Actuarial Applications of Multifractal Modeling

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Actuarial Applications of Multifractal Modeling

• Part IITime Series ApplicationsYakov Lantsman,
  Ph.D.NetRisk, Inc.

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Financial Time Series Existing Solutions

• Modeling financial time series are based on
  assumptions of Markov chain stochastic processes
  (rejection of long-term correlation).
• Efficient Market Hypothesis (EMH) and Capital
  Assets Pricing Model (CAPM).
• Lognormal distribution framework is prevailing to
  model uncertainty.
• Existing models possess large set of parameters
  (ARIMA, GARCH) which contribute to high degree of
  instability and uncertainty of conclusions.

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Financial Time Series Proposed Approach

• Multifractal modeling framework to model
  financial time series interest rate, CPI,
  exchange rate, etc.
• Multiplicative Levy cascade as a mechanism to
  simulate multifractal fields.
• Application of Extreme Value Theory (EVT) to
  model probabilities of extreme events.

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Some References on Multifractal Modeling

• Multifractal Analysis of Foreign Exchange Data,
  Schmitt, Schertzer, Lovejoy.
• Multifractality of Deutschemark / US Dollar
  Exchange Rates, Fisher, Calvet, Mandelbrot.
• Multifractal Model of Asset Returns, Mandelbrot,
  Fisher, Calvet.
• Volatilities of Different Time Resolutions,
  Muller, et al.
• Chaotic Analysis on US Treasury Interest Rates,
  Craighead
• Temperature Fluctuations, Schmitt, et al.

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Financial Time Series Modeling Hierarchy

• Continuous time diffusion models
• one-factor (Cox, Ingersoll and Ross)
• multi-factor (Andersen and Lund)
• Discrete time series analysis
• ARIMA
• GARCH
• ARFIMA, HARCH (Heterogeneous)
• MMAR (Multifractal Model of Asset Return).

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Financial Time Series MMAR

• Information contained in the data at different
  time scales can identify a model.
• Reliance upon a single scale leads to
  inefficiency and forecasts that vary with the
  time-scale of the chosen data.
• Multifractal processes will be defined by a
  restrictions on the behavior in their moments as
  the time-scale of observation changes.

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Three Pillars of MMAR

• MMAR incorporates long (hyperbolic) tail, but not
  necessarily imply an infinite variance (additive
  Levy models)
• Long-dependence, the characteristic feature of
  fractional Brownian motion (FBM)
• Concept of trading time that is the cumulative
  distribution function of multifractal measure.

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MMAR Definition

• P(t) 0 ? t ? T price of asset and
  X(t)Ln(P(t)/P(0))
• Assumption
• X(t) is a compound process X(t) ? BH ? (t), BH
  (t) is FBM with index H, and ? (t) stochastic
  trading time
• ? (t) is a multifractal process with continuous,
  non-decreasing paths and stationary increments
  satisfies
• BH (t) and ? (t) are independent.
• Theorem
• X(t) is multifractal with scaling function ?X (q)
  ? ?? (Hq) and stationary increments.

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MMAR Statistical Properties (Structure Function)

• Self-Similarity
• Universality
• Link to Power Spectrum

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Q-Q Plots for Error Term Distributions
Treasury Yields (Normal)
Industrial B1 Bond Yields (Normal)
Industrial B1 Bond Yields (t-distribution)
Treasury Yields (t-distribution)
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Interest Rate Modeling
3-month Treasury Bill Rate (weekly observations)
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Interest Rate Modeling

• K(q) function

log-log plot of power spectrum function
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Exchange Rate Modeling
/DM spot rate (weekly observations)
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Exchange Rate Modeling
log-log plot of power spectrum function

• K(q) function

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Actuarial Applications of Multifractal Modeling

• Part IITime Series ApplicationsYakov Lantsman,
  Ph.D.NetRisk, Inc.