The Generalized Extreme Value (GEV) Distribution, Implied Tail Index and Option Pricing

Black and Scholes assumed the returns for an option’s underlying stock followed a lognormal diffusion, which produces a lognormal probability density for the stock price at option expiration. This was mathematically very convenient but, unfortunately, not very satisfactory empirically. Observed return distributions for actual stocks, stock indexes, and most other kinds of assets exhibit distinctly non-lognormal shapes, with negative skewness and fat tails. Numerous non- Black–Scholes pricing models have been developed to try to capture these important features of real-world returns. Here, Markose and Alentorn propose the use of the generalized extreme value (GEV) distribution in place of the lognormal. Tail shape in the GEV distribution is governed by a single parameter, and it can be proven that the extreme tails of any plausible return density resemble those from a GEV. The authors develop a pricing model under GEV returns, whose form is very like Black–Scholes, along with equations for its Greek letter risks. They then demonstrate its superior ability to fit FTSE option prices from 1997 to 2009 and explore its behavior around critical market events, such as the 1997 Asian crisis, the 9/11 attack in 2001, and the collapse of Lehman Brothers in 2008.

[1]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  A. Jenkinson.  The frequency distribution of the annual maximum (or minimum) values of meteorological elements , 1955 .

[3]  B. Mandlebrot.  The Variation of Certain Speculative Prices , 1963 .

[4]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[5]  Douglas T. Breeden,et al.  Prices of State-Contingent Claims Implicit in Option Prices , 1978 .

[6]  M. Rubinstein..  Nonparametric tests of alternative option pricing models using all reported trades and quotes on the , 1985 .

[7]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[8]  R. Jarrow,et al.  APPROXIMATE OPTION VALUATION FOR ARBITRARY STOCHASTIC PROCESSES , 1982 .

[9]  R. J. Ritchey,et al.  CALL OPTION VALUATION FOR DISCRETE NORMAL MIXTURES , 1990 .

[10]  E. Ziegel.  Extreme Value Theory and Applications , 1994 .

[11]  Laurens de Haan,et al.  Safety First Portfolio Selection, Extreme Value Theory and Long Run Asset Risks , 1994 .

[12]  Andrew W. Lo,et al.  Nonparametric estimation of state-price densities implicit in financial asset prices , 1995, Proceedings of 1995 Conference on Computational Intelligence for Financial Engineering (CIFEr).

[13]  William R. Melick,et al.  Using options prices to infer PDF'S for asset prices: an application to oil prices during the Gulf crisis , 1996 .

[14]  Tie Su,et al.  SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES , 1996 .

[15]  Salih N. Neftçi,et al.  An Introduction to the Mathematics of Financial Derivatives , 1996 .

[16]  M. Rubinstein.,et al.  Recovering Probability Distributions from Option Prices , 1996 .

[17]  P. Buchen,et al.  The Maximum Entropy Distribution of an Asset Inferred from Option Prices , 1996, Journal of Financial and Quantitative Analysis.

[18]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[19]  Tie Su,et al.  Implied Volatility Skews and Stock Index Skewness and Kurtosis Implied by S&P 500 Index Option Prices , 1997 .

[20]  Michael Rockinger,et al.  Density-embedding Functions , 1997 .

[21]  Jón Dańıelsson,et al.  Tail Index and Quantile Estimation with Very High Frequency Data , 1997 .

[22]  William R. Melick,et al.  Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis , 1997, Journal of Financial and Quantitative Analysis.

[23]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[24]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[25]  William R. Melick.  Results of the Estimation of Implied PDFs from a Common Dataset , 1999 .

[26]  Nikolaos Panigirtzoglou,et al.  Testing the Stability of Implied Probability Density Functions , 2002 .

[27]  Didier Sornette,et al.  On Rational Bubbles and Fat Tails , 1999, cond-mat/9910141.

[28]  A. McNeil.  Extreme Value Theory for Risk Managers , 1999 .

[29]  Carmela Quintos,et al.  Structural Change Tests in Tail Behaviour and the Asian Crisis , 1999 .

[30]  Confidence Intervals and ConstantMaturity Series for Probability Measures Extracted from Options Prices , 1999 .

[31]  Jens Carsten Jackwerth,et al.  Option Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review , 1999 .

[32]  S. Resnick,et al.  Extreme Value Theory as a Risk Management Tool , 1999 .

[33]  R. Savickas.  A Simple Option‐Pricing Formula , 2002 .

[34]  How Useful Are Implied Distributions? Evidence from Stock Index Options , 2000 .

[35]  A. Lo,et al.  Nonparametric Risk Management and Implied Risk Aversion , 2000 .

[36]  Paul Embrechts,et al.  Extreme Value Theory: Potential And Limitations As An Integrated Risk Management Tool , 2000 .

[37]  From Skews to a Skewed-t , 2000 .

[38]  C. Corrado.  Option Pricing Based on the Generalized Lambda Distribution , 2001 .

[39]  Ramazan Gençay,et al.  EVIM: A Software Package for Extreme Value Analysis in MATLAB , 2001 .

[40]  Nikolaos Panigirtzoglou,et al.  A new approach to modeling the dynamics of implied distributions: Theory and evidence from the S&P 500 options , 2004 .

[41]  K. Dowd.  Measuring Market Risk , 2002 .

[42]  Katherine Campbell.  Statistical Analysis of Extreme Values , 2002, Technometrics.

[43]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[44]  Stephen Figlewski,et al.  Assessing the Incremental Value of Option Pricing Theory Relative to an Informationally Passive Benchmark , 2002 .

[45]  Turan G. Bali.  The generalized extreme value distribution , 2003 .

[46]  S. Satchell,et al.  Forecasting Volatility in Financial Markets : A Review , 2004 .

[47]  R. Bliss,et al.  Option-Implied Risk Aversion Estimates , 2004 .

[48]  R. Savickas.  EVIDENCE ON DELTA HEDGING AND IMPLIED VOLATILITIES FOR THE BLACK‐SCHOLES, GAMMA, AND WEIBULL OPTION PRICING MODELS , 2005 .

[49]  C. Schwab,et al.  Wavelet Galerkin pricing of American options on Lévy driven assets , 2005 .

[50]  Stephen Figlewski.  Estimating the Implied Risk Neutral Density , 2008 .

[51]  Peter Winker,et al.  Computational methods in financial engineering : essays in honour of Manfred Gilli , 2008 .

[52]  S. Markose,et al.  Generalized Extreme Value Distribution and Extreme Economic Value at Risk (EE-VaR) , 2008 .

[53]  G. Skiadopoulos,et al.  The Informational Content of Financial Options for Quantitative Asset Management: A Review , 2009 .

[54]  Tim Bollerslev,et al.  Volatility and Time Series Econometrics , 2010 .

[55]  M. Rocco.  Extreme Value Theory for Finance: A Survey , 2012 .

[56]  Long Kang,et al.  Volatility and time series econometrics: essays in honor of Robert Engle , 2011 .