Discussion of Mandelbrot Themes: Alpha (Tail Index) and Scaling (H) Prepared by Sheri Markose, Amadeo Alentorn and Vikentia Provizionatou WEHIA 2005 Implications for Option Pricing and Risk Management: Market Implied Tail Index and Scaling Value At Risk

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

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

The GEV distribution The standardized GEV distribution is given by: where: – μ is the location parameter – σ is the scale parameter – ξ is the shape parameter

The GEV for different values of ξ

Density functions for GEV returns

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.

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:

Results: Pricing bias (90 days) The GEV model removes the pricing bias (Price bias = Market price – Calculated price )

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 )

Results: Implied tail index Shape parameter ξ for put options from GEV returns 1997 – 2003

RNDs before and after 9/11 events

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.

PART 2: Scaling Value At Risk

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.

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.

Self-similarity and VaR VaR q is the q–quantile of the portfolio return distribution, and the scaling law that applies to the distribution of returns F(R k ) also applies to the q- quantile. From this it follows that the scaling exponent is:

Empirical scaling for VaR The first empirical scaling rule (H est ) 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 (H num ) for the q-quantile one day returns and the q-quantile of n>1 returns.

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

Empirical scaling numerical exponent Numerical scaling law results for the 0.99 VaR (Left Tail) of FTSE-100. The data sample is to

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

Backtesting results: using charts

Conclusions Data determined scaling exponents are time- variant. Empirical scaling based on both the H num and H est 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.