Original paper

SSRN-id2620112.pdf2409.3KB

Abstract

Mandelbrot has significantly contributed in many ways to the area of finance. He was one of the first who criticized the oversimplifications centered around the early stochastic process models of Bachelier utilizing normal distribution. In his view, markets were fractal and much wilder than classical theory suggests. Additionally, he was a profound critic of the efficient markets hypothesis. Particularly, his work of fractional Brownian motion showed that the independence claim made by that hypothesis is not valid; in addition, he proposed a multi-fractal asset model to reconcile for effects observed in the market. However, it is also known that his vision of fractal markets used fractal trends. Recently, we were able to show that the scaling behaviour of trends, as defined by a specific trend decomposition using wavelets, are the root cause for the momentum effect.

Additionally, we were able to show that these trends have fractal characteristics. In this work, we will revisit Mandelbrot’s vision of fractal markets. We will show that the momentum effect discussed heavily in literature can be modeled by the so-called Mandelbrot Market-Model. Additionally, this model shows, from the risk side, that markets are wilder because of trend structures compared with classical models. In conclusion, we derive what Mandelbrot always knew: There are no efficient markets.

Keywords: Mandelbrot, Momentum effect, Wavelets, Efficient Markets. Model, Trends

Trading rules

Target the top quintile by size from NYSE, NASDAQ, and AMEX.
Choose the upper quintile stocks based on their 12-month momentum.
Construct a portfolio with uniform weights for the picked stocks.
Adjust the portfolio on a monthly basis.

Python code

Backtrader

import backtrader as bt
import backtrader.feeds as btfeeds
import datetime

class Momentum(bt.Strategy):
    params = (
        ('momentum_period', 252),
        ('rebalance_interval', 21),
        ('num_stocks', 0.2),
    )

    def __init__(self):
        self.add_timer(
            bt.timer.SESSION_END, monthdays=[1], monthcarry=True
        )  # Rebalance monthly
        self.inds = {}
        for d in self.datas:
            self.inds[d] = bt.indicators.ROC(d.close, period=self.params.momentum_period)

    def prenext(self):
        self.next()

    def notify_timer(self, timer, when, *args, **kwargs):
        if self._getminperstatus() < 0:
            return
        stocks = list(self.datas)
        stocks.sort(key=lambda d: self.inds[d][0], reverse=True)
        selected_stocks = stocks[:int(len(stocks) * self.params.num_stocks)]
        for d in self.datas:
            target = self.broker.getvalue() * (1 / len(selected_stocks)) if d in selected_stocks else 0
            self.order_target_value(d, target)

cerebro = bt.Cerebro()

# Add data for NYSE, NASDAQ, and AMEX stocks
# Replace 'NYSE.csv', 'NASDAQ.csv', and 'AMEX.csv' with appropriate files
for market in ['NYSE.csv', 'NASDAQ.csv', 'AMEX.csv']:
    data = btfeeds.GenericCSVData(
        dataname=market,
        fromdate=datetime.datetime(2000, 1, 1),
        todate=datetime.datetime(2020, 12, 31),
        dtformat=('%Y-%m-%d'),
        openinterest=-1,
        nullvalue=0.0,
        plot=False,
    )
    cerebro.adddata(data)

cerebro.broker.setcash(1000000.0)
cerebro.broker.setcommission(commission=0.001)  # 0.1% commission
cerebro.addstrategy(Momentum)

print('Starting Portfolio Value: %.2f' % cerebro.broker.getvalue())
results = cerebro.run()

print('Final Portfolio Value: %.2f' % cerebro.broker.getvalue())
cerebro.plot()

Make sure to replace ‘NYSE.csv’, ‘NASDAQ.csv’, and ‘AMEX.csv’ with the appropriate files containing the stock data for the mentioned exchanges. This code uses the Backtrader framework to implement the trading strategy as described.

Mandelbrot Market-Model and Momentum

Original paper

Abstract

Trading rules

Python code

Backtrader