Abstract
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.
Highlights
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
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; we investigate the optimal execution problem by taking into account the trading signal and the execution risk simultaneously
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
Summary
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. 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. 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. 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.
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