Discrete Dynamics in Nature and Society

Volume 2018, Article ID 4721596, 11 pages

https://doi.org/10.1155/2018/4721596

## Optimal Trade Execution under Jump Diffusion Process: A Mean-VaR Approach

School of Systems Science Beijing Normal University, Beijing 100875, China

Correspondence should be addressed to Tianmin Zhou; moc.liamtoh@82_txdzdsl and Handong Li; nc.ude.unb@dhl

Received 2 February 2018; Revised 3 May 2018; Accepted 8 May 2018; Published 1 October 2018

Academic Editor: Xiaohua Ding

Copyright © 2018 Tianmin Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the classical optimal execution problem, the basic assumption of underlying asset price is Arithmetic Brownian Motion (ABM) or Geometric Brownian Motion (GBM). However, many empirical researches show that the return distribution of assets may have heavy tails than those of normal distribution. The uncertain information impact on financial market may be considered as one of the main reasons for heavy tails of return distribution. To introduce this information impact, our paper proposes a Jump Diffusion model for optimal execution problem. The jumps in our model are described by the compound Poisson process where random jump amplitude depicts the information impact on price process. In particular, the model is simple enough to derive closed-form strategies under risk neutral and Mean-VaR criterion. Simulation analysis of the model is also presented.

#### 1. Introduction

The algorithmic trading in equities and other asset classes has been greatly developed over the past decade. The key problem of algorithmic trading is to decrease execution cost, which is the difference in the value between the ideal trade and what was actually done [1]. The execution cost can be decomposed into explicit costs and implicit costs [2]. Explicit costs are the direct costs of trading, such as commissions, fees, and taxes. Implicit costs are indirect costs, which are mainly due to price impact of trading and quite significant in large trade [3]. To decrease price impact, traders can break up a large trade into a number of smaller blocks. However, a longer execution horizon may increase the exposure to the timing risk.

In pair of seminal works, Bertsimas and Lo [4] first obtained a closed-form solution of optimal execution problem in discrete time. They defined the information price model under the assumption that the unaffected price process follows ABM and added price impact on the execution price of the trade. Almgren and Chriss [5, 6] extended the work by taking a tradeoff between expect execution cost and risk. In their trading model, a closed-form solution to the optimal execution problem is given in continuous time, and the price impact is divided into temporary impact and permanent impact. Temporary impact mainly comes from supply-demand imbalances at the moment of trading. In contrast, permanent impact refers to a price change persisting for the whole period. Both temporary and permanent impact function are considered as linear in trade size.

Based on the pioneering works, there is a large number of literatures on the optimal execution problem. Schied and Schöneborn [7] obtained the optimal execution strategy through utility maximization. Gatheral and Schied [8] derived the closed-form solution under the assumption of using GBM as unaffected price process. In their corresponding model, the time-averaged VaR is chosen to quantify the risk associated with trade. Related work along this line with robust solution can be found in Schied [9]. In contrast to the traditional price impact mode, Forsyth [10] considered that the price impact function depended on both trade size and current price, and the numerical method was used in solving mean-variance problem. Forsyth et al. [11] suggested that using quadratic variation as risk measure was time-consistent in the dynamic programming. Cheng et al. [12] summarized the risk criterions and obtained closed-form strategies with uncertain order fills in Almgren–Chriss framework.

A large number of literatures about optimal execution are mainly based on the assumption of ABM or GBM. The assumption fails to capture the information impact on the asset price. The information impact includes changing market condition and sudden availability of some information about the asset. There are relatively few studies on the optimal execution problem under the information impact. Bertsimas and Lo [3] first used first-order autoregressive behavior to describe the information impact on unaffected price process. Huberman and Stanzl [13] extended the work by incorporating uncertainty liquidity information. Apart from these studies, few literatures focus on the jump phenomenon in optimal execution problem. The phenomenon describes that the asset price process is often interrupted by some sudden information. These interruptions cause significant discontinuity of asset prices process, the so-called jump phenomenon [14–16], which explains that the asset return probability distribution has heavy tails [17–19]. Moazeni et al. [3] first studied the optimal execution problem under Jump Diffusion process and used Monte Carlo simulation to obtain the minimum CVaR (conditional value at risk) execution strategy. In their model, the compound Poisson process is used to explain the price impact of other large trades. In contrast, we use the compound Poisson process to depict the uncertain information impact and derive the analytic expression for the VaR (value at risk) evaluation. Then we use discrete stochastic dynamic programming method introduced by Bellman [20] and Bertsimas and Lo [4] to obtain a closed-form solution to optimal execution problem under risk neutral and Mean-VaR criterion.

The remainder of this paper is organized as follows. In Section 2, we introduce the basic definitions of optimal execution problems. Section 3 derive closed-form solutions for the optimal execution strategies under risk neutral criterion and Mean-VaR criterion. We provide the influence of different parameters on the optimal strategies and carry out the Monte Carlo simulation to compare different execution cost in Section 4. The last section concludes some remarks of our model, and the proofs are given in the appendix.

#### 2. Problem Description and Notations

This paper assumes that we hold large shares of an asset which we plan to completely sell over a fixed time interval . We divide into intervals of length and let , where for . During the transaction, the deterministic initial asset price is denoted by , and the dynamics of include two distinct components: the Brownian motion and the compound Poisson process. Then the law of motion for is expressed aswhere represents the volatility of the asset, is independent random variables with zero mean and unit variance, and is the increment of Brownian motion. The uncertain information impact is the compound Poisson process with the arrival rate and the jump amplitude , where is the jump number count on .

We specify the trade list which represent the number of the shares that we seek to sell between times and . Let denote the holding shares at time and satisfy the following expression: . We assume that our initial holding shares are and final holding shares are . Similar to Bertsimas and Lo [4], the discrete price process with permanent impact can be expressed aswhere satisfies , and is the linear permanent impact coefficient which measures the permanent impact of trade size. In addition to permanent impact, the execution price may decrease between and due to temporary imbalances in supply and demand, and the effect does not appear in the next price . According to the Linear-percentage temporary price impact model [4] and temporary market impact model [6], we assume that the temporary impact is the function of trading rate and only affects the execution price during the interval to . The execution price is given bywhere represents the linear temporary impact and .

Under the assumption above, we define that the total amount at the end of the time horizon is . The main objective function is to maximize the expected total amount concerned with the given risk measure. Hence the objective function in general form can be described as follows:where is the risk measure of execution under the risk aversion parameter and is the unaffected price increment in (1). We compute the risk measure associated with at each time point. In next section, we first consider the optimization problem under risk neutral and then take the execution risk into the optimization.

#### 3. Optimal Solution of Problems

In this section, the optimizations under two conditions will be described in detail.

##### 3.1. Optimal Execution Strategy under Risk Neutral

We first maximize the expected total amount at the end of the time horizon and our optimization problem can be described as follows:which aims to find an execution strategy to maximize the expected total amount. The trading strategy needs to completely sell total shares at time . Then we get a typical optimal control problem, and we can solve (5) by the discrete stochastic dynamic programming method and the closed-form solution can be expressed as follows:where , , is the optimal execution strategy, is the optimal-value function and is the number of holding shares for . The corresponding proof is provided in Appendix A.

When , the optimal execution strategy turns toand the optimal-value function isSubstituting the initial conditional into (8) yields optimal execution strategy size . By the recursive law, we can getwhich shows that total shares are sold at a constant rate over the time horizon . The optimal execution strategy is called TWAP (Time Weighted Average Price) or Naïve strategy, and the remainder shares are . Under the linear impact function, the TWAP strategy has the least price impact during transaction.

When , in contrast to the TWAP strategy, the optimal execution strategy (6) depends on the price impact and expected jump size per unit time. While the TWAP strategy never buys for sell execution, the optimal execution strategy may include buying in some periods, and this is also provided in literature [3]. For , the decrease speed of the holding shares is smaller than TWAP strategy at the beginning of the trade, and the rate is larger than the TWAP strategy near the expiration. For , the optimal execution strategy reduces to the TWAP strategy. For , the decrease speed of the holding shares is larger than TWAP strategy at the beginning of the trade and the rate is smaller than the TWAP strategy near the expiration.

##### 3.2. Optimal Execution Strategy under Mean-VaR

Apart from the expected execution amount, a risk-averse investor also needs to consider the execution risk. He will take a tradeoff between price impact and risk. That is to say, investors always want to pursue a strategy that maximizes the expected execution amount under a given risk preference. Therefore, we add a risk measure into the objective function and the optimized objective function becomeswhere is the risk exposure of our holding shares at confidence level . Under the assumption of Jump Diffusion process, , and is the cumulative distribution function of unaffected price increment in (1), which can be expressed asHere is a Gaussian Mixture Model (GMM). A proof for (12) is given in Appendix B. Next, the closed-form solution of problem (11) iswhere , , is the optimal execution strategy, is the optimal-value function, and is the number of holding shares for *. *We provide a corresponding proof in Appendix C.

When , the unaffected price process obeys the ABM, and the optimal execution strategy is where is the increment of Brownian motion. The optimal-value function isThe optimal execution strategy depends on the price impact and the risk exposure of our holding shares. The decrease speed of the holding shares is larger than TWAP strategy at the beginning of the trade, and the rate is smaller than the TWAP strategy near the expiration.

When , the unaffected price process obeys the Jump Diffusion process, and the optimal execution strategy is (13). Because of adding jump into ABM, the optimal execution strategy depends on and . For , the expectation of the compound Poisson process covers the risk exposure, and the optimal execution strategy trades slowly except near expiration. The optimal execution strategy reduces to the TWAP strategy, for . For , the optimal execution strategy trades quickly except near expiration. The comparison among these strategies is illustrated in next section.

#### 4. Numerical Simulation

In this section, we first illustrate the influences of parameters variations on optimal execution strategies. Then we show the comparison among transaction costs of different execution strategies. The initial parameters are given as follows.

We have a single asset with current market price =$100, and the initial holdings are shares. The whole shares are expected to sell in one day and the interval between trade is . The daily volatility of stock is , the volatility in every time interval is , the temporary impact coefficient is , and the permanent impact coefficient is . The range of arrival rate is and the jump amplitude is the normal distribution with and , and the range of preference for risk is (see Table 1).