Discussion of Mandelbrot Themes: Alpha Tail Index and Scaling H

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Discussion of Mandelbrot Themes Alpha (Tail
Index) and Scaling (H)
Implications for Option Pricing and Risk
Management Market Implied Tail Index and Scaling
Value At Risk

• Prepared by Sheri Markose, Amadeo Alentorn and
  Vikentia Provizionatou
• WEHIA 2005

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PART 1 Market Implied Tail Index and Option
Pricing with the GEV distribution
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Objectives

• To use the Generalized Extreme Value (GEV)
  distribution in a new option pricing model to
• Remove pricing biases associated with
  Black-Scholes
• Capture the stylized facts of the price implied
  Risk Nuetral Density Function(RND)
• Left skewness
• Excess kurtosis (fat tail)
• Obtain a closed form solution for the European
  option price
• Extract the market implied tail index for asset
  returns

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The GEV distribution

• The standardized GEV distribution is given by
• where
• µ is the location parameter
• s is the scale parameter
• ? is the shape parameter

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The GEV for different values of ?
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GEV based density functions
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The call option closed form solution

• The closed form solution of the call option
  pricing equation under GEV returns is
• where
• We obtain a similar equation for put options.

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Methodology of RND estimation

• For a given day, we have a set of N traded option
  prices with the same maturity, but different
  strikes.
• We use a non-linear least squares algorithm to
  find the set of parameters that minimize the sum
  of squared errors

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Results Pricing bias (90 days)

• The GEV model removes the pricing bias
• (Price bias Market price Calculated price )

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Results Pricing bias (10 days)

• For short time horizons, both models improve, but
  the GEV model has a smaller error.
• (Price bias Market price Calculated price )

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Results Implied tail index

• Shape parameter ? for put options from GEV
  returns 1997 2003

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RNDs before and after 9/11 events
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Conclusions

• Modelling negative returns with the GEV yields an
  accurate option pricing model, which removes the
  pricing biases of the Black-Scholes model.
• Implied RNDs and the implied tail index reflect
  the market sentiment of increased probability of
  downward moves, specially after crisis events,
  but do not predict them.
• Future work will consist on calculating Economic
  Value At Risk from the GEV based RNDs, and
  assessing the hedging performance of the model.

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PART 2 Scaling Value At Risk
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Scaling VaR for regulatory purposes

• The regulatory standard involves reporting the
  10-day Value-at-Risk at 99 per cent confidence
  level on trading portfolios of banks.
• The current common practice is to use the
  daily-VaR, routinely calculated using the banks
  internal models, and scale it up to the 10-day
  VaR using the square-root-of-time rule. The
  latter, which is appropriate for Gaussian
  distributions, has been criticized on the grounds
  that asset returns data is far from Gaussian.

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Scaling and self-similarity

• Self-similarity refers to the property that the
  increments of X at scale tau kv has the same
  distribution as any other increment under
  appropriate rescaling.
• A stochastic process is self-similar, if there
  exists H ? 0, such that for any k gt0,
  
• H is referred to as the scaling exponent, though
  for historical reasons it is also called the
  Hurst coefficient.  

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Self-similarity and VaR

• VaRq is the qquantile of the portfolio return
  distribution, and the scaling law that applies to
  the distribution of returns F(Rk) also applies to
  the q-quantile.
• From this it follows that the scaling exponent is

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Empirical scaling for VaR

• The first empirical scaling rule (Hest) assumes a
  pseudo scale invariant measure of the scale
  exponent, which is derived by the gradient of the
  linear regression of the q-quantile of the
  returns with different holding periods in a
  log-log plot.
• The second empirical scaling rule involves the
  numerical local determination of scale variant
  exponents (Hnum) for the q-quantile one day
  returns and the q-quantile of ngt1 returns.

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The law of risk? Don't bet on itReviewed by
Howard Davies THE (MIS)BEHAVIOUR OF MARKETS A
Fractal View of Risk, Ruin and Reward

• http//www.timesonline.co.uk/article/0,,923-139623
  5,00.html
• .. but the practical implications are
  straightforward. If markets are more volatile
  than we think and the seas are rougher than they
  were, banks need to hold more capital to ensure
  they can survive wild price swings. Regulators
  should require them to hold more, and to test
  their portfolios against the possibility of
  extreme events.

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Empirical scaling estimated exponent

• Estimated scaling exponents for the FTSE-100
  time series 1500-days sample (1998-2002). Results
  of the regression analysis are presented for the
  left tail quantiles ranging from a size of 0.70
  to 0.99.

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Empirical scaling numerical exponent

• Numerical scaling law results for the 0.99 VaR
  (Left Tail)
• of FTSE-100. The data sample is 02-01-86 to
  03-06-02.

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Backtesting results reporting violations

• Average Violations reported for the
  square-root-of-time rule (SQRT) and the scaling
  law exponent (H) at different holding periods (k)
  and VaR (left tail) quantiles

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Backtesting results using charts
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Conclusions

• Data determined scaling exponents are
  time-variant.
• Empirical scaling based on both the Hnum and Hest
  is significantly different than the
  square-root-of-time rule.
• The backtesting shows that the application of the
  empirically determined scaling rules outperforms
  the square-root-of-time rule and leads to a
  significant amount of saving in banks capital.