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Key GMM (Kelly Optimization) Paper Explaining the Formula

September 21, 2010

There exists some confusion between the kelly criterion that was originally applied to sports betting (discrete bets) and the  kelly criterion for financial markets by Ed Thorp that is based on continuous time.  The financial kelly application is derived from the latter to account for the lack of defined payoffs or losses that exist in the markets. The method for applying the financial kelly application for portfolio optimization shares some similarities with conventional mean-variance optimization in that it uses the covariance matrix to account for differences in the relationship between assets . However, the kelly method is designed to maximize the compounded return while classic optimization maximizes the arithmetic sharpe ratio. Based on several inquiries, I have decided to provide a helpful link beyond the original Thorp paper: Javier Estrada wrote a key paper about the Kelly optimization that I like to call “GMM” or geometric mean maximization. The paper can be found here: . Suffice to say the math is not terribly difficult but certainly not like programming a technical indicator and requires the use of covariance matrices. This is only for the tech and quant saavy, but can be more easily replicated using statistical software such as “R” (which is what we used) or MATLAB.

Nonetheless, the method that we used for sentiment spreads was innovative and unique based on the specific application. Clever individuals can certainly replicate what was done to a large extent. The purpose of the article on creating an intermarket ensemble was to demonstrate how classical portfolio theory can assist with the adaptive signal aggregation process. This method is not linear and helps to account for the relationships between signals that regression often fails to address explicitly via a least squares method. As a consequence the output is superior and presents a malleable framework for more sophisticated usage.

6 Comments leave one →
  1. September 21, 2010 7:08 pm

    Very good clarification. I’ll see if I don’t have a sample file for you to post. Best, JP

  2. Alexandre permalink
    September 22, 2010 1:59 pm

    Interesting, but I wouldn’t call it GMM. My first reaction was to think it was about Generalised Method of Moments, an estimation technique in Econometrics. But great reference, wasn’t aware of this paper at all and Estrada is a nice academic.

  3. EmilT permalink
    December 1, 2014 3:29 pm

    What kind of optimization routine did you use to solve the GMM problem? Considering only small daily returns, a simple expansion would turn the problem into n linear equations with a closed form solution.

    • david varadi permalink*
      December 1, 2014 3:33 pm

      hi Emil it was a closed form solution using a system of equations (minverse and the covariance matrix and returns).

      • EmilT permalink
        December 2, 2014 7:43 am

        Thanks! Did you keep the restriction that all weights should add up to one? Since the test was allocating 100% long/short based on if the sum was positive/negative this would boost the returns (by solving n+1 equations given by the Lagrange multiplier method) over using the weights derived without any restriction on leverage.

  4. david varadi permalink*
    December 2, 2014 1:05 pm

    hi Emil, yes– the implementation had an absolute capital sum restriction of 100%. you are correct,

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