# Optimal multi-asset trading with linear costs: a mean-field approach

## TL;DR

This paper introduces a mean-field approach to solve multi-asset trading problems with L1 transaction costs, providing approximate solutions and insights into optimal trading strategies under risk aversion.

## Contribution

It develops a novel mean-field method to reduce multi-asset trading with L1 costs to a single-asset problem, offering analytical solutions for Ornstein-Uhlenbeck predictors.

## Key findings

- Optimal strategy is of the 'bang-bang' type.
- Trading threshold depends linearly on global position with computable slope.
- Numerical simulations support analytical results.

## Abstract

Optimal multi-asset trading with Markovian predictors is well understood in the case of quadratic transaction costs, but remains intractable when these costs are $L_1$. We present a mean-field approach that reduces the multi-asset problem to a single-asset problem, with an effective predictor that includes a risk averse component. We obtain a simple approximate solution in the case of Ornstein-Uhlenbeck predictors and maximum position constraints. The optimal strategy is of the "bang-bang" type similar to that obtained in [de Lataillade et al., 2012]. When the risk aversion parameter is small, we find that the trading threshold is an affine function of the instantaneous global position, with a slope coefficient that we compute exactly. We relate the risk aversion parameter to the desired target risk and provide numerical simulations that support our analytical results.