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Deep Reinforcement Learning (Deep RL) has been explored for a number of applications in finance and stock trading. In this paper, we present a practical implementation of Deep RL for trading natural gas futures contracts. The Sharpe Ratio obtained exceeds benchmarks given by trend following and mean reversion strategies as well as results reported in literature. Moreover, we propose a simple but effective ensemble learning scheme for trading, which significantly improves performance through enhanced model stability and robustness as well as lower turnover and hence lower transaction cost. We discuss the resulting Deep RL strategy in terms of model explainability, trading frequency and risk measures.

Over the past half-century, the empirical finance community has produced vast literature on the advantages of the equally weighted S\&P 500 portfolio as well as the often overlooked disadvantages of the market capitalization weighted Standard and Poor's (S\&P 500) portfolio (see \cite{Bloom}, \cite{Uppal}, \cite{Jacobs}, \cite{Treynor}). However, portfolio allocation based on Tukey's transformational ladde have, rather surprisingly, remained absent from the literature. In this work, we consider the S\&P 500 portfolio over the 1958-2015 time horizon weighted by Tukey's transformational ladder (\cite{Tukey2}): $1/x^2,\,\, 1/x,\,\, 1/\sqrt{x},\,\, \text{log}(x),\,\, \sqrt{x},\,\, x,\,\, \text{and} \,\, x^2$, where $x$ is defined as the market capitalization weighted S\&P 500 portfolio. Accounting for dividends and transaction fees, we find that the 1/$x^2$ weighting strategy produces cumulative returns that significantly dominates all other portfolios, achieving a compound annual growth rate of 18\% over the 1958-2015 horizon. Our story is furthered by a startling phenomenon: both the cumulative and annual returns of the $1/x^2$ weighting strategy are superior to those of the $1/x$ weighting strategy, which are in turn superior to those of the 1/$\sqrt{x}$ weighted portfolio, and so forth, ending with the $x^2$ transformation, whose cumulative returns are the lowest of the seven transformations of Tukey's transformational ladder. The order of cumulative returns precisely follows that of Tukey's transformational ladder. To the best of our knowledge, we are the first to discover this phenomenon., Comment: 32 pages

We empirically show the superiority of the equally weighted S\&P 500 portfolio over Sharpe's market capitalization weighted S\&P 500 portfolio. We proceed to consider the MaxMedian rule, a non-proprietary rule designed for the investor who wishes to do his/her own investing on a laptop with the purchase of only 20 stocks. Rather surprisingly, over the 1958-2016 horizon, the cumulative returns of MaxMedian beat those of the equally weighted S\&P 500 portfolio by a factor of 1.15., Comment: 11 pages, 5 figures