Multifractal models of stock market volatility with applications to the leverage effect

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Multi-fractal models of stock market volatility
with applications to the leverage effect

• Zoltán Eisler

Budapest University of Technology and
Economics Budapest, Hungary
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Outline

• Stock price basics
• Stochastic volatility models
• The Lux model
• Parameter fitting and possible extensions

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Notations

• Asset price
• log-price (p(t))
• log-returns (x?t(t))
• This will be the process to model

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How to get started?

• Lots of observations (stylized facts)

• Lots of parameters
• We need a simple model to account for as many as
  possible
• Phenomenological models

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Stochastic volatility models

• Discrete time steps (?t1)
• Two independent terms amplitude (volatility) and
  sign
• The sign is usually modeled as 1 or as N(0,1)
  (no correlation for the direction of the price
  change)

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The Lux model

• The amplitude is a product of independent random
  variables (multipliers)
• Each accounts for volatility on a given time
  scale
• Where the average lifetime of is 2(i-k).

1 Thomas Lux The Multi-Fractal Model of Asset
Returns..., Working paper (2003)
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The Lux model

• The average lifetime of is 2(i-k).
• This corresponds to subordinated time scales in
  autocorrelations
• The distribution of is lognormal with a
  parameter (?)
• If k?8 scale-invariance emerges multi-fractality

1 Thomas Lux The Multi-Fractal Model of Asset
Returns..., Working paper (2003)
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Autocorrelation properties of log-returns

• Multi-fractal property (lack of time scale,
  idealized case)
• The t(q) scaling function can be measured by
  Detrended Fluctuation Analysis (MF-DFA)

• 3 C.-K. Peng, S. V. Buldyrev, S. Havlin, M.
  Simons, H. E. Stanley, A. L. Goldberger Phys.
  Rev. E 49, pp. 1685-1689 (1994)

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• can be used to fit the distribution of
  multipliers (?)

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Fitting the parameters to empirical data

• ? parameter governing multi-fractal
  autocorrelations
• can be fitted through t(q) (analytical formula)
• s standard deviation of price changes
• k number of multipliers
• can be fitted through squared log-return
  autocorrelations (analytical formula known)

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• the effective length of power-law memory

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Fitting the parameters to empirical data

• ? parameter governing multi-fractal
  autocorrelations
• can be fitted through t(q) (analytical formula
  known)
• s standard deviation of price changes
• k number of multipliers
• can be fitted through squared log-return
  autocorrelations (analytical formula known)
• the effective length of power-law memory

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The extended Lux model (1)

• A possible extension of the Lux model
  non-vanishing third moment

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The extended Lux model (1)

• A possible extension of the Lux model
  non-vanishing third moment
• Solution
• skewed sign
• with with probability
  ½e
• and with
  probability ½-e

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The extended Lux model (1)

• A possible extension of the Lux model
  non-vanishing third moment
• Solution
• skewed sign
• time averaged third moment of log-returns

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The extended Lux model (2)

• Another possible extension leverage
  autocorrelations

• 4 Jaume Masoliver, Josep Perello, Int. J. Th.
  App. Fin. 5, pp. 541-562 (2002)

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The extended Lux model (2)

• Another possible extension leverage
  autocorrelations
• As long as sign and amplitude are uncorrelated

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The extended Lux model (2)

• Combination of a trick by Pochart and Bouchaud
  with the skewed sign

• 5 Benoit Pochart, Jean-Philippe Bouchaud,
  arXivcond-mat/0204047 (2002)

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The full set of parameters

• s standard deviation of price changes
• ? parameter governing multi-fractal
  autocorrelations
• k number of multipliers
• e skewness parameter
• K(t) leverage autocorrelations

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Why do we care?

• We have to have predictions, so why not have the
  best
• These models contain much arbitrariness, but they
  are computationally efficient
• Limitation underestimates fluctuations

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Why do we care?

• We have to have predictions, so why not have the
  best
• These models contain much arbitrariness, but they
  are computationally efficient
• Limitation underestimates fluctuations
• Applications
• option pricing (asymmetric smiles)
• volatility forecasting

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Thank you!