A Note on Portfolio Optimization with Quadratic Transaction Costs

The general approach for introducing liquidity management in the mean-variance optimization model of Markowitz (1952) is to assume fixed bid-ask spreads. We then obtain the linear transaction cost model, which can be solved using an augmented quadratic programming problem (Scherer, 2007). However, as shown by Lecesne and Roncoroni (2019a, 2019b), unit transaction costs may be a linear function of the trading size, implying that a model with quadratic transaction costs may be more appropriate. In this article, we investigate this approach and show how linear and quadratic transaction costs modify the mean-variance optimized framework. In particular, we do not obtain a standard QP problem when transaction costs are quadratic, because the budget constraint is no longer linear. In this case, we obtain a quadratically constrained quadratic program (QCQP), which is an NPhard problem. However, using the ADMM framework, we are able to derive an efficient algorithm that solves this issue. Finally, we use this algorithm to illustrate the impact of transaction costs on optimized portfolios and Markowitz efficient frontiers. The authors are grateful to Jules Roche for his helpful comments.