Constrained Optimal Execution in Limit Order Book Market with Power-shaped Market Depth

Instead of using the classical block-shaped market depth to build the optimal execution model, this work studies the constrained optimal execution problem in a limit order book (LOB) market with a power-shaped market depth. Different from the linear price impact derived from the framework of block-shaped market depth, the price impact generated from the framework of our power-shaped market depth becomes a nonlinear function, which is consistent with the empirical studies. We also consider a class of state-dependent upper and lower bound constraints on trading strategies, which includes non-negative constraint (or non-short selling constraint) as its special case. Even though both the power-shaped market depth and trading strategy constraints make it hardly to solve such an optimal execution problem analytically, we still develop some significant properties on the optimal execution policy and optimal execution cost of our model. From some illustrative examples, we find that the optimal execution policy derived from our model is quite different from the one generated from the model with the block-shaped market depth when the market exhibits finite resiliency. However, when the market resilience is infinite, the obtained result becomes different, i.e., the optimal execution policies derived from these two kind of models are equivalent. For a special model with the stochastic block-shaped market depth and infinite market resilience, we successfully derive the analytical solution for such an optimal execution problem by utilizing the state separation property induced from its structure. The revealed optimal execution strategy is a piece-wise affine function with respect to the current remaining position, which can be computed off-line efficiently by solving two coupled equations. Finally, due to its explicit solution, we utilize this optimal execution model to demonstrate that the model admits no price magnification opportunity for the two-sided trading strategy