Index tracking with differentiate asset selection

Partial index tracking aims to replicate the performance of a given benchmark index with a small number of its constituents. It can be formulated as a sparse regression problem, but remains challenging due to several practical constraints, especially the fixed number of assets in the portfolio. In this paper, we propose a differentiable relaxation for asset selection, such that we can construct a portfolio with exactly K assets, where the objective function can be optimised efficiently via vanilla gradient descent. Our method is backtested with S&P 500 index data from 2002 to 2020. Empirical results demonstrate that our model achieves excellent tracking performance compared with some widely used approaches.