# Mandelbrot Fractals applied in Eco-Finance, a synthesis.

Until the late 1990s, the Markowitz model (1954) and the theories of perfectly efficient markets (1960) with portfolios ranging from risk-free diversification to volatile assets were consistent with linear stock prices, simply normalizing them in Black-Scholes-Merton pricing (1973). Since 1990, stock market movements have been represented in a chaotic (1963), fractal (1963), non-linear world, in exponential terms of the law scale discounted at each volatility node. Wild Market Turbulence (mandelbrot fractals).

Fractal movements within or outside the range of fluctuations correspond to the historical phases of globalization of international economies and the”3D” of globalization (Disintermediation, Decompartmentalization, Deregulation) and visible on the S&P, DJIA, Russell2000, Nasdaq, CAC40, DAX, Footsie, Nikkei, etc. stock indices.

Referring to the recent 2013 Nobel Prizes in Economics, there are three contemporary “grandes écoles” on fractals:

– “Chicago School”, IL, USA

M. Friedman, F. Black-M. Scholes-R. Merton, H. Markovitz, B. Malkiel, E. Fama,

L. Hansen, W. Sharpe

Continuous Efficient Markets Model in Pure and Perfect Competition

– “Harvard School”, MA, USA

L. Bachelier, P. Samuelson, J. Simons, Gordon-Shapiro, I. Fisher, N. N. Taleb

Possible discontinuous gap model in monopolistic competition

– “Yale School”, CT, USA

B. Mandelbrot, R. Shiller, F. Modigliani, F. Hayek, J. Schumpeter, R. Dalio

Fractal model complementary to the Harvard model. And the school model

in Chicago is irrational about modern stock prices.

The Monte-Carlo modeling corrected for Fractals with p= α the Hurst Coefficient H and (1-p)= 1-α or (1-p)= 1-H, or explicitly Lévy’s law L-stable, allows in probabilistic mathematics to demonstrate academically the difference of these schools.

The exponential used in Monte-Carlo is the continuously discounted rate of return of future r interest rates or discounted Cash Flows in discounted future pay-offs; compared to the discrete discounting of the Gordon-Shapiro and Miller-Modigliani model.

Thus the growth of pay-offs in a probability of p= 0.6 independent data on a Monte-Carlo model of thousands (5000) of CDO situations, ABS for example normalized and securitized in a pool of similar credit connections says average identical in this p= 0.6 by these calculations of correlations between these 5000 similar cases.

If the “Chicago School” was sufficient and fairly effective in explaining the periods of DJIA stock prices in stable range, for example 1814-1834, 1844-1851, 1863-1925 or 1961-1982; the “Harvard School” will explain the inconsistencies and limitations of the “Chicago School” effectively in the 1789-1818, 1818-1844 period ranges,

of 1834-1843, 1937-1961 or 1988-2000; and finally, the “Yale School” will further demonstrate the inconsistencies and limitations of the “Chicago School” effectively in the period ranges of 1851-1863, 1925-1937, 1982-1988 or 2000 to 2014.

As the aggregate money supply of the National Central Banks, USFed, ECB, Bank of Japan BoJ, Bank of England BoE or Bank of China BoC, from the most liquid or less liquid, M1 would be “the Chicago school”, M2 would be M1 + “the Harvard school” and M3 would be M2 + “the Yale school”.

To return to the Monte-Carlo illustration and the logical link with this Dow Jones DJIA chart, if the Samuelson, Black-Scholes-Merton, Markowitz, Malkiel, Fama, “Chicago School” model determines a perfect random walk (Play on Wall Street between a monkey and a pool of confirmed market operators by choosing performance stocks) by using the Monte-Carlo tree structure at p= 1/2 and 1-p= 1/2, that is, the independence of continuous market asset prices.

With the Mandelbrot Fractals.

If H=α= 1/2 = β, “Chicago School” and “Harvard School” match perfectly

to the Monte Carlo scenarios:

– Differential calculations: surface area of population density; Brownian motion,

stochastic; regularity in irregularity: invariance

→ Self-affining fractals

If H=α≠ 1/2 = β, “Harvard School” corresponds with discrete discontinuous gaps in the

Monte-Carlo scenarios:

Endomorphimes, isomorphisms, nucleus and surface transposed to the same

characteristics; roughness: abstraction of dimensions, dim 3, dim 4; reduction

of size without deformation

→ Similar fractals

If H=α≠ 1/2 ≠ β, “Yale School” corresponds discontinuously to the Monte-

Carlo:

– Ellipse, Garch, Chi-Garch, Cauchy, ARMA, Koch; econometric aid: diagram

iterative in chaos

→ Random and multifractal fractals

Mandelbrot’s fractals also concluded and distinguished the standardized thick distribution tails into a uniform law (from 96%) with an extremely rare appearance of 1/1million to 1/1billion years that the “Harvard School” had also already succeeded in demonstrating and imposed as an academic standard:

•Power law, bronwian sequelae, by Cauchy, by Lévy

– Law of scale, economies of scale

– Market bias finance, discontinuous jumps

– Brownian economic and political cycles

→ Noah’s Effect (work of the financial analyst and hedger) & Joseph’s Effect

(work of the trader, market maker, trader and trader in Forex)

As a general conclusion, we can remember Benoît Mandelbrot’s book (1924-

2010) the Trading Plan of the Mandelbrot Fractals as follows:

– 1. Fractals are turbulent movements

– • 2. Financial markets are very risky

– • 3. Market timing in small intervals

– • 4. Market prices jump, they don’t slip

– • 5. Time, time series, is flexible

– • 6. Markets operate identically

– • 7. Uncertain markets and inevitable bubbles

– • 8. Financial markets are misleading, no gold average

– • 9. Estimate the probabilities of future volatilities

– • 10. The notion of Value has a limited value