Exact solutions for optimal execution of portfolios transactions and the Riccati equation

TL;DR

This paper presents two methods to find exact solutions to the Almgren-Chriss optimal execution model, demonstrating its equivalence to a Riccati equation and identifying conserved quantities in certain cases.

Contribution

It introduces two novel methods for solving the Almgren-Chriss equation exactly and links it to Riccati equations, advancing analytical solutions in optimal trading models.

Findings

Two exact solutions for the Almgren-Chriss equation.

Reduction of the model to known solvable equations.

Identification of a conserved quantity in the reduced form.

Abstract

We propose two methods to obtain exact solutions for the Almgren-Chriss model about optimal execution of portfolio transactions. In the first method we rewrite the Almgren-Chriss equation and find two exact solutions. In the second method, employing a general reparametrized time, we show that the Almgren-Chriss equation can be reduced to some known equations which can be exactly solved in different cases.For this last case we obtain a quantity conserved. In addition, we show that in both methods the Almgren-Chriss equation is equivalent to a Riccati equation.