We start by reviewing some matrix algebra in order to understand the notation later on.  

We can now simulate some cross correlated stocks with stochastic changes in
expected return as follows:

We can now use quadratic optimization to find the optimal long/short weights that
maximize our risk adjusted expected portfolio returns. The objective function and
the corresponding constraints are given by:



   

where W is a column vector containing the portfolio weights, T is the transpose notation
ie convert a column vector to a row vector, ER is a column vector with expected
returns, Q is the covariance matrix and S is a column vector of 1's.

Note that  is the expected portfolio return and  is the portfolio variance.

The benchmark will be an simple equal weighted portfolio.