Abstract
We extend the self-exciting model by assuming that the temporary market impact is nonlinear and the coefficient of the temporary market impact is an exponential function. Through optimal control method, the optimal strategy satisfies the second-order nonlinear ordinary differential equation. The specific form of the optimal strategy is given, and the decreasing property of the optimal strategy is proved. A numerical example is given to illustrate the financial implications of the model parameter changes. We find that the optimal strategy of a risk-neutral investor changes with time and investment environment.
Highlights
In the financial field, the problem of optimal liquidation is widely studied
We suppose that the coefficient of temporary impact component like the exponential function is widely used in economic activities. us, the coefficient of temporary impact component is assumed to be ea+b(x− Xt), where a > 0 and b > 0
Since the coefficient of temporary impact component is ea+b(x− Xt), there exists a unique strategy for mean optimization. e strategy is the unique solution of the following differential equation:
Summary
The problem of optimal liquidation is widely studied. In 1998, Bertsimas and Lo [1] study the minimum transaction completion in the case of fixed trading time dynamic trading strategy. Based on the original scholar’s model, Almgren and Chriss [2] consider the expected costs and risks of execution and propose a simple market impact model. It includes the following three parts: unaffect price process, temporary market impact, and permanent. From equation (24) and Figures 2 and 3, when the temporary market impact is a power function and the coefficient of temporary market impact is a linear function, the investor realizes that they will face large execution costs with bigger a and smaller b so that they speed up liquidation early and slow down the trading speed later. “Optimal execution of execution costs,” Journal of Finance Markets, vol 1, pp. 1–50, 1998
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