University of Pisa Faculty of Economics

Bank, Stock Exchange and Insurance2009/2010

Fractal Finance applied to financial

markets Candidate

SupervisorPierpaolo Cassese Ch.mo

Prof. Piero Bellandi

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Summary

The fractal structure of equity market

Hurst’s ratio and R/S AnalysisResults Building an investment strategy

using Elliott’s Theory Concluding Remarks

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The fractal structure of equity market

• Finance proposed a big contribute for developing the fractal theory such as forecast form of risk.

• Bachelier’s brownian motion as cornerstone of fractal theory.

• The Bachelier’s theory maintained which the series of return was i.i.d and plotted with a bell shape.

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Bachelier Theory

Is it possible to define normal and independent the asset returns?

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Bachelier Theory

Is the volatility zero ?

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The Fractal Theory

• Lévy Stable’s distribution

• Long Dependence or Fractional Brownian Motion

• Multifractal Process with multiplicative cascade or trading time

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Lévy distribution“The variation of certain of speculative prices” (1963)

• Mandelbrot has considered for his studies the Cauchy’s curve with fat tails

• Cauchy’s function gives to the price variations higher and rarer a major probability of success

• The Cauchy's distribution reflects more risk on the markets

• The Lèvy Stable are characterized by scale invariance

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Lond Dependence e Fractional brownian motion“Multifractal Model of Asset Returns” (1997)

• The natural phenomena do not trace a random walk as Bachelier argued.

• It exists a long term correlation which slowdown in the future that it influences the movement of future price

• The variation of stock price doesn’t appare in indipendent manner towards previous fluctuations

• MFB has a self similarity characteristic

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The Hurst’s Ratio and the R/S Analysis

• In 1951 Hurst sought some phenomena with long dependence studing the data of Nilo River overflow

• The result kept by the search has modified the view respect of Random Walk Series (i.i.d)

• Hurst has introduced a costant K which measures the bias of FBM

• in 1968 Mandelbrot has seen in the no-randomized series the requirement of fractality

• Fractal dimension D = 2 – H

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Hurst’s Ratio

• for 0 < H < 0.5 the series is non-persistent and the risk of asset becomes strong.

• for H = 0.5 the historical series plots a path defined Random Walk which it is indipendent by the previous phenomena verified

• For 0.5 <H <1 the series is called persistent or historical events influence the future trend of the title.

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Rescaled Range Analysis

It is the first method of analysis which it uses a non parametric test where the unique variables gove from X series and its standard deviation.

the R/S formula misures simply if, within a different lenght range, the different between max an min values is superior or not towards it that forecasting when each data is indipendent from the data estimated.

It’s useful to compute if a data series presents a long term dependence.

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Results of research conducted on 4 assets listed on FTSE Mib

Results of R/S Analysis for Eni

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Results of R/S Analysis for Fiat

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Results of R/S Analysis for StMicroelectronics

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Results of R/S Analysis for Unicredit

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Building an investment strategy using Elliott’s Theory

• Elliott’s Theory like fractal application to financial markets

• How to foresee the cycle of stock across the Waves Theory

• Alternating phases bullish or bearish signals intended as input or output from the market

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Building an investment strategy using Elliott’s Theory

Case Study : Unicreditasset UNICREDIT

ENTRY PRICE

7/7/2010

1,80 €

STOP LOSS 1,75 €

TARGET PROFIT

14/07/2010

2,11 €

BUDGET 10.000 €

N°of share UNICREDIT

5550

Tot.cost 9990,00 €

GAIN %

GUADAGNO POT.

16.7%

1650 €

LOSS%

PERDITA POT.

2,7%

277,50 €

ENTRY PRICE

20/7/2010

2 €

STOP LOSS 1,95 €

TARGET PROFIT 2,4 €

Bid

5/8/2010 2,20 €

GAIN % 10%

LOSS % 2,3%

Expected profit

1210 €

Expected loss 275 €

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Concluding Remarks

• The theory of efficient markets is eclipsed by the fractal theory and analytical models as predictive analysis technique.

• Given the values of Hurst ratio of the titles studied, other than 0.5 can be clearly demonstrated that the stock market has a fractal structure. For low values of H has been able to show how the disorder has a high number, the more random pattern and much more volatile, while high values of H in the series show a low noise, less volatile and therefore less risky.

• Possibility of adopting graphical analysis with the intention of building trading systems using Elliott Wave and Fibonacci oscillators as well as to identify the cycles in the prices of financial instruments and for the study of stochastic volatility.