Stochastic portfolio theory with price impact

Abstract

We develop a framework for stochastic portfolio theory (SPT), which incorporates modern nonlinear price impact and impact decay models. Our main result is the derivation of the celebrated master formula for additive functional generation of trading strategies in a general high-dimensional market model with price impact. We also derive formulas for an investor's relative wealth with respect to the market portfolio, conditions that guarantee positive observed market prices and a stochastic differential equation governing the dynamics of the observed price, the investor's holdings and the price impact state processes. As an application of these results, we develop conditions for relative arbitrage in the price impact setting analogous to previously obtained results for the frictionless setting. We then apply our framework to backtest the quadratic and entropy generating functions on historical US equity data, illustrating how price impact can negatively affect portfolio performance.