Optimal Execution considering Trading Signal and Execution Risk Simultaneously

In this paper, we study the optimal execution problem by considering the trading signal and the transaction risk simultaneously. We propose an optimal execution problem by taking into account the trading signal and the execution risk with the associated decay kernel function and the transient price impact function being of generalized forms. In particular, we solve the stochastic optimal control problems under the assumptions that the decay kernel function is the Dirac function and the transient price function is a linear function. We give the optimal executing strategies in state-feedback form and the Hamilton‐Jacobi‐Bellman equations that the corresponding value functions satisfy in the cases of a constant execution risk and a linear execution risk. We also demonstrate that our results can recover previous results when the process of the trading signal degenerates.


Introduction
It is known that when traders execute a large order in a short time, it will cause severe effect on the stock price in the stock market. is effect is called the price impact or the market impact in academia. e price impact is often adverse for traders because they liquidate or build a large position in a short time with worse average price compared to the initial price. Hence, traders or financial institutions often bear an extra cost due to the price impact except for some fixed cost charged by exchange. Consequently, the topic on how to reduce the cost caused by the price impact has received much attention. e problem about reducing the cost caused by the price impact is always expressed as an optimal execution problem in the literature, i.e., looking for an optimal executing strategy to minimize the expected cost due to the price impact. Plenty of works have been done on this topic. Bertsimas and Lo [1] studied a discrete time model of price impact with linear impact function and derived dynamic optimal trading strategies to minimize the expected cost. Almgren and Chriss [2] considered the continuous time case of Bertsimas and Lo [1]. ey choose the trade-off between the expectation and the variance of the impact cost as the optimization objective and then gave the explicit solution by the variation method. In addition, they proposed the concept of L-VaR. Almgren [3] further considered nonlinear impact functions and added risk terms in the temporary impact process on the basis of Almgren and Chriss [2]. Obizhaeva and Wang [4] (an early work published later) studied the optimal executing strategy given the dynamic structure of the demand and supply of the equity. Alfonsi et al. [5] extended the model of Obizhaeva and Wang [4] by allowing for a time-dependent resilience rate with more generalized equilibrium dynamics for bid and ask price. Alfonsi et al. [6] considered more general shape of the LOB on the basis of Obizhaeva and Wang [4] and gave the explicit form of optimal executing strategies. ey also illustrated the robustness of the optimal strategies with respect to the shape function and resilience type. Gatheral and Schied [7] assumed the asset price followed a geometric Brownian motion and gave the explicit optimal executing strategy with risk aversion. Almgren [8] assumed the market liquidity and volatility were stochastic and time varying, then proposed the HJB equation of the optimal execution problem, and tried to solve it numerically.
Gatheral et al. [9] studied the optimal execution problem in the frame of transient price impact model. Under the assumption of linear impact function, they characterized the optimal executing strategy as the solution of a generalized Fredholm integral equation of the first kind. ey also studied the existence problem. Cheridito and Sepin [10] studied the discrete time case of the price impact model with stochastic volatility and stochastic market liquidity. Schoneborn [11] discussed three approaches that remedied the flaw of the optimal executing strategy under the meanvariance framework that big order and small order have the same executing pattern. Cartea and Jaimungal [12] studied the optimal execution problem by taking into account the order flow of all other agents and gave the explicit solution with linear impact function. Cheng et al. [13] considered the execution risk under the framework of Almgren and Chriss and solved the optimal execution problem with different risk aversions. Jin [14] studied the optimal execution problem with an optimization objective of loss probability and talked about the liquidity adjusted VaR. Curato et al. [15] studied the transient impact model with nonlinear impact function and solved the optimal executing problem numerically. e previous works on the optimal execution problem are mainly based on the framework of Almgren and Chriss [2] or Gatheral et al. [9]. In the framework of Almgren and Chriss [2], the price impact has two kinds of definitions, i.e., the permanent impact and the temporary impact. Orders to be executed are thought to contain some fundamental information about the stock. is information is absorbed into the stock price in the trading and leads to a permanent impact on the intrinsic price of the stock. is is described by a permanent impact function of trading rate in the intrinsic price process of the stock. In addition, the price that we can observe in the market is affected by the trading at the moment and is described by the intrinsic price plus a temporary impact function of trading rate. In the framework of Gatheral et al. [9], the intrinsic stock price is set to be a martingale, i.e., the orders executed have no impact on the intrinsic price. Besides, the observed price is not only affected by instant trading but also affected by historical trading through a decay kernel function and a transient impact function of trading rate. In this paper, we combine the frameworks of Almgren and Chriss [2] and Gatheral et al. [9]. More specifically, we suppose trading gives rise to a permanent impact on the intrinsic price of the stock, and the observed price is affected by historical trading through a decay kernel function and a price impact function of trading rate.
In the view of some practitioners in trading, the market may not always be so efficient. ere exsit signals in the stock market with which the traders can predict the future return of the stock to some extent. Indeed, some hedge funds make profit with trading signals founded by technical analysis or other ways. Cartea and Jaimungal [12] studied the optimal executing strategy by incorporating order flow. In this work, the trading rate of all other traders can be treated as a signal of the intrinsic stock return. Motivated by these, we propose a trading signal term in the intrinsic price process. In addition, the experiences from practitioners indicate that a trading signal usually has the properties of stationarity and mean reversion. erefore, we assume that the trading signal follows an Ornstein-Uhlenbeck process. Besides, Cheng et al. [13] suggested that the order delivered by traders may not be filled fully, i.e., the traders can face the execution risk; therefore, we investigate the optimal execution problem by taking into account the trading signal and the execution risk simultaneously. More specifically, we propose an optimal execution problem with a generalized kernel function and a generalized transient impact function. To solve this optimal execution problem, we set the kernel function to be the Dirac function, which is compatible with the framework suggested in Almgren and Chriss [2], and the transient impact function to be a linear function. In this setting, we give analytical solutions to the optimal execution problems with a constant execution risk and a linear execution risk, respectively. Moreover, we prove that our results can recover the results in Cheng et al. [13] if the trading signal process degenerates, i.e., the mean reversion speed of the trading signal degenerates to 0. Our results can provide some insights for the hedge funds that possess some trading signals to design their trading scheme. e rest of this paper is organized as follows. In Section 2, we describe our model. In Section 3, we propose our optimal execution problem. In Section 4, we solve the optimal execution problem with a constant execution risk and a linear execution risk, respectively, and discuss the solutions. In Section 5, we conclude this paper and point out some directions for further work.

Model Settings
Suppose that we have a scheme of liquidating X shares of stock in time interval [t 0 , T], and t 0 � 0. At time t, the amount of stock remaining to be liquidated is denoted as x t , and thus x t 0 � X.
We suppose the intrinsic stock price, which cannot be observed directly in the market, follows the stochastic process below: where L t is trading signal and is defined in (4), B t is a standard Brownian motion, and θ, ρ, and σ 1 are constant parameters with ρ > 0, σ 1 > 0. In the setting of (1), we suppose that trading has a permanent impact on the intrinsic stock price, and the permanent impact function is set to be a linear function. is setting follows the existing works based on the framework of Almgren and Chriss [2], such as Gatheral and Schied [7], Cartea and Jaimungal [12], Cheng et al. [13], Jin [14], and so on. Besides, we further suppose that the intrinsic price of the stock is also affected by other factors, such as trading signals and the trading rate of other traders.
In addition, we suppose the observed price of the stock can be expressed as

Mathematical Problems in Engineering
where G(·) is the decay kernel function, g(·) is the transient impact function, and v s is the trading rate.
In the setting of (2), we follow the framework of Gatheral et al. [9]. e form of (2) indicates that the trading before time t has a decayed effect on the observed price at time t. We note that when the decay kernel function is set to be the Dirac function δ 0 (·), this model of observed price degenerates to the framework of Almgren and Chriss [2]. Indeed, we have which is just the form in Almgren and Chriss [2]. e experiences from practitioners indicate that a trading signal usually has the properties of stationarity and mean reversion. Motivated by this, we propose a signal process L t in (1) and suppose L t follows an Ornstein-Uhlenbeck process as below: where c is the speed of mean reversion such that c > 0 and W t is a standard Brownian motion.
In practice, an order may not be executed fully due to the shortage of liquidity in the market or some other technical reasons. So, traders may face an execution risk. Some previous works have talked about this topic, and we follow the setting in Cheng et al. [13].
We suppose the executing process follows the stochastic process below: where x t is the amount of stock that remains to be liquidated, v t is the trading rate, m(·) is a function that affects the diffusion of this process, and Z t is a standard Brownian motion.
For mathematical tractability, we suppose the standard Brownian motions B t , W t , and Z t are independent.

Optimal Execution Problem
With the setting in previous parts, we propose our optimal execution problem in this section.
Following the setting in Cheng et al. [13], we define our PnL as the difference between the realized value by trading and the initial intrinsic value of our position. Indeed, a selling order always pushes the stock price down, so we obtain lower price than the initial price, and thus the PnL is always negative. Naturally, we wish the PnL to be larger. Specifically, we define So, at time T, the PnL is Note that at time T, we may have x T > 0 due to the execution risk. In this case, we need to liquidate the remaining shares immediately, so we put a punishment on the remaining shares. We denote the punishment as λ(x T ).
In the setting of Cheng et al. [13], the punishment function is quadratic, i.e., λ(x) � − αx 2 with α > 0, which is compatible with the results of Almgren and Chriss [2]. In our work, we also follow this treatment. Hence, we define the adjusted PnL as With all the settings above, we can get the specific form of the adjusted PnL, which is illustrated in Proposition 1.

Proposition 1.
With the settings of (1), (2), (4), (5), (7), and (8) and the assumption that the Brownian motions B t , W t , and Z t are independent, the adjusted PnL defined in (8) has the following expression: Proof. Applying Itô's formula and with (1) and (5), we have By integrating, we get On the other hand, with (2), we have Mathematical Problems in Engineering 3 Combining (7), (8), (11), and (12), we get Applying Itô's formula to (5), we have By integrating, we get Substituting (1), (5), and (15) into (13), we get expression (9). □ e expression in (9) indicates that the randomness of the adjusted PnL comes from three stochastic sources, i.e., Brownian motions B t , W t , and Z t . In addition, it is worth noting that the last two terms of (9) are Itô integrations and thus are martingales. Hence, the expectation of the adjusted PnL can be expressed as Note that E t (·) represents E(·|x t � x, L t � l) in here and the other parts of the following context.
With the specific form of the adjusted PnL, it is natural for us to propose an optimal execution problem. More specifically, we look for an optimal trading rate process v t to maximize the expected utility of the adjusted PnL with a utility function. Here we choose the identity utility function and formulate our optimal execution problem as follows: where E t 0 (PnL _ adj) satisfies (16). So far, we have proposed our optimal execution problem described in (17). In this optimal execution problem, we assume the decay kernel function G(·) and the transient price impact function g(·) are of generalized forms. e topic about the form of these two functions has been widely discussed in the literature, such as Gatheral [16]. We note that the choices of this two functions may lead to different types of optimal execution problems, thus requiring different techniques to solve the corresponding optimal execution problems. In the next section, we choose some special decay kernel functions and transient price impact functions so that the optimal execution problem becomes a standard stochastic optimal control problem, and we solve it under different cases of execution risk.

Optimal Executing Strategy
In this section, we appropriately choose the decay kernel function and the transient price impact function to solve the optimal execution problem (17).
To make the problem more tractable, we choose the Dirac function as the kernel decay function. As mentioned in Section 2, this case conforms to the setting of Almgren and Chriss [2]. In addition, we set price transient impact function g(·) as a linear function; more specifically, where η is a constant and η > 0. Besides, we follow the treatment about the punishment function λ(·) in Cheng et al. [13], i.e., λ(x) � − αx 2 with α > 0. Hence, the expectation of the adjusted PnL in (16) becomes Note that under the conditions above, the optimal execution problem (17) has become a standard stochastic optimal control problem. We define the value function as

Substituting (21) into (20), we finally get
So far, the only undefined term in our optimal execution problem is the function m(·). Note that the form of m(·) determines the execution risk, and we follow the setting of Cheng et al. [13]. More specifically, we solve the optimal execution problem (17) with a constant execution risk and a linear execution risk, respectively.
Applying Itô's formula, we have With equation (5), we have Taking similar procedures, we have Mathematical Problems in Engineering 5 For convenience, we define J(t, x, l) as and hence (32) Note that F 1 (T) � 0, F 2 (T) � 0, G 1 (T) � 0, G 2 (T) � 0, and H(T) � 0; then, we have With the conclusions above, we have To use the method of completing square, we compare the first integration term above with where g 1 (s), g 2 (s) are functions of s. Matching coefficients, we can get the following ODE system: with terminal conditions F 1 (T) � 0, F 2 (T) � 0, G 1 (T) � 0, G 2 (T) � 0, and H(T) � 0.
To solve F 1 (s), we eliminate g 1 (s) in the first row of (36) and get the ODE below: Solving this special Riccati equation with the terminal condition, we have where β � (2η/2α − θ). Consequently, we have To solve H(s), we eliminate g 2 (s) in the third row of (36) and get the ODE below: Substituting (39) into this linear ODE, we solve it to have Consequently, we have 6 Mathematical Problems in Engineering Substituting (42) into the ODE in the second row of (36), we solve the ODE to get In addition, it is obvious that F 2 (t) ≡ 0 and G 2 (t) ≡ 0 from (36).
With the results above, we have e inequality above indicates that v s � g 1 (s)x s + g 2 (s)L s is the unique solution to maximize J(t, x, l) and thus the unique solution to the optimal execution problem (17). As a consequence, the value function V(t, x, l) has the following expression: Taking the partial derivatives of V(t, x, l), we have en, it is straightforward to verify that V(t, x, l) satisfies the HJB equation below: □ is theorem indicates that the optimal executing strategy v t is a linear combination of the remaining position x t and the trading signal L t and thus a dynamic executing strategy. Note that when ρ � 0 and letting α ⟶ + ∞, we have v * t � (x t /T − t), which is an adaptive VWAP strategy. Furthermore, we denote the weight of L t in (20) as w(c), i.e., and provide the following results of Corollary 1.

Corollary 1.
With the assumptions in eorem 1, the weight w(c) of the trading signal L t in the expression of the optimal executing strategy (25) is monotonic increasing with respect to the mean reversion speed c of the trading signal for c ∈ (0, +∞). In addition, when limiting c to 0, the limitation of w(c) exists and can be expressed as Proof. We prove the monotonicity first. For convenience, we define w(c) as and hence w(c) � − (ρ/2η(T − t + β))w(c).
Taking the derivative of w, we have Mathematical Problems in Engineering (51) Further, we define a(c) as and hence w ′ (c) � (e − c(T− t) /c 3 )a(c) and a(0) � 0.
In addition, given the Taylor expansion of e − c(T− t) as we then have Substituting this to (48) and taking the limit, we get (49).

□
We remark that the corollary above indicates our result can recover the result of Cheng et al. [13] with the mean reversion speed c of the trading signal L t degenerating to 0.

Linear Execution Risk.
For the linear execution risk, the function m(·) in (5) is a linear function, i.e., where m 0 > 0. In this case, the execution risk is related to the trading rate. Specifically, the faster we trade, the bigger the probability that our orders cannot be fully filled. is is in line with our intuition and reality in the market. Indeed, the liquidity of market is limited. If we trade very fast, our orders may merely be filled partially. Now the value function (22) is of the following form: en, we solve the optimal execution problem (17) and get the following theorem. en, the optimal execution problem (17) has a unique solution with state-feedback form: where A 1 (t) � E(t) + α − (θ/2) and E(t) is defined as where q(x) � log|x| − (k/x), k � (θ/2) − (η/m 2 0 ). In addition, the value function V(t, x, l) defined as (58) satisfies the HJB equation below: with terminal condition V(T, x, l) � − αx 2 .
Moreover, V(t, x, l) can be expressed as follows: where B 1 (t) and C(t) are defined as (79) and (77).