We propose a simple approach to dynamic multi-period portfolio choice with quadratic transaction costs. The approach is tractable in settings with a large number of securities, realistic return dynamics with multiple risk factors, many predictor variables, and stochastic volatility. We obtain a closed-form solution for a trading rule that is optimal if the problem is restricted to a broad class of strategies we define as "linearity generating strategies" (LGS). When restricted to this parametric class the highly non-linear dynamic optimization problem reduces to a deterministic linear-quadratic optimization problem in the parameters of the trading strategies. We show that the LGS approach dominates several alternative approaches in realistic settings. In particular, we demonstrate large performance differences when there is a dynamic factor structure in returns or stochastic volatility (i.e., when the covariance matrix is stochastic), and when transaction costs covary with return volatility.