I'm trying to solve portfolio problem with minimising its Expected shortfall, assuming the returns follow a stable distribution. If I'm able to calculate MLE fit to the series, calculate expected shortfall of that instrument, then how would I optimise the portfolio? I've read for instance here it is then linear programming problem.
I've tried to calculate ES of each of my instrument and then optimize the objective function
weights*ES, but I always get minimum for vector weights 0, which makes sense, that's when its lowest. But that's of course not what I want, what is it that I'm missing? Is it good idea to take code which calculates this for normally distributed returns and just replace the function which calculates the actual ES?
Thanks a lot