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 of the paper

• 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
  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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Density functions for GEV returns
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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 t kv has the same
  distribution as any other increment t under
  appropriate rescaling.
• A stochastic process is self-similar, if there
  exists such that for any ,
  
• 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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Empirical scaling estimated exponent

• Estimated scaling exponents for the FTSE-100
  time series at the 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.