# Asset Pricing with General Transaction Costs: Theory and Numerics

## TL;DR

This paper analyzes how general convex transaction costs influence risk-sharing equilibria, demonstrating that equilibrium returns exhibit mean-reversion and providing numerical methods for complex models, with implications for liquidity premia and volatility.

## Contribution

It introduces a comprehensive framework for modeling risk-sharing with general convex costs, including numerical solutions for complex models with endogenous volatilities.

## Key findings

- Equilibrium returns mean-revert around frictionless levels.
- Quadratic costs lead to Ornstein-Uhlenbeck dynamics.
- Calibration shows liquidity premia increase volatility moderately.

## Abstract

We study risk-sharing equilibria with general convex costs on the agents' trading rates. For an infinite-horizon model with linear state dynamics and exogenous volatilities, we prove that the equilibrium returns mean-revert around their frictionless counterparts - the deviation has Ornstein-Uhlenbeck dynamics for quadratic costs whereas it follows a doubly-reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully-coupled forward-backward SDEs. These fall outside the scope of known wellposedness results, but can be solved numerically using the simulation-based deep-learning approach of Han, Jentzen and E (2018). In a calibration to time series of prices and trading volume, realistic liquidity premia are accompanied by a moderate increase in volatility. The effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.