Mandelbrot Market-Model and Momentum

Abstract

Mandelbrot was one of the first who criticized the oversimplifications in finance modeling. In his view, markets have long-term memory, were fractal and thus much wilder than classical theory suggests. Recently, we were able to show that the scaling behaviour of trends, as defined by a specific trend decomposition using wavelets, are causing the momentum effect. In this work, we will show that this effect can be modeled by fractal trends. The so-called Mandelbrot Market-Model shows that markets are wilder compared with classical models. In conclusion, we derive what Mandelbrot always knew: There are no efficient markets.Keywords: efficient market hypothesis, momentum effect, fractal markets, trends Benoît Mandelbrot is considered by some as the father of quantitative finance as described by Dempster (2011). Among his extraordinary contributions in different disciplines (see Peitgen (2010) for a summary), he has contributed to the field of finance in many different areas. First, he was among the first who criticised normal distribution's usage to model asset returns as originally proposed by Bachelier (1900). Second, Mandelbrot (1963) observed that asset returns exhibit scaling properties, which may indicate that distributions of asset returns have infinite variance. Third, he was able to model "trending" and "mean reversion" effects by introducing, with van Ness in Mandelbrot (1968), the so-called fractional Brownian motion. Finally, Mandelbrot (1997) proposed a combined model using fractional Brownian motion, which is deformed in time by using multiplicative cascades. Although he was able to address fat tails, the non-stationarity in volatility and trending/mean-reversion effects in that model (a summary has been published in Quantitative Finance, refer to his series Mandelbrot (2001, I-IV), he also, in Mandelbrot (2001-III), sketched out a vision of "cartoons", i.e., a fractal trend compositions scheme (by using midpoint replacement techniques) constituting what he called a fractal market. From this viewing angle and particularly supported by his models, he concluded that there are no efficient markets, as assumed by his former student Eugene Fama in Fama (1970). Although many authors after Mandelbrot, and particularly Lo (1998), have shown that there is a long-term dependence of asset returns (violating the weak form of the efficient market hypothesis), there are counter arguments claiming that no one can exploit these dependencies structurally over long time horizons. Particularly, the size and value effect, e.g., reported in Fama (2012), are said to represent a higher insolvency or illiquidity risk. In that regard, the momentum effect, originally reported in Jegadeesh (1993) and also confirmed in Fama (2008) earlier, is of special importance, because it is reported by using simple management rules that leads to structural outperforming portfolios in terms of risk and return compared to a market index.Recently, in Berghorn (2015), we were able to show that the momentum effect is caused by trends in...

Trend Momentum II: Driving Forces of Low Volatility and Momentum

“…The cohort of high volatility stocks (in combination with momentum) yields almost twice of the risk adjusted returns if compared to to the cohort of low volatility stocks. The same conclusion can be drawn for cohorts of assets with higher Hurst exponents and lower Hurst exponents (where the conclusion is even stronger): Berghorn and Otto (2017a), the momentum effect can be mimicked by Monte Carlo simulations using Fractional Brownian Motions based on the Hurst exponent. This leads us to the assumption that Hurst exponents measure trending effects rather than long-term memory as mentioned by Mandelbrot and Van Ness (1968).…”

“…Therefore, we estimate the drift of the last trend segment visible at a given wavelet scale. For the mathematical definition, please refer to Berghorn and Otto (2017a).…”

“…To identify the key drivers of momentum, we rather need a deeper statistical view, based on an estimation of Hurst exponents and the so-called rescaled range (R/S)analysis (introduced by Mandelbrot) used in, e.g. Berghorn and Otto (2017a). According to this approach, we can show that almost every asset of the Bloomberg World Ex China Index has a Hurst exponent of greater than 0.5 (see Appendix B).…”