A drawdown formula for stock trading via linear feedback in a market governed by Brownian Motion

Abstract

Control of drawdown is one of the greatest concerns to both stock traders and portfolio managers. That is, one typically monitors “drops in wealth” over time from highs to subsequent lows, and investors often shy away from funds with a past history of large drawdowns. With this motivation in mind, this paper addresses the analysis of drawdown when a feedback control is employed in stock trading in an idealized market with prices governed by Geometric Brownian Motion. We begin with a result in the applied probability literature which is applicable to cases involving buy-and-hold. Subsequently, after modifying this result via an Ito correction to account for geometric compounding of the daily stock price, we consider the effect on drawdown when a simple pure-gain feedback control is used to vary the investment I(t) over time. That is, letting V (t) denoting the trader's account value at time t ≥ 0, when a feedback control I(t) = KV (t) is used to modify the amount invested, the buy-and-hold result no longer applies. Our first result is a formula for the expected value for the maximum drawdown in logarithmic wealth log(V (t)). This formula is given in terms of the feedback gain K, the price drift μ, the price volatility σ and terminal time T. Subsequently, using a fundamental relationship between logarithmic and percentage drawdowns, we obtain an estimate for the expected value of the maximum percentage drawdown of V (t). This paper also includes an analysis of the asymptotic behavior of this drawdown estimate as T → ∞ and Monte Carlo simulations aimed at validation of our estimates.