When Trend Beats Noise: A Taylor Expansion Theory of Moving Averages and Positive-Expectancy Trading

This paper presents a mathematical framework for interpreting moving-average trading systems through Taylor expansion. We show that moving averages encode lagged approximations of local derivatives of log price: the slope of a smoothed price process approximates local return velocity, while changes in this slope approximate local acceleration. We then introduce a stochastic trend-field model in which log returns are decomposed into a latent local trend component and a zero-mean stochastic disturbance with time-varying volatility. Applying Taylor expansion to the latent trend field over a finite horizon yields a sufficient condition under which the expected cumulative return, net of approximation error and trading cost, is positive. The main result is a conditional existence theorem: if local trend drift, represented by velocity and acceleration terms, dominates Taylor remainder error, stochastic volatility, and transaction costs, then a positive-expectancy trading system exists on that set of market states. The theorem does not imply universal market predictability or unconditional profitability. Rather, it formalizes the weaker claim that positive-expectancy trading is mathematically possible when local trend structure is sufficiently strong relative to noise and implementation frictions.