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

Abstract

We propose two methods to obtain exact solutions for the AlmgrenChriss 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.