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I am looking for an equation for the optimal fractional bet sizing for N number of simultaneous correlated bets.

I am looking specifically for an equation for binary bets, but an equation for bets with distributional profit and loss outcomes would be welcome as well.

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    $\begingroup$ Can you tell us about what you came across in your own research and why they did not work for you? $\endgroup$
    – Alper
    Oct 9 at 11:11
  • $\begingroup$ Kelly Criterion for a single or even multiple independent binary bets is easy to find, however; I can't really find anything regarding specifically what I asked for. $\endgroup$
    – lucmad
    Oct 9 at 19:29
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A natural question which was likely studied in academic literature (even though Kelly is not particularly popular among portfolio managers). I guess you could generalize Eq. (6.87) of this book. If $f(R)$ is the joint probability density of the returns of your $N$ assets (so $R$ is $N$-dimensional vector), for example Gaussian, $$ f(R)\propto\exp(-(1/2)\sum_{ij}C^{-1}_{ij}(R_i-\mu_i)(R_j-\mu_j)), $$ and $x$ is an $N$-dimensional vector of your bets, the Kelly log utility is $$ U(x)=\int\log(1 + x\cdot R)f(R)dR. $$ The utility is maximized when $\partial U/\partial x_i=0$, or $$ 0 = \int\frac{R_i}{1+x\cdot R}f(R)dR. $$ I don't know if this can be solved in a closed form, but maximizing $U(x)$ numerically seems doable. Note that in the classical Kelly case you need to consider only the slab are $0\le\sum_ix_i\le1$, but if you are long/short and levered, the constraint is lifted.

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