Author(s): Jonathan Scott Image by the author. Brownian Motion in Multifractal Time I must thank my good friend Daniel Luftspring for his contributions and guidance throughout the development of this project. Short on the Assessment of Risk At the core of the financial market theory, there is a large and unstable assumption. The assumption is that the risk of a financial instrument can be determined by its volatility and that this measure scales with the square root of time. To understand the origins of this assumption, you must travel back before the subprime mortgage crisis of 2008, the dot-com bubble of 2000, Black Monday in 1987, the Wall Street Crash of 1929, and then another one hundred years. Here, you will find Robert Brown looking through a microscope at a speck of pollen suspended in water. He observed that the speck danced erratically. In 1900, this observation was applied to finance with advanced mathematics by Louis Bachelier in his dissertation, Théorie de la spéculation, on modeling the stock market. With the assumption that the price of a stock moved in the same random manner as the speck of pollen, he was able to calculate a price for the stock’s option. In 1905, Albert Einstein studied the same process and provided a way to mathematically describe the movement of the particle. Einstein proved that the movement was attributed to the individual water molecules bombarding the speck of pollen. Additionally, he provided mathematics to show that the distance the speck travels from its origin scales at the square root of the time that has elapsed. In honor of Robert Brown, a process which moves in a statistically similar way to the speck of pollen is called Brownian motion. To this day, financial market theory is based on the work provided by Bachelier and Einstein in the early 1900s. The distance the speck of pollen travels is comparable to the volatility of a financial asset, wherein the price range from the minimal value to the maximum value of a mean adjusted cumulative deviation series within a given length of time should scale with the square root of the length of that time period. In other words, the longer the time period, the larger the expected price range. A proper assessment of risk is necessary, as it is the basis for valuing financial assets and constructing efficient portfolios. A Brownian motion’s movement is described by a Standard Normal Distribution. This famous distribution conveniently describes many complex systems. Therefore, it is understandable that Bachelier would assume the stock market could be described by this distribution. This is, however, a misclassification. Einstein proved the distance a Brownian motion process travels from its origin scales at the square root of time; however, it is commonly shown that a stock market’s variability will scale faster than the square root of time. Edgar E. Peters provides excellent empirical evidence of the true scaling rate in Fractal Market Analysis. This is to say that the market is more volatile than a Brownian motion suggests. Volatility has long been considered the essence of risk, and without a proper assessment of risk, financial instruments cannot be valued properly and investors will be blindsided by ruinous events in the market. It is much like a theory of sea waves that forbids their swells to exceed six feet. – Mandelbrot, Benoit. How Fractals Can Explain What’s Wrong with Wall Street. Benoit B. Mandelbrot Benoit Mandelbrot recognized the flaws of financial market theory models and attempted to provide an alternative — one which more accurately assessed risk. The model would generate a realistic time series with large swings in price and clusters of volatility. These were not accounted for by existing models. Mandelbrot’s contributions to the mathematical community are vast and impactful, but perhaps his most notable contribution is his work on fractal geometry. A fractal, which is more formally described in the next sections, has a common characteristic: the structure is random locally and deterministic globally. Edgar E. Peters provides the example in Fractal Market Analysis of mammalian lung bronchi. The branching of bronchi at a microscopic level is random, but the global structure of the lungs is almost always deterministic. Similarly, the micro-movements of a stock appear random within short spans of time, but there is a more deterministic structure globally. Mandelbrot believed a simple fractal shape could explain the complex structure of the market. With apologies to him and his colleagues, this article attempts to describe his greatest method for emulating a financial market: Brownian Motion in Multifractal Time. A Fractured Market A fractal is a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole. – Mandelbrot, Benoit. Scaling in Financial Prices: III. Cartoon Brownian motions in multifractal time. Upon observing that the market looks similar at all scales, he was confident that a proper fractal could simulate the market and capture its key attributes: large swings and clustered volatility. Fractal Price Generator Figure 1: Initiator and Generator of the financial fractal The root of a fractal is an initiator and a generator. In the case of Mandelbrot’s financial fractal, which is represented in Figure 1, the initiator is a straight line representing the price change, and the generator is a lightning bolt shape with two turning points and proceeds in an up-down-up fashion. The up-down-up procession allows for the up and down periods in the market. Each of the three straight line segments of the generator becomes the next iteration of initiators. Through infinite iteration, the fractal builds. Figure 2 iterates this process four times. Figure 2: Iterating a fractal price generator four times The model can be extended by randomizing each of the three segments of the generator as shown in Figure 3. Figure 3: Randomizing segments of the fractal price generator There is special generator for a Brownian motion process — the process from which financial theory builds. It is symmetric with the first turning point at y=2/3 and x=4/9. Iterate the Brownian motion generator such that each of the three segments is the same proportional width and height and the process will always be Brownian motion. Iterating the Brownian motion generator creates a visually realistic picture of the market, as seen in Figure 4. Figure 4: Brownian motion time series This is a pretty picture which may pass a visual test, but the statistical properties of the Brownian motion generator are starkly different than that of a true market. It is often easier to look at the price differences between moments than the price chart when determining the fitness of the model. For example, Figure 4 passes the visual test of what a market’s price chart looks like, but it is visually clear that its moment to moment price changes, as represented in Figure 5, do not represent that of a true market (see Figure 6). The moment to moment price changes of the Brownian motion, represented in Figure 5, spread uniformly throughout time and the amplitude of very few changes fall out of the norm. Figure 5: Brownian motion (white noise) volatility A true market’s moment to moment price changes will look like Figure 6. This figure displays clusters of volatility, unlike the uniform volatility in Figure 5. In Figure 6 it is more likely for a
