ABSTRACT
This chapter defines a class of price impact models initiated by Obizhaeva and Wang and derives their mathematical properties. It introduces two base variables: prices and trades. Because of market microstructure, these variables come in multiple flavors. The different price descriptions lead to formal definitions of price impact and instantaneous transaction costs. The chapter describes the trader's P&L. It then covers the frictionless case used in the seminal work by Black and Scholes. In that setting, the new observed price equals the last traded price. This assumption reflects the trading data of the time: transaction prices were public, but there was no continuously quoted order book providing market information on fills. Optimal execution minimizes an order's arrival slippage. A different approach replaces the exponential decay with a general time kernel. For instance, empirical results indicate that impact decays with a power-law over long timescales.
