"Recently, in Berghorn (2015, Trend Momentum, Quantitative Finance, Volume 15, Issue 2, 261-284), we were able to show that the momentum effect is caused by trends in asset price data. Trends, are not mathematically defined in general and the term „trend“ is often used ambiguously. In Berghorn (2015, Trend Momentum, Quantitative Finance, Volume 15, Issue 2, 261-284) we used a special wavelet trend decomposition scheme to model price data into up- and downward trends. Particularly, we were able to show that, in accordance with Mandelbrot’s scaling analysis, so defined trends in price data are scaled with respect to the granularity level of analysis of that particular wavelet decomposition scheme.

Furthermore, the average trend sizes, as computed by the decomposition scheme, are following a power law. When we compared the power law exponents (measuring the average trend sizes) of random processes and the real world data, we were able to show that real world data structurally exhibits higher exponents. By mathematical experiments, we showed that the higher (so-called momentum) exponent (in real world data) is causing the momentum effect. Because the momentum strategy did not exhibit higher risks and had a significant higher return, we concluded that there are no efficient markets because the independence of asset returns is non-existent, and momentum strategies provide simple mechanisms to generate excess returns.

In the following, we will revisit some key components of Berghorn (2015, Trend Momentum, Quantitative Finance, Volume 15, Issue 2, 261-284). We use these to establish additional Monte Carlo simulations. Particularly, we test certain random process models by comparing the outcome of a classic momentum strategy with what we have observed with real world data. We will show, in particular, that non-stationary random walks (with piecewise varying drifts and volatilities) do not model momentum well. We will then use trend bootstrapping based on a recursive trend decomposition introduced in Berghorn (2015, Trend Momentum, Quantitative Finance, Volume 15, Issue 2, 261-284) and show that this type models the momentum effect quite well. We will analyse the trend statistics of that decomposition and will model price data as being constructed by a superimposition of trends. Under this model (and after taking the logarithm of the prices), the slopes and the sizes of the trends are drawn from a lognormal distribution. This is a natural extension of continuous returns and allows us to derive an analytical form for this type of process. We compare this construction and verify that it (although technically different) fulfils Mandelbrot’s vision of fractal markets. This so-called Mandelbrot Market-Model describes the mathematical characteristics enabling the momentum effect. Additionally, we show that market data under this model is wilder than what is usually assumed. Particularly, the shortfall risks of a buy-and-hold investment are significantly higher when we use this model compared to classical approaches. In that regard, we conclude that this model is better suited than classical approaches in literature, such as random walk or fractional brownian motion. Consequently, and in accordance with Mandelbrot’s analysis, markets are wilder and exhibit trending (modelled by that model) and, therefore, cannot be efficient."

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