Note Quant - Optimizing the Trading Cost in presence of Market Impact

It's well known that most fund managers don't outperform the "market". This is mainly due to transaction costs. Especially the cost of trade impact, which is called market impact.
In every market, the quoted price is associated with a given (stable) number of buys and sells, called liquidity. Every time you want to buy or sell a share, you disrupt the stability of this liquidity (either by increasing or reducing it). You'll then face market resistance, which takes the form of a surcharge. This additional cost is proportional to the quantity of shares traded.
The liquidity of a share (or asset) is the ability to buy or sell it without causing an unfavorable price movement. The market impact of a given set of trades is the amount of liquidity this transaction will require...[...]