What is the return of risky asset in direct utility optimization probem?

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?

Your question is very confusing. But let's take it by parts:

1. You say you have power utility so your utility is: $\frac{W_{t+1}^{1-\gamma}}{1-\gamma}$
2. You have a risk-free rate number
3. You have an option implied distribution for stock returns so that should give you a two vectors one with returns $r_{t+1}$ and another with probabilities $dF(r_{t+1}$).
4. Given the non parametric nature of the problem (as you have a distribution of returns) you need to solve the problem numerically. Bellow a dummy example using matlab. Where I assume a risk-free, a gamma, a distribution for returns.
5. The result for that calibration is to allocate 0.62 to the risky asset and the remaining to the risk free.
clearvars
gamma = 10;
rf = 0.02;
ret = -0.02:0.01:0.07; %10 possible returns between -0.02 and 0.07
prob = 1/size(ret,2)*ones(size(ret,2),1); %Same probabilities each

% Now the maximization problem

alpha = (0.00:0.01:1.0)'; %grid for alpha

ExpUtility = zeros(size(alpha,1),1);
for i=1:size(prob,1)

ExpUtility = ExpUtility + prob(i)*(((1+alpha*ret(i) + (1-alpha)*rf)).^(1-gamma))/(1-gamma);

end

[maximum, index] = max(ExpUtility);

sum(ret'.*prob)
alpha(index)
• Thanks for this precise answer; I understand the problem better now. But there is still a problem. Your example takes the probabilities as given. When we have a nonparametric distribution function f(r), the probability of each return (as a point) to happen is zero (because distribution is continues). So I can't have a 'prob' vector like you. How should I tackle this problem? I think we should take integral of distribution instead of the sum you use here, but I don't know exactly how. May 5 '18 at 20:39
• If you have a functional form for $dF(r_{t+1}$ with which you can take a derivative then you can solve the problem using first order conditions. You should have it. You either have a vector of for $dF(r_{t+1}$ and corresponding probabilities or a functional form. May 5 '18 at 20:42

I wrote this code in R which I think is better for a continuous distribution function.

Utility <- function(r_t)    (-1/ARA) *  exp( (1 + a_t*r_t + (1-a_t) * r_f) * (-ARA) )

MaxiProb <- function(r_t)  Utility(r_t) * realPDF(r_t)

alpha <- seq(-1,2 , 0.01)
ExpUtility <- rep(0 , length(alpha))

for (i in seq_along(alpha)) {
a_t <- alpha[i]
ExpUtility[i] <- integral(MaxiProb , -Inf , Inf )
}

First I define the utility function and the product of utility and PDF to find $U(r_{t+1}) dF({t+1})$. Then I define alpha and expected utility which is negative exponacial, and I calculate the integral for each alpha to find the possible expected utilites. Then I can find the alpha which gives the largest expected utility here:

alpha[which.max(ExpUtility)]

Of course, the code is not reproducible since the PDF is not available here, but it gives an understanding of what am I talking about.