Log-Returns Structure for Long and Short Positions

This paper extends the log-return structural framework to a unified analysis of both long and short positions. The framework is based on the decomposition of any logarithmic return series into two non-negative components: the Positive component (), defined as the cumulative sum of all positive log returns, and the Negative component (), defined as the cumulative absolute sum of all negative log returns. This decomposition is made possible by the additivity of logarithmic returns and allows the return process to be analyzed as a structure rather than only as a final outcome. The paper first establishes the structural layer of the asset, in which and define the asset’s return DNA. From these components, several structural measures are derived, including total movement or volatility (), net asset return (), the Log-Bias Ratio (), and the edge measures , , and . These metrics form a closed system that describes the relationship between return, volatility, risk, and directional bias. The central contribution of the paper is the distinction between structural symmetry and economic asymmetry. In logarithmic price space, a short position appears to be the mirror image of a long position. However, when the analysis is transferred from asset returns to realized wealth, this symmetry breaks down. A passive short position is constrained by capped profit and theoretically unlimited loss, creating a structurally different wealth dynamic from that of a long position. To formalize this distinction, the paper derives the Short-Wealth Formula for a passive 100% notional short position: or equivalently: This formula identifies the wipeout point, where the asset price doubles and the short investor’s initial equity is erased. Beyond this point, logarithmic wealth returns become undefined and the position enters a debt zone. The paper concludes by proposing a two-layer framework: Asset Structure, which describes the objective movement of the instrument, and Wealth Dynamics, which describes the realized capital process of the investor. This distinction provides a basis for more accurate long/short strategy analysis, structural backtesting, and future development of adjusted return models that compare strategies under normalized risk or activity conditions.