On Stock Trading: Can a Trend Follower Expect to Win?

Abstract

Our answer to this question is a qualified “yes.” In an idealized market with perfect liquidity, continuous trading and no transaction costs, if a trend follower is a small enough player so as to be a price-taker with adequate resources to avoid margin calls, we rigorously prove that a positive expected value of the trading gain can be guaranteed. Surprisingly, the trend-following trading strategy which we describe, called Simultaneous Long-Short (SLS), is model-free. That is, the trader requires neither an “internal model” for the underlying stock prices nor a behavioral model for the actions of other players in the marketplace. While there are a number of papers in the literature which demonstrate the efficacy of such model-free strategies in an empirical data context via back-testing, to our knowledge, there is a little or no literature which includes a purely theoretical rationale to explain why such methods may work well in practice. The SLS strategy, motivated by ideas in feedback control theory, has the following salient feature: Rather than use of a model of any sort to adjust the amount invested over time, profits and losses are the only factor used for adaptation. More precisely, the number of dollars invested I(t) at time t is adapted as a function of the cumulative gains or losses g(t) on [0, t]. Subsequently, the “power” of our trading scheme is demonstrated using Geometric Brownian Motion for the prices. We include a rather arbitrary time-varying drift μ(t) and a time-varying volatility (t) as well. The class of admissible time variations negates the efficacy of many obvious trading strategies based on estimation of the direction in which the prices are drifting. Despite this lack predictability, we rigorously prove that the SLS trader “expects” to make money on every trade. That is, for any non-trivial price variation and any t > 0, the expected value of the trading gain satisfies E[g(t)] > 0. To our knowledge, such a remarkable property, attainment of performance robustness in a framework without any model to reveal the directional movement of prices, has never been uncovered. We believe that these results may be the “tip of the iceberg” in terms of new research directions aimed at revealing a theory which explains the empirically observed success of trend-following strategies beyond the one considered here.