fractal presentation

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FractalsFractalsA Tribute to Benoît Mandelbrot

Andrew D Smith and Paul Sweeting

© 2010 The Actuarial Profession www.actuaries.org.uk

2011 Life Conference

21st November 2011

Agenda

• Fractals and self-similarity

• Fractal behaviour in financial markets

• Fitting fractals – Hausdorff Dimension

• Fractional Brownian motion

• Lévy stable processes

• Conclusions

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What is a fractal?

• A shape or pattern...

• ...that can be broken down into components...

• ...each of which resemble the whole

• Key property is self-similarity

Self-similarity

• Self similarity can be near-exact– e.g. for a fern...

• ...or more approximate– e.g. for a coastline...

– ...or for financial returns

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Agenda






• Conclusions

How are financial returnsself-similar?

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• Consider the log of equity

• By considering charts using different timescales...

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y g g

• ...we see that they look interchangeable...

• ...but how are they affected by scale?

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Power Laws for Self-Similarity

• Consider the risk (of a change in equities or interest rates) measured over a time interval t

• Traditionally assume the dispersion of the change is proportional to t½, based on Brownian motion

• More generally, can consider power law processes where the dispersion grows like t2-d for some d.

6© 2010 The Actuarial Profession www.actuaries.org.uk

Agenda





• Levy stable processes

• Conclusions

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Measuring the fractional dimension

• The Hausdorff dimension measures the rate at which a measurement increasesincreases...

• ...as the scale of measurement reduces

• Given by d in N= c/rd

Hausdorff dimension for some common shapes

• Straight line: 1

• Curve: >1

• West coast of England: 1.25

• Financial time series?

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Calculating the Hausdorff Dimension for UK Equities

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Agenda






• Conclusions

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Fractional Brownian MotionMandelbrot & van Ness (1968)

Standard Brownian MotionZ G i

Fractional Brownian MotionZ G i• Zero mean Gaussian process

• Var(Xt-Xs) = |t-s|

• Independent increments

• Zero mean Gaussian process

• Var(Xt-Xs) = |t-s|4-2d

• Long-range positive autocorrelation


d = 1.5 d = 1.4

Fractional Brownian MotionPractical Issues

• Permits arbitrage (if you know the dimension d)

• Not a Markov process

– Need to know the entire history of the process to project it

• Retains Gaussian assumption – no fat tails or jumps

• Has not seen much application in serious financial models


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Agenda






• Conclusions

Lévy Stable Processes (Mandelbrot, 1964)Infinite Variance Central Limit Theorem

Standard Brownian Motion Lévy Stable Process• Independent Gaussian

Increments

• Continuous paths

• Independent non-Gaussian increments

• Has jumps


d = 1.5 d = 1.4

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Lévy Stable ProcessesPractical Issues

• Four parameters: location, scale, asymmetry and tail exponent

• Parameter estimation is difficult

– Likelihood function not known on closed form

– Second and higher moments do not exist (apart from Normal)

– Methods using characteristic functions are notoriously unstableunstable

• Infinitely many jumps on any finite interval

• Generally imply the depressing conclusion that extreme events are more probable than previously thought


Agenda






• Conclusions

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Hausdorff Dimension Measuresand Value-at-Risk Time Scaling

HausdorffDimension

Brownian motion d = 1.5

Empirical in range 1.3 to 15. In this table we use d = 1.4

Gross-up for annual VaR from monthly VaR

Sqrt(12) = 3.46 4.44

Gross-up for Sqrt(52) = 7.21 10.71


Gross up for annual VaR from weekly VaR

Sqrt(52) 7.21 10.71

Mandelbrot’s Solutions to Dimension d < 1.5A Summary

Independent Correlated Increments increments

Continuous paths Brownian motion Fractional Brownian motion

Jump processes Lévy Stable processes

Since the 1960’s, many other possible explanations


have emerged to account for scaling properties not being exactly 1.5. These include alternative time series models (time varying drift or volatility) as well as sampling biases in the estimation of the Hausdorff dimension.

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Recent DevelopmentsTempering the Tails of Lévy Processes

• Recent surge in empirical work

• Invention of versions without the fat tails

– KoBoL processes (Koponen, 1995, extended by Boyarchenko & Levendorskiĭ, 2000)

– Independently constructed by Carr, Madan, Géman & Yor, (2002) and also known as the CGMY model.

• Retains fractional dimension but a change of measure is• Retains fractional dimension but a change of measure is needed to construct the scaling property.

• Extension to stochastic volatility models (Barndorff-Nielsen & Shephard, 2006)


Using fractals for risk management

• Applying fractals to one financial series is straightforward...

• ...but most investment risk relates to portfolios of assets

• Two approaches could be used

– Fit a fractal to the historical performance of the portfolio

– Fit a fractal to independent factors of the returns using factor analysis

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Questions or comments?

Expressions of individual views by members of The Actuarial Profession and its staff are encouraged.

The views expressed in this presentation are those of the presenter.


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