A probabilistic analysis of the trading the line strategy

We provide analytic models for which the appropriate statistics of the trading the line strategy, Nh, can be derived in closed form. In particular, we provide closed-form expressions concerning the average duration of the open position, E(Nh), the variance of the open duration, Var(Nh), the average of the stopped log price, E(SNh), the variance of the stopped log price, Var(SNh), the correlation, Corr(Nh, SNh), and the Laplace transform, E(e-sNh). These results are obtained, in discrete time settings, for binomial and other price scenarios. Furthermore, when analytic results are not possible, such as the case of a normal distribution for log returns, we show by simulation that our general conclusions still hold. Using these statistics we point out some of the subtle features of the trailing stops strategy.