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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.

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 [

In pair of seminal works, Bertsimas and Lo [

Based on the pioneering works, there is a large number of literatures on the optimal execution problem. Schied and Schöneborn [

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 [

The remainder of this paper is organized as follows. In Section

This paper assumes that we hold large shares

We specify the trade list

Under the assumption above, we define that the total amount at the end of the time horizon

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

We first maximize the expected total amount at the end of the time horizon

When

When

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 becomes

When

When

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

Simulation parameters.

Parameters | Values |
---|---|

Initial price | $100/share |

Total position | |

Trading time | 1day |

The permanent impact coefficient | |

The temporary impact coefficient | |

The daily volatility | |

The number of trades | 240 |

The interval between trade | 1/240 |

Arrival rate of jump | |

Mean of jump amplitude | |

Variance of jump amplitude | |

Appetite for risk | |

The optimal execution strategy varying with the parameter

The execution trajectories under different

The optimal execution strategy under ABM is well-studied. Figure

The execution trajectories under different price processes.

Next, we analyze the influence of parameters variations on optimal strategy. For

The influence of parameters

Assume that the stock price process follows the Jump Diffusion process, the temporary impact and permanent impact are linear function of trading size for each time interval. Let the price processes

Optimal execution results.

Strategy | Average of cost |
---|---|

Random | $13,199,249.89 |

TWAP | $11,569,787.27 |

Risk neutral | $10,871,389.31 |

Mean-VaR | $21,321,881.81 |

Mean-Va | $19,111,757.60 |

From the results, we find that the random strategy has the largest cost. Because of taking jump process into consideration, the Risk neutral strategy and the

To compare the differences among these costs in statistics, we carry out a paired-samples t-test to analyze the distinction of the transaction costs. The paired-samples t-test is widely used for determining whether there is a systematic deviation between paired test data; that is, if the difference between paired test data is significant, the difference can be always observed under different conditions. Since the observation data are from different objects, they are not sample data, and the test method is as follows.

Considering

Then we use 5

Optimal trading results.

Mean | Standard deviation | 95% confidence interval | T | df | Sig. | ||
---|---|---|---|---|---|---|---|

lower limit | upper limit | ||||||

Risk neutral versus Random | -2,327,861.00 | 2,870,953.99 | -2,407,457.99 | -2,248,263.99 | -57.33 | 4999 | .000 |

Risk neutral versus TWAP | -698,397.97 | 2,799,931.52 | -776,025.44 | -620,770.44 | -17.64 | 4999 | .000 |

Mean-Va | -2,210,124.22 | 770,966.15 | -2,231,499.09 | -2,188,749.34 | -202.71 | 4999 | .000 |

From Table

Most published literatures on the optimal execution problem are typically solved in Arithmetic Brownian Motion or Geometric Brown motion. In this paper, we use Jump Diffusion process to capture the uncertain information impact, which reflects more realistic phenomenon of modern financial market. The proposed model includes a compound Poisson process and a closed-form solution to the optimal execution strategy is derived by stochastic dynamic programming method. The optimal execution strategy does not depend on the asset price volatility under risk neutral. In contrast, under Mean-VaR, the optimal execution strategy depends on price distribution. In addition, we illustrate the influence of parameters changes on the optimal execution strategy. The paired t-test is applied to perform statistical analysis and shows that the paired execution costs are obviously different. Traders can determine an optimal execution strategy according to an appropriate optimization model. Although we provide a closed-form solution to the optimal execution strategy, there are many issues that need to be modified. For example, our optimization does not incorporate a nonnegative constrain, Jump Diffusion process in our model may give negative value, etc. Therefore, various further works need to be extended.

According to the iterative form of the value function of dynamic programming, we can get

According to the definition of the distribution function, there are

In our model, a compound Poisson process is introduced into ABM. Thus, the distribution of price increment depends on

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.