# Kelly Criterion in correlated stocks

I would like to ask if there exist any mathematical proof or model which addresses how the Kelly criterion can be applied to find portfolio weights when the stocks are correlated.

Luenberger's book has a discussion on growth-optimal (i.e. Kelly) portfolios, also for the multivariate case with correlated assets.

@BOOK{Luenberger1998,
title        = {Investment Science},
publisher    = {Oxford University Press},
year         = 1998,
author       = {David G. Luenberger}
}

• That books gets very mixed reviews on Amazon and is not cheap. Do you have it and, if so, do you recommend it. I won't blame you if I get it and end up not liking it :). Thanks. May 23, 2020 at 15:14

This post provides a model for the Kelly Criterion under no leverage and no short constraints, and yields the following quadratic program:

$$\max_f g = r + \sum_{i=1}^n f_i(\mu_i - r) - \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n f_i f_j \hat{\Sigma}_{ij}$$

$$\textrm{s.t.} \sum_{i=1}^n f_i \leq 1$$

$$f_i \in [0, 1]$$

They also have code examples in Python showing how to solve it with Pyomo and IPOpt.

They cite this paper, which looks like it has a derivation of this model.