  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.
