Option Pricing and the Implied Tail Index with the Generalized Extreme Value (GEV) Distribution
AbstractThe 1987 stock market crash, the LTCM debacle, the Asian Crisis, the bursting of the high technology Dot-Com bubble of 2001-2 with 30% losses of equity values, events such as 9/11 and sudden corporate collapses of the magnitude of Enron - have radically changed the view that extreme events have negligible probability. The well known drawback of the Black-Scholes model is that it cannot account for the negative skewness and the excess kurtosis of asset returns. Since the work of Jackwerth and Rubinstein (1996) which demonstrated the discontinuity in the implied skewness and kurtosis across the divide of the 1987 stock market crash - a large literature has developed, which aims to extract the risk neutral probability density function from traded option prices so that the skewness and fat tail properties of the distribution are better captured than in the case of lognormal models. This paper argues that the use of the Generalized Extreme Value Distribution (GEV) for asset returns provides not just a flexible framework that subsumes as special cases a number of classes of distributions that have been assumed to date in more restrictive settings â€“ but also delivers the market implied tail index for the assets returns. Under the postulation of the GEV distribution in the Risk Neutral Density (RND) function for the asset returns, we obtain an original analytical closed form solution for the Harrison and Pliska (1981) no arbitrage equilibrium price for the European call option. The implied GEV parameters and RND are estimated from traded option prices for the period from 1997 to 2003. The pricing performance of the GEV option pricing model is compared to the benchmark Black-Scholes model and found to be superior at all time horizons and at all levels of moneyness. We explain how the implied tail index extracted from traded put prices are efficacious at identifying the fat tailed behaviour of losses or negative returns and hence of the skew in the left tail of the RND function for the underlying price. The GEV implied RNDs before and after special events such as the Asian Crisis, the LTCM crisis and 9/11 are also analyzed
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by Society for Computational Economics in its series Computing in Economics and Finance 2005 with number 397.
Date of creation: 11 Nov 2005
Date of revision:
Risk neutral probability density function; Generalized Extreme Value Distribution; Implied Tail Index.;
Find related papers by JEL classification:
- G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
- G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies
This paper has been announced in the following NEP Reports:
- NEP-ALL-2005-11-19 (All new papers)
- NEP-FIN-2005-11-19 (Finance)
- NEP-FMK-2005-11-19 (Financial Markets)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Jarrow, Robert & Rudd, Andrew, 1982. "Approximate option valuation for arbitrary stochastic processes," Journal of Financial Economics, Elsevier, vol. 10(3), pages 347-369, November.
- Yacine Aït-Sahalia & Andrew W. Lo, 1998.
"Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices,"
Journal of Finance,
American Finance Association, vol. 53(2), pages 499-547, 04.
- Yacine Ait-Sahalia & Andrew W. Lo, 1995. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," NBER Working Papers 5351, National Bureau of Economic Research, Inc.
- Yacine Aït-Sahalia & Andrew W. Lo, . "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," CRSP working papers 332, Center for Research in Security Prices, Graduate School of Business, University of Chicago.
- Karim Abadir & Michael Rockinger, . "Density-Embedding Functions," Discussion Papers 97/16, Department of Economics, University of York.
- Ritchey, Robert J, 1990. "Call Option Valuation for Discrete Normal Mixtures," Journal of Financial Research, Southern Finance Association & Southwestern Finance Association, vol. 13(4), pages 285-96, Winter.
- Jackwerth, Jens Carsten, 1999. "Option Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review," MPRA Paper 11634, University Library of Munich, Germany.
- Robert Savickas, 2002. "A Simple Option-Pricing Formula," The Financial Review, Eastern Finance Association, vol. 37(2), pages 207-226, 05.
- Gençay Ramazan & Selçuk Faruk & Ulugülyagci Abdurrahman, 2001. "EVIM: A Software Package for Extreme Value Analysis in MATLAB," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 5(3), pages 1-29, October.
- Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
- Robert R. Bliss & Nikolaos Panigirtzoglou, 2004. "Option-Implied Risk Aversion Estimates," Journal of Finance, American Finance Association, vol. 59(1), pages 407-446, 02.
- Lux, Thomas & Sornette, Didier, 2002.
"On Rational Bubbles and Fat Tails,"
Journal of Money, Credit and Banking,
Blackwell Publishing, vol. 34(3), pages 589-610, August.
- Quintos, Carmela & Fan, Zhenhong & Phillips, Peter C B, 2001. "Structural Change Tests in Tail Behaviour and the Asian Crisis," Review of Economic Studies, Wiley Blackwell, vol. 68(3), pages 633-63, July.
- Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
- Bali, Turan G., 2003. "The generalized extreme value distribution," Economics Letters, Elsevier, vol. 79(3), pages 423-427, June.
- Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
- Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
- Panigirtzoglou, Nikolaos & Skiadopoulos, George, 2004. "A new approach to modeling the dynamics of implied distributions: Theory and evidence from the S&P 500 options," Journal of Banking & Finance, Elsevier, vol. 28(7), pages 1499-1520, July.
- William R. Melick & Charles P. Thomas, 1996. "Using options prices to infer PDF'S for asset prices: an application to oil prices during the Gulf crisis," International Finance Discussion Papers 541, Board of Governors of the Federal Reserve System (U.S.).
- Jong, C.M. de & Huisman, R., 2000. "From Skews to a Skewed-t," Research Paper ERS-2000-12-F&A, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus Uni.
- Buchen, Peter W. & Kelly, Michael, 1996. "The Maximum Entropy Distribution of an Asset Inferred from Option Prices," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 31(01), pages 143-159, March.
- Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. " Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-32, December.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum).
If references are entirely missing, you can add them using this form.