This paper studies trend filtering methods. These methods are widely used in momentum strategies, which correspond to an investment style based only on the history of past prices. For example, the CTA strategy used by hedge funds is one of the best-known momentum strategies. In this paper, we review the different econometric estimators to extract a trend of a time series. We distinguish between linear and nonlinear models as well as univariate and multivariate filtering. For each approach, we provide a comprehensive presentation, an overview of its advantages and disadvantages and an application to the S&P 500 index. We also consider the calibration problem of these filters. We illustrate the two main solutions, the first based on prediction error, and the second using a benchmark estimator. We conclude the paper by listing some issues to consider when implementing a momentum strategy.

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The performance of trend following strategies can be ascribed to the difference between long-term and short-term realized variance. We revisit this general result and show that it holds for various definitions of trend strategies. This explains the positive convexity of the aggregate performance of Commodity Trading Advisors (CTAs) which -when adequately measured -turns out to be much stronger than anticipated. We also highlight interesting connections with so-called Risk Parity portfolios. Finally, we propose a new portfolio of strangle options that provides a pure exposure to the longterm variance of the underlying, offering yet another viewpoint on the link between trend and volatility.

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We revisit the "Smile Dynamics" problem, which consists in relating the implied leverage (i.e. the correlation of the at-the-money volatility with the returns of the underlying) and the skew of the option smile. The ratio between these two quantities, called "Skew-Stickiness Ratio" (SSR) by Bergomi (2009), saturates to the value 2 for linear models in the limit of small maturities, and converges to 1 for long maturities. We show that for more general, non-linear models (such as the asymmetric GARCH model), Bergomi's result must be modified, and can be larger than 2 for small maturities. The discrepancy comes from the fact that the volatility skew is, in general, different from the skewness of the underlying. We compare our theory with empirical results, using data both from option markets and from the underlying price series, for the S&P 500 and the DAX. We find, among other things, that although both the implied leverage and the skew appear to be too strong on option markets, their ratio is well explained by the theory. We observe that the SSR indeed becomes larger than 2 for small maturities, signalling the presence of non-linear effects.