Optimal liquidation with temporary and permanent price impact, an application to cryptocurrencies

Paper: arXiv 2303.10043

Authors: Hugo E. Ramirez, Julián Fernando Sanchez

Abstract

This paper studies the optimal liquidation of stocks in the presence of temporary and permanent price impacts, and we focus in the case of cryptocurrencies. We start by presenting analytical solutions to the problem with linear temporary impact, and linear and quadratic permanent impact. Then, using data from the order book of the BNB cryptocurrency, we estimate the functional form of the temporary and permanent price impact in three different scenarios: underestimation, overestimation and average estimation, finding different functional forms for each scenario. Using finite differences and optimal policy iteration, we solve the problem numerically and observe interesting changes in the optimal liquidation policy when applying calibrated linear and power forms for the temporary and permanent price impacts. Then, with these optimal policies, we identify optimal liquidation trajectories and simulate the liquidation of initial inventories to compare the performance among the optimal strategies under different parametrizations and against a naive strategy. Finally, we characterize the optimal policies based on the functional form of the inventory and find that policies generating the highest revenue are those starting with a low trading rate and increasing it as time passes.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides solid analytical derivations and numerical PDE methods (finite differences, policy iteration) for optimal execution with non-linear price impact. Its empirical application to cryptocurrency order book data (BNB) bridges theory and market microstructure, though backtesting depth could be expanded.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Cryptocurrency Liquidation<br/>with Temporary & Permanent Price Impact] --> B[Data Input: Limit Order Book Data<br/>BNB Cryptocurrency]
    B --> C[Impact Estimation: Calibrate Temporary & Permanent Impacts<br/>Under, Over, & Average Estimation]
    C --> D[Numerical Optimization: Solve HJB Equations<br/>via Finite Differences & Policy Iteration]
    D --> E[Simulation & Evaluation: Test Optimal Trajectories<br/>vs. Naive Liquidation Strategy]
    E --> F[Key Outcome: Highest Revenue Strategies Start<br/>with Low Trading Rates and Accelerate Over Time]