# An Optimal Execution Problem with Market Impact

**Authors:** Takashi Kato  
**arXiv:** [0907.3282](https://arxiv.org/abs/0907.3282)  
**Date:** 2009-07-20  
**Summary page:** https://www.summarizepaper.com/en/paper/0907.3282v8/

## Key points

- Study by Takashi Kato focuses on optimal execution problem in continuous-time market model with market impact
- Problem formulated as stochastic control problem
- Research explores properties of corresponding value function
- Link between right-continuity at time origin and strength of market impact for large sales
- Value function remains continuous in other cases
- Demonstrates semi-group property (Bellman principle) and characterizes value function as viscosity solution of Hamilton-Jacobi-Bellman equation
- Examples show how optimal strategies influenced by trader's security holdings
- In Black-Scholes type market with nonlinear market impact, optimal strategies involve gradual liquidation rather than block liquidation even with risk-neutral behavior
- Research provides insights on how market impact and trader characteristics influence optimal execution strategies

## Summary

The study "An Optimal Execution Problem with Market Impact" by Takashi Kato focuses on analyzing the optimal execution problem in a continuous-time market model that considers market impact. The problem is formulated as a stochastic control problem, and the research explores various properties of the corresponding value function. One key finding is the link between right-continuity at the time origin and the strength of market impact for large sales, while in other cases, the value function remains continuous. Additionally, the study demonstrates the semi-group property (Bellman principle) and characterizes the value function as a viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Examples are introduced to showcase how optimal strategies can be significantly influenced by factors such as trader's security holdings. Interestingly, in scenarios involving a Black-Scholes type market with nonlinear market impact, optimal strategies do not involve block liquidation but rather gradual liquidation, even when traders exhibit risk-neutral behavior. Overall, this research sheds light on how market impact and varying trader characteristics can influence optimal execution strategies, providing valuable insights for decision-making in financial markets.