supply and demand caused by the trading of investors. Secondly, the permanent impact shows the change in the equilibrium due to trading which will last for the life of liquidation. Finally, in the linear Almgren-Chriss framework, the price process is assumed to be(3)SX≔S0t+γXt-X0+kX˙t,t∈0,T,where the parameters γ>0 and k>0 are constants and represent permanent and temporary price impact components, respectively. The permanent impact component is a linear function of the variable share number. The temporary impact component is the linear function of trading speed. Huberman and Stanzl [12] consider that the permanent market impact must be linear in the price process in order to rule out any price manipulation strategies.When the coefficient of temporary impact is a linearly decreasing function, the function satisfiesρ-λ0t, where λ0>0. We denote it by λ(t)=ρ-λ0t, where λ0>0. The price process becomes(4)SX≔S0t+γXt-X0+λtX˙t,t∈0,T.At each time t∈[0,T], the infinitesimal amounts of -X˙(t)dt shares are sold at price SX(t). So the revenue of each time is the product of the amount of shares and the price. Therefore, the total revenues are the sum of revenues at each time generated by the strategy X∈X(x,T). It is represented by(5)RX≔-∫0TX˙tSXtdt.So the optimal trade execution problem becomes the maximization of expected revenues. We only need to solve the maximization of the function of expected revenues:(6)maximizeERX.Bertsimas and Lo [25] proposed problem (6). Carmona and Joseph Yang [18] use (6) to deal with the problem of the maximization. But the volatility risk is not considered in the maximization of expected revenues. Therefore, the maximization of the mean-variance criterion is used in practice: (7)maximizeERX-α2var⁡RX,whereα is a nonnegative risk aversion parameter. In the frame of Almgren and Chriss, problem (7) is usually discussed on the subclass Xdet(x,T) of deterministic strategies (see Almgren and Chriss [10] and Almgren [11]). At the same time, some authors study the maximization of the optimal execution which is based on the utility function. The result that the maximization mean-variance of expected revenues over deterministic strategies is equivalent to the maximization of the expected utility of revenues is investigated by Schied et al. [19]:(8)maximizeEuαRXover all strategies inX(x,T), where(9)uα≔1α1-e-αx,ifα>0,x,ifα=0is a utility function with constant absolute risk aversions.


### 2.2. Nash Equilibrium

We suppose that all participators are active in the market. Theith investor uses the strategies Xi. The price process S0 will be impacted when the investor has the strategy Xi. Following variable time t, the ith investor will be influenced by others’ strategies. From (3), the market price impact model is made by the aggregated price impact; namely,(10)SX1,…,Xn≔S0t+γ∑j=1nXjt-Xj0+k∑j=1nX˙t,t∈0,T.When the coefficient of temporary impact is λ(t)=ρ-λ0t, the above equation becomes(11)SX1,…,Xn≔S0t+γ∑j=1nXjt-Xj0+λt∑j=1nX˙t,t∈0,T.Theith agent’s strategy is denoted by X-i≔X1,Xi-1,…,Xi+1,Xn. So, the revenue of ith investor is defined by(12)RXi∣X-i=-∫0TX˙itSX1,…,Xntdt.Therefore, when every investor gets the optimal strategy, the market problem is whether there exists a Nash equilibrium.Definition 1 (see [23]).
Suppose that every agent’s initial asset positions and risk aversion parameters arex1,…,xn∈R and α1,…,αn, respectively, where risk aversion parameter is nonnegative:(a)
If there exists a collectionX1∗,…,Xn∗ of deterministic strategies such that, for each i and X-i∗=X1∗,Xi-1∗,…,Xi+1∗,Xn∗, the strategy Xi∗∈Xdet(xi,T) maximizes the mean-variance functional(13)ERX∣X-i∗-αi2var⁡RX∣X-i∗over allX∈Xdet(xi,T), the collection X1∗,…,Xn∗ is defined by a Nash equilibrium for mean-variance optimization.(b)
If there exists a collectionX1∗,…,Xn∗ of admissible strategies such that, for each i, the strategy Xi∗∈X(xi,T) maximizes the expected utility (14)EuαiRX∣X-i∗over allX∈X(xi,T), the collection X1∗,…,Xn∗ is defined by a Nash equilibrium for CARA utility maximization.The equilibrium strategies of utility optimization are considered as adapted when the optimal strategies satisfy the mean-variance optimization. Next, we will prove the existence and uniqueness of Nash equilibrium which are based on the mean-variance maximization. In this paper, we discuss the situation where the coefficient of temporary impact decreases linearly in a fixed time.Definition 2 (see [26]).
LetD be the class of functions which is constructed by all continuously differentiable function y(x) in [a,b], where y(x) satisfies the boundary conditions y(a)=A and y(b)=B. A function F(x,y,y˙) which has three variables is twice continuously differentiable about every variable. In D, the functional J[y(x)] is defined by(15)Jyx=∫abFx,yx,y˙xdx.Lemma 3 (see [26]).
If there is a functiony=y0(x) in D, so that (15) has an extremum, the function y=y0(x) is the solution of the differential equation(16)F˙y-ddxF˙y˙=0.


## 2.1. Introduction of Models

In this paper, we use the model which is a continuous-time framework of Almgren and Chriss [10]. In this framework, every investor is active in a fixed time period [0,T]. An investor in this framework is required to hold x shares at the initial position and completely trade at the terminal time T. Investors know all market information. The information flow is considered as a filtration (Ft)t>0 on a given probability space (Ω,F,P). The investor’s trading strategy is defined by X=(X(t))t∈[0,T]. The trading strategy satisfies the following properties:(i)
X is absolutely continuous and bounded with derivative X˙(t) and satisfies the liquidation constraint X(T)=0.(ii)
X is adapted to the filtration (Ft)t>0. There is a progressively measurable process X˙(t) which satisfies ∫0T(X˙(t,ω))2dt<∞. X satisfies(1)Xt,ω=X0,ω+∫0tX˙s,ωds,t∈0,T.The class of all strategies is denoted byX(x,T). X(x,T) satisfies X(0)=x for given x∈R. The subclass of all strategies is denoted by Xdet(x,T). It does not depend on ω and is in X(x,T). S0 is defined by the unaffected price process. It describes the fluctuations of asset prices which are perceived by an investor in the symmetric information market during the time interval [0,T]. In the Almgren-Chriss model, it is considered as the price of the risky asset which follows the Bachelier model [24]. Further, an extra drift in addition to the current model is based on the Bachelier model [24]. Therefore, we have (2)S0t=S0+σWt+∫0tbsds,where S0 is the initial price, W(t) is the standard Brownian motion, the positive constant σ is the volatility of unaffected price process, and b(t) is an extra drift which is deterministic and continuous.The investor chooses a trading strategy that is described byXt which is the number of shares at time t. The market impact is the adverse feedback effect about the quoted price of a stock caused by the investor’s own trading and it is divided into three components: unaffected price process and permanent and temporary impact components. Firstly, the temporary impact shows the temporary imbalances between supply and demand caused by the trading of investors. Secondly, the permanent impact shows the change in the equilibrium due to trading which will last for the life of liquidation. Finally, in the linear Almgren-Chriss framework, the price process is assumed to be(3)SX≔S0t+γXt-X0+kX˙t,t∈0,T,where the parameters γ>0 and k>0 are constants and represent permanent and temporary price impact components, respectively. The permanent impact component is a linear function of the variable share number. The temporary impact component is the linear function of trading speed. Huberman and Stanzl [12] consider that the permanent market impact must be linear in the price process in order to rule out any price manipulation strategies.When the coefficient of temporary impact is a linearly decreasing function, the function satisfiesρ-λ0t, where λ0>0. We denote it by λ(t)=ρ-λ0t, where λ0>0. The price process becomes(4)SX≔S0t+γXt-X0+λtX˙t,t∈0,T.At each time t∈[0,T], the infinitesimal amounts of -X˙(t)dt shares are sold at price SX(t). So the revenue of each time is the product of the amount of shares and the price. Therefore, the total revenues are the sum of revenues at each time generated by the strategy X∈X(x,T). It is represented by(5)RX≔-∫0TX˙tSXtdt.So the optimal trade execution problem becomes the maximization of expected revenues. We only need to solve the maximization of the function of expected revenues:(6)maximizeERX.Bertsimas and Lo [25] proposed problem (6). Carmona and Joseph Yang [18] use (6) to deal with the problem of the maximization. But the volatility risk is not considered in the maximization of expected revenues. Therefore, the maximization of the mean-variance criterion is used in practice: (7)maximizeERX-α2var⁡RX,whereα is a nonnegative risk aversion parameter. In the frame of Almgren and Chriss, problem (7) is usually discussed on the subclass Xdet(x,T) of deterministic strategies (see Almgren and Chriss [10] and Almgren [11]). At the same time, some authors study the maximization of the optimal execution which is based on the utility function. The result that the maximization mean-variance of expected revenues over deterministic strategies is equivalent to the maximization of the expected utility of revenues is investigated by Schied et al. [19]:(8)maximizeEuαRXover all strategies inX(x,T), where(9)uα≔1α1-e-αx,ifα>0,x,ifα=0is a utility function with constant absolute risk aversions.


## 2.2. Nash Equilibrium

We suppose that all participators are active in the market. Theith investor uses the strategies Xi. The price process S0 will be impacted when the investor has the strategy Xi. Following variable time t, the ith investor will be influenced by others’ strategies. From (3), the market price impact model is made by the aggregated price impact; namely,(10)SX1,…,Xn≔S0t+γ∑j=1nXjt-Xj0+k∑j=1nX˙t,t∈0,T.When the coefficient of temporary impact is λ(t)=ρ-λ0t, the above equation becomes(11)SX1,…,Xn≔S0t+γ∑j=1nXjt-Xj0+λt∑j=1nX˙t,t∈0,T.Theith agent’s strategy is denoted by X-i≔X1,Xi-1,…,Xi+1,Xn. So, the revenue of ith investor is defined by(12)RXi∣X-i=-∫0TX˙itSX1,…,Xntdt.Therefore, when every investor gets the optimal strategy, the market problem is whether there exists a Nash equilibrium.Definition 1 (see [23]).
Suppose that every agent’s initial asset positions and risk aversion parameters arex1,…,xn∈R and α1,…,αn, respectively, where risk aversion parameter is nonnegative:(a)
If there exists a collectionX1∗,…,Xn∗ of deterministic strategies such that, for each i and X-i∗=X1∗,Xi-1∗,…,Xi+1∗,Xn∗, the strategy Xi∗∈Xdet(xi,T) maximizes the mean-variance functional(13)ERX∣X-i∗-αi2var⁡RX∣X-i∗over allX∈Xdet(xi,T), the collection X1∗,…,Xn∗ is defined by a Nash equilibrium for mean-variance optimization.(b)
If there exists a collectionX1∗,…,Xn∗ of admissible strategies such that, for each i, the strategy Xi∗∈X(xi,T) maximizes the expected utility (14)EuαiRX∣X-i∗over allX∈X(xi,T), the collection X1∗,…,Xn∗ is defined by a Nash equilibrium for CARA utility maximization.The equilibrium strategies of utility optimization are considered as adapted when the optimal strategies satisfy the mean-variance optimization. Next, we will prove the existence and uniqueness of Nash equilibrium which are based on the mean-variance maximization. In this paper, we discuss the situation where the coefficient of temporary impact decreases linearly in a fixed time.Definition 2 (see [26]).
LetD be the class of functions which is constructed by all continuously differentiable function y(x) in [a,b], where y(x) satisfies the boundary conditions y(a)=A and y(b)=B. A function F(x,y,y˙) which has three variables is twice continuously differentiable about every variable. In D, the functional J[y(x)] is defined by(15)Jyx=∫abFx,yx,y˙xdx.Lemma 3 (see [26]).
If there is a functiony=y0(x) in D, so that (15) has an extremum, the function y=y0(x) is the solution of the differential equation(16)F˙y-ddxF˙y˙=0.


## 3. Main Results

Theith agent’s admissible strategies and other strategies are denoted by Xi∈X(xi,T) and X-i≔X1,Xi-1,…,Xi+1,Xn for i=1,…,n, respectively. Let Y=Xi(t); the revenue function is defined by(17)RY∣X-i=-∫0TY˙tS0+σWt+∫0tbsds+γ∑j=1nXjt-Xj0+λt∑j=1nX˙jtdt=-∫0TY˙tS0dt-∫0TY˙tσWtdt-∫0TY˙t∫0tbsdsdt-γ∫0TY˙t∑j=1nXjt-Xj0dt-∫0TY˙tλt∑j=1nX˙jtdt=S0y+σ∫0TYtdWt+∫0TYtbtdt+γ∑j≠in∫0TYtX˙jtdt+12γy2-γy2-∑j≠in∫0TλtY˙tX˙jtdt-∫0TλtY˙2tdt.The mean and variance of revenue function are(18)ERY∣X-i=S0y-12γy2+∫0TYtbtdt+γ∑j≠in∫0TYtX˙jtdt-∑j≠in∫0TλtY˙tX˙jtdt-∫0TλtY˙2tdt,var⁡RY∣X-i=σ2∫0TY2dt,respectively. From Definition 1, the mean-variance function is(19)RY∣X-i-αi2varRY∣X-i=S0y-12γy2+∫0TYtbtdt+γ∑j≠in∫0TYtX˙jtdt-∑j≠in∫0TλtY˙tX˙jtdt-∫0TλtY˙2tdt-αi2σ2∫0TY2tdt=S0y-12γy2+∫0TYtbt+γ∑j≠inYtX˙jt-∑j≠inλtY˙tX˙jt-λtY˙2t-αiσ22Y2t.When all agents’ strategies are deterministic, one has(20)RY∣X-i-αi2varRY∣X-i=c+∫0TLit,Yt,Y˙t∣X-idt,where c=S0y-1/2γy2 and the Lagrangian Li is given by(21)Lit,Yt,Y˙t∣X-i=Ytbt+γ∑j≠inX˙jt-αiσ22Y2t-λt∑j≠inY˙tX˙jt+Y˙2t.Theorem 4.
Forn∈N, α1,…,αn⩾0, and xn,…,xn, there exists a unique Nash equilibrium X1∗,…,Xn∗ for mean-variance optimization. The equilibrium strategy is the unique solution of the following second-order system of differential equation:(22)αiσ2Yt-2λ˙tY˙t-2λtY¨t=bt+γ∑j≠inX˙jt+λ˙t∑j≠inX˙jt+λt∑j≠inX¨jtwith two-point boundary conditions(23)Xi0=xi,XiT=0for i=1,…,n.Before proving Theorem4, we give two lemmas.Lemma 5.
In the context of Theorem4, there exists at most one Nash equilibrium for mean-variance optimization.Proof.
We prove this lemma by contradiction and assume that there exist two different Nash equilibria strategiesX10,…,Xn0 and X11,…,Xn1. For β∈[0,1], let Xiβ≔βXi1+(1-β)Xi0 and define(24)fβ≔∑i=1n∫0TLit,Xiβt,X˙iβt∣X-i0+Lit,Xi1-βt,X˙i1-βt∣X-i.
Assume that maximization of the functionalY↦∫0TLi(t,Y(t),Y˙(t)∣X-i)dt is given by Xik for k=0,1. Therefore, we get f(β)⩽f(0) for β>0, which implies(25)ddββ=0+fβ⩽0.Then, it becomes(26)ddββ=0+fβ=∑i=1n∫0TγXi1t-Xi0t∑j=1nX˙j0t-X˙j1t-γXi1t-Xi0tX˙i0t-X˙i1t+αiσ2Xi1t-Xi0t2+λtX˙i1t-X˙i0t∑j=1nX˙j1t-X˙j0t+λtX˙i0t-X˙i1tdt.Since (27)∫0TXi1t-Xi0tX˙i0t-X˙i1tdt=12Xi1T-Xi0T2-12Xi10-Xi002=0,∫0TXi1t-Xi0tX˙j0t-X˙j1tdt=-∫0TXj1t-Xj0tX˙i0t-X˙i1t,∑i=1n∑j=1n∫0TXi1t-Xi0tX˙j0t-X˙j1tdt=0,we get(28)ddββ=0+fβ=∫0Tαiσ2∑i=1nXi1t-Xi0t2+λt∑i=1nX˙i0t-X˙i1t2+λt∑i=1nXi1t-Xi0t2dt>0,which contradicts (25).Lemma 6.
There exists at most one maximizer inXdet(y,T) of the function ∫0TLi(t,Y(t),Y˙(t)∣X-i)dt for i=1,…,n. If X1,…,Xn∈C2[0,T] exist, the equilibrium strategy is the unique solution of the two-point boundary value problem(29)αiσYt-2λ˙tY˙t-2λtY¨t=bt+γ∑j≠inX˙jt+λ˙t∑j≠inX˙jt+λt∑j≠inX¨jt,Y0=y,YT=0.Proof.
From the strict concavity of the LagrangianLi and the convexity of the set Xdet(y,T), there exists at most one maximizer in Xdet(y,T).
Next, the existence of a maximizer under the additional assumptionX1,…,Xn∈C2[0,T] is proven. Under the assumption and from Definition 2 and Lemma 3, the Lagrangian equation becomes(30)Lqit,Yt,Y˙t∣X-i=bt+γ∑j≠inX˙jt-αiσ2Yt,Lpit,Yt,Y˙t∣X-i=-λt∑j≠inX˙jt-2λtY˙t,dLpit,Yt,Y˙t∣X-idt=-λ˙t∑j≠inX˙jt-λt∑j≠inX¨jt-2λ˙tY˙t-2λtY¨t;then(31)αiσ2Yt-2λ˙tY˙t-2λtY¨t=bt+γ∑j≠inX˙jt+λ˙t∑j≠inX˙jt+λt∑j≠inX¨jt.Let the right-hand side of (31) be represented by f(t), λ(t)=ρ-λ0t, where λ(t)>0, ρ>0, and λ0>0. The homogeneous equation of (31) is(32)ρ-λ0tY¨t-λ0Y˙t-αiσ22Yt=0.From the results of [27], the solution of the above equation is(33)Yt=C1Y1t+C2Y2t,where(34)Y1t=∑k=0∞αiσ2ρ-λ0t/2λ02kk!Γk+1,Y2t=-∑k=0∞αiσ2ρ-λ0t/2λ02kk!Γk+1ln⁡αiσ2ρ-λ0t2λ02+12∑k=0∞αiσ2ρ-λ0t2λ02kψk+1k!,where Γ(k) is the gamma function. ψ(1)=-C; ψ(n)=-C+∑k=1n-1k-1. ψ(t)=ln⁡Γtt′ is the logarithmic derivative of the gamma function. C=0.5772… is the Euler constant. The Wronskian determinant is W(Y1,Y2)=-2/παiσ2(ρ-λ0t)/2λ2. So the solution of (31) can be represented by(35)Yt=C1Y1+C2Y2+Y2∫0tY1fsλsdsW-Y1∫0tY2fsλsdsW.C1 and C2 in (35) are determined by the boundary conditions Y(0)=y and Y(T)=0. Now, we prove that Y∗ is indeed the maximizer of our problem. From the concavity of (21) and the solution Y∗ of (31), we get(36)Lit,Y∗t,Y˙∗t∣X-i-Lit,Yt,Y˙t∣X-i⩾Lqit,Y∗t,Y˙∗t∣X-iY∗t-Yt+Lpit,Y∗t,Y˙∗t∣X-iY˙∗t-Y˙t=ddtLpit,Y∗t,Y˙∗t∣X-iY∗t-YtLpit,Y∗t,Y˙∗t∣X-iY˙∗t-Y˙t=ddtLpit,Y∗t,Y˙∗t∣X-iY∗t-Yt.Then, we obtain(37)∫0TLit,Y∗t,Y˙∗t∣X-idt-∫0TLit,Yt,Y˙t∣X-idt⩾∫0TddtLpit,Y∗t,Y˙∗t∣X-iY∗t-Ytdt=0,where Y∗(0)=Y(0) and Y∗(T)=Y(T).Proof of Theorem4.
From Lemma5, it is shown that there exists at most one Nash equilibrium. Lemma 6 tells us that every Nash equilibrium strategy must be the solution of the following system:(38)αiσYt-2λ˙tY˙t-2λtY¨t=bt+γ∑j≠inX˙j∗t+λ˙t∑j≠inX˙j∗t+λt∑j≠inX¨j∗t,Y0=y,YT=0.From Lemma 6, we get the solution of homogeneous equation for the above equation. Because(39)Δ=Y10Y20Y1TY2T≠0,the above nonhomogeneous differential equation has a unique solution. So there exists a unique Nash equilibrium X1∗,…,Xn∗ for mean-variance optimization.
The maximization of (20) and (21) will become the linear-quadratic differential game with state constraints. We use the method of Euler-Lagrange. The problem is considered to prove the existence and uniqueness of solution for the second-order linear differential equation with variable coefficients and two-point boundary conditions. The existence and uniqueness of solutions are equal to the existence and uniqueness of the Nash equilibrium. We will prove that the Nash equilibrium of mean-variance optimization is the CARA utility optimization.Corollary 7.
Forn∈N, α1,…,αn⩾0, and x1,…,xn, the Nash equilibrium of mean-variance optimization in Theorem 4 is also a Nash equilibrium for CARA utility maximization.Proof.
Suppose that the unique Nash equilibrium with mean-variance optimization in Theorem4 is X1∗,…,Xn∗. When Xi∗=X1∗,…,Xi-1∗,Xi+1∗,…,Xn∗, the agent i thinks that the price process(40)SX-i∗≔S0t+γ∑j=inXjt-Xj0+λt∑j=inX˙jtis unperturbed. So, it can be simplified by(41)SX-i∗≔S0+σWt+∫0tbitdt,where bi(t) is a deterministic and continuous function. From the study of Schied et al. [19], the unperturbed price process SX-i∗ has all exponential moments; that is, EeβSX-i∗<∞ for all β∈R. Under the above assumption and the given positive constant αi, we have(42)supX∈Xxi,T⁡EuαiRXi∣X-i=supX∈Xdetxi,T⁡EuαiRXi∣X-i.From Theorem 4, the optimal strategy X-i∗ must be deterministic and unique as long as it exists. We get that the strategy Xi∗∈Xdet(xi,T) satisfies(43)EuαiRXi∣X-i=1αi1-e-αiERXi∣X-i+αi/2var⁡RXi∣X-i.It means that the maximization of CARA utility is equivalent to maximization of the mean-variance function. That is, the problem of optimal CARA utility can be simplified by the mean-variance optimization. When αi=0, the result is also right.Let(F~t)t⩾0 be any subfiltration of (Ft)t⩾0. From the proof of Corollary 7, we get that the Nash equilibrium for mean-variance optimization in Theorem 4 is also a Nash equilibrium for CARA utility. The mean-variance optimization is equivalent to the maximization of CARA utility. At the same time, the optimal strategies are adapted to (F~t)t⩾0.We will simplify the model based on the fact that all agents have the same risk aversion.Corollary 8.
Under assumption of Theorem4, suppose that α1=⋯=αn=α⩾0. Then(44)∑t≔∑i-1nXi∗tis the unique solution of the following differential equation with two-point boundary value problem:(45)ασ2∑t+-n+1λ˙t+1-nγ∑˙t-n+1λt∑¨t=nbt,∑0=∑inxi,∑T=0.
For∑(t), each equilibrium strategy Xi∗ is equal to the unique solution of the following differential equation with two-point boundary value problem:(46)ασ2Yt-λ˙tY˙t+γY˙t-λtY¨t=bt+γ∑˙t+λ˙t∑˙t+λt∑¨t,Xi0=xi,XiT=0.Proof.
Let∑(t)≔∑j=1nXj(t); (31) can be written in the form(47)ασ2Yt-λ˙tY˙t+γY˙t-λtY¨t=bt+γ∑˙t+λ˙t∑˙t+λt∑¨t.Next, we discuss a special situation in whichb=0.Corollary 9.
Under assumption of Corollary8, additionally, suppose that b=0 and α>0; the equilibrium strategy ∑(t) satisfies the form (48)ασ2∑t+-n+1λ˙t+1-nγ∑˙t-n+1λt∑¨t=0.When (n-1)γ/(n+1)λ0 is not an integer, the solution of (48) is(49)∑t=2ασ2ρ-λ0tn+1λ02n-1γ/n+1λ0/2C1in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02+C2in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02cos⁡n-1γπn+1λ0--1n-1γ/n+1λ0in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02sin⁡n-1γπn+1λ0-1.
When(n-1)γ/(n+1)λ0 is an integer, the solution of (48) is(50)∑t=2ασ2ρ-λ0tn+1λ02n-1γ/n+1λ0/2C1in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02+C2in-1γ/n+1λ02π-1n-1γ/n+1λ0+1In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02ln⁡2ασ2ρ-λ0t/n+1λ022+12∑k=0n-1γ/n+1λ0-1n-1γ/n+1λ0-k-1!k!+12-1n-1γ/n+1λ0∑k=0∞2ασ2ρ-λ0t/n+1λ0222k+n-1γ/n+1λ0Ψk+1+Ψn-1γ/n+1λ0+k+1k!n-1γ/n+1λ0+k!,where Iν(x)=∑k=0∞(x/2)2k+ν/k!Γ(ν+k+1) and ψ(x) is the logarithmic derivative of the gamma function.Proof.
From the setting of Corollary8, (31) becomes(51)ασ2∑t+-n+1λ˙t+1-nγ∑˙t-n+1λt∑¨t=0.When λ(t)=ρ-λ0t, let ∑(t)=y; the above equation becomes(52)ρ-λ0ty¨+-λ0+n-1γλ0n+1y˙-ασ2λ0n+1y=0.
Citing the results in [27], we obtain the solutions of (52) in the following form. When (n-1)γ/(n+1)λ0 is not an integer, the solution of (52) is(53)y=2ασ2ρ-λ0tn+1λ02n-1γ/n+1λ0/2C1in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02+C2in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0t/n+1λ02cos⁡n-1γπ/n+1λ0--1n-1γ/n+1λ0in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0t/n+1λ02sinn-1γπ/n+1λ0.When (n-1)γ/(n+1)λ0 is an integer, the solution of (52) is(54)y=2ασ2ρ-λ0tn+1λ02n-1γ/n+1λ0/2C1in-1γ/n+1λ0In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02+C2in-1γ/n+1λ02π-1n-1γ/n+1λ0+1In-1γ/n+1λ02ασ2ρ-λ0tn+1λ02ln⁡2ασ2ρ-λ0t/n+1λ022+12∑k=0n-1γ/n+1λ0-1n-1γ/n+1λ0-k-1!k!+12-1n-1γ/n+1λ0∑k=0∞2ασ2ρ-λ0t/n+1λ0222k+n-1γ/n+1λ0Ψk+1+Ψn-1γ/n+1λ0+k+1k!n-1γ/n+1λ0+k!,
whereIν(x)=∑k=0∞(x/2)2k+ν/k!Γ(ν+k+1) and ψ(x) is the logarithmic derivative of the gamma function.Corollary 10.
Under assumption of Corollary9, let n=2. Then the form of ∑(t) satisfies(55)ρ-λ0t∑¨t+-λ0+γ3λ0​​∑˙t-ασ23λ0∑t=0.When γ/3λ0 is not an integer, the solution of (55) is (56)∑t=2ασ2ρ-λ0t3λ02γ/3λ0/2C1iγ/3λ0Iγ/3λ02ασ2ρ-λ0t3λ02+C2iγ/2λ0Iγ/3λ02ασ2ρ-λ0t3λ02cos⁡γπ3λ0--1γ/3λ0iγ/3λ0Iγ/3λ02ασ2ρ-λ0t3λ02sin⁡γπ3λ0-1.
Whenγ/3λ0 is an integer, the solution of (55) is(57)∑t=2ασ2ρ-λ0t3λ02γ/3λ0/2C1iγ/3λ0Iγ/3λ02ασ2ρ-λ0t3λ02+C2iγ/3λ02π-1γ/3λ0+1Iγ/3λ02ασ2ρ-λ0t3λ02ln⁡2ασ2ρ-λ0t/3λ022+12∑k=0γ/3λ0-1γ/3λ0-k-1!k!+12-1γ/3λ0∑k=0∞2ασ2ρ-λ0t/3λ0222k+γ/3λ0Ψk+1+Ψγ/3λ0+k+1k!γ/3λ0+k!,where Iν(x)=∑k=0∞(x/2)2k+ν/Γ(k+1)Γ(ν+k+1) and ψ(x) is the logarithmic derivative of the gamma function.Proof.
Whenn=2, from Corollary 9, (52) becomes(58)ρ-λ0ty¨+-λ0+γ3λ0y˙-ασ23λ0y=0.From the results of [27], when γ/3λ0 is not an integer, the solution of (58) is(59)y=2ασ2ρ-λ0t3λ02γ/3λ0/2C1iγ/3λ0Iγ/3λ02ασ2ρ-λ0t3λ02+C2iγ/2λ0Iγ/3λ02ασ2ρ-λ0t3λ02cos⁡γπ3λ0--1γ/3λ0iγ/3λ0Iγ/3λ02ασ2ρ-λ0t3λ02sin⁡γπ3λ0-1.When γ/3λ0 is an integer, the solution of (58) is(60)y=2ασ2ρ-λ0t3λ02γ/3λ0/2C1iγ/3λ0Iγ/3λ02ασ2ρ-λ0t3λ02+C2iγ/3λ02π-1γ/3λ0+1Iγ/3λ02ασ2ρ-λ0t3λ02ln⁡2ασ2ρ-λ0t/3λ022+12∑k=0γ/3λ0-1γ/3λ0-k-1!k!+12-1γ/3λ0∑k=0∞2ασ2ρ-λ0t/3λ0222k+γ/3λ0Ψk+1+Ψγ/3λ0+k+1k!γ/3λ0+k!,where Iν(x)=∑k=0∞(x/2)2k+ν/Γ(k+1)Γ(ν+k+1) and ψ(x) is the logarithmic derivative of the gamma function.Corollary 11.
Under assumption of Corollaries8 and 9, the equilibrium strategy Δ(t)=X1(t)-X2(t) satisfies the form(61)ρ-λ0tΔ¨t-λ0+γΔ˙t-ασ2Δt=0.When -γ/λ0 is not an integer, one has(62)Δt=2ασ2ρ-λ0tλ02-γ/2λ0C1i-γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1+C2i-γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1cos⁡-γλ0π-iγ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γγ/λ0+k+1sin-γλ0π-1.When -γ/λ0 is an integer, one gets(63)Δt=2ασ2ρ-λ0tλ02-γ/2λ0C1i-γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1+C2i-γ/λ0-1γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1ln⁡ασ2ρ-λ0tλ0+12∑k=0γ/λ0-1-1kασ2ρ-λ0tλ02k-γ/λ0γ/λ0-k-1!k!+12-1γ/λ0∑k=0∞ασ2ρ-λ0tλ02k+γ/λ0ψγ/λ0+k+1+ψk+1k!γ/λ0+k!.Proof.
SinceΔ(t)=X1(t)-X2(t), (46) becomes(64)ασ2Δt-λ˙tΔ˙t+γΔ˙t-λtΔ¨t=0.Let y=Δ(t) and λ(t)=ρ-λ0t; (64) becomes(65)ρ-λ0ty¨-λ0+γy˙-ασ2y=0.From the results of [27], we get the solution of (65) in the following form. When -γ/λ0 is not an integer, one has(66)y=2ασ2ρ-λ0tλ02-γ/2λ0C1i-γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1+C2i-γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1cos⁡-γλ0π-iγ/λ∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γγ/λ0+k+1sin⁡-γλ0π-1.When -γ/λ0 is an integer, we have (67)y=2ασ2ρ-λ0tλ02-γ/2λ0C1i-γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1+C2i-γ/λ0-1γ/λ0∑k=0∞ασ2ρ-λ0t/λ0Γk+1Γ-γ/λ0+k+1ln⁡ασ2ρ-λ0tλ0+12∑k=0γ/λ0-1-1kασ2ρ-λ0tλ02k-γ/λ0γ/λ0-k-1!k!+12-1γ/λ0∑k=0∞ασ2ρ-λ0tλ02k+γ/λ0ψγ/λ0+k+1+ψk+1k!γ/λ0+k!,where Δ(0)=x1-x2 and Δ(T)=0.Corollary 12.
Under assumption of Corollaries10 and 11, the equilibrium strategies X1∗(t) and X2∗(t) are X1∗(t)=∑∗(t)+Δ∗(t)/2 and X2∗(t)=Δ∗(t)-∑∗(t)/2, respectively.Proof.
From Corollaries10 and 11, ∑∗(t)=X1∗(t)+X2∗(t) and Δ(t)=X1∗(t)-X2∗(t); we have X1∗(t)=∑∗(t)+Δ∗(t)/2 and X2∗(t)=Δ∗(t)-∑∗(t)/2.


## 4. Conclusion

The existence and uniqueness of Nash equilibrium are proven in the previous sections. We get the concrete form of an agent’s optimal strategy in a special case. Next, we will discuss the variation of strategy when market volatility and the coefficient of temporary market impact change.From Figures1–3, we get certain information. When σ increases from 0.5 to 1.5, agent 1 realizes that the market volatility risk is increasing. The agent’s revenue will have a bigger fluctuation following the market volatility. Therefore, the agent will speed up liquidation in the former half of the time [0,T] so that the agent can get more revenue. At the same time, he will decrease liquidation so that the cost decreases. From Figure 3, it shows that the trading speed increases following the increasing of coefficient λ0. When the coefficient λ0 increases, the influence of the temporary impact will decrease. The trading cost will decrease so that the agent gets the maximization of revenue. When the coefficient of temporary market impact decreases with time, the agent realizes that the influence of temporary impact is decreasing and slows down the trading speed.Figure 1
α
=
1; σ=1; ρ=1; λ0=0.2; γ=0.3; X1(0)=3 (dashed); X2(0)=2 (solid).Figure 2
α
=
1; σ1=1.5 (dashed); σ2=1 (solid); σ3=0.5 (thick); ρ=1; λ0=0.2; γ=0.3; X1(0)=3; X2(0)=2.Figure 3
α
=
1; σ3=0.5; ρ=1; λ01=0.2 (thick); λ02=0.34 (solid); λ03=0.44 (dashed), γ=0.3; X1(0)=3; X2(0)=2.



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