I am trying to do this portfolio optimization for a one-month investment between S&P 500 as a risky asset and one risk-free asset:
Assume that I have a power utility function, a risk-free rate interpolated for one month, and an option implied distribution function of next month returns. To find the two alpha as optimal weights of my portfolio, I need to know the return of the risky asset, i.e. $r_{t+1}$. What should I use for it?
And when I want to maximize the utility, I should take it as a constant in the $dF(r_{t+1})$? i.e. $dF$ is a constant number that will not play any role in the maximization problem?