[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/p/sce/scecf5/397.html
   My bibliography  Save this paper

Option Pricing and the Implied Tail Index with the Generalized Extreme Value (GEV) Distribution

Author

Listed:
  • Sheri Markose
  • Amadeo Alentorn
Abstract
The 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 – 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

Suggested Citation

  • Sheri Markose & Amadeo Alentorn, 2005. "Option Pricing and the Implied Tail Index with the Generalized Extreme Value (GEV) Distribution," Computing in Economics and Finance 2005 397, Society for Computational Economics.
    • Handle: RePEc:sce:scecf5:397
    as

    Download full text from publisher

    File URL: http://privatewww.essex.ac.uk/~aalent/GEV/AmadeoAlentorn_GEV_Option_Pricing.pdf
    File Function: main text
    Download Restriction: no
    File URL: http://repec.org/sce2005/up.727.1107213766.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. 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.).
    3. 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.
    4. de Jong, C.M. & Huisman, R., 2000. "From Skews to a Skewed-t," ERIM Report Series Research in Management 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 University Rotterdam.
    5. C. J. Corrado & Tie Su, 1997. "Implied volatility skews and stock return skewness and kurtosis implied by stock option prices," The European Journal of Finance, Taylor & Francis Journals, vol. 3(1), pages 73-85, March.
    6. Bali, Turan G., 2003. "The generalized extreme value distribution," Economics Letters, Elsevier, vol. 79(3), pages 423-427, June.
    7. Robert Savickas, 2002. "A Simple Option‐Pricing Formula," The Financial Review, Eastern Finance Association, vol. 37(2), pages 207-226, May.
    8. 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 621-651, October.
    9. 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(1), pages 143-159, March.
    10. 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.
    11. Karim Abadir & Michael Rockinger, "undated". "Density-Embedding Functions," Discussion Papers 97/16, Department of Economics, University of York.
    12. Jackwerth, Jens Carsten, 1999. "Option Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review," MPRA Paper 11634, University Library of Munich, Germany.
    13. Paul Embrechts & Sidney Resnick & Gennady Samorodnitsky, 1999. "Extreme Value Theory as a Risk Management Tool," North American Actuarial Journal, Taylor & Francis Journals, vol. 3(2), pages 30-41.
    14. 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 285-296, Winter.
    15. 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 637-654, May-June.
    16. 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.
    17. Robert JARROW & Andrew RUDD, 2008. "Approximate Option Valuation For Arbitrary Stochastic Processes," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 1, pages 9-31, World Scientific Publishing Co. Pte. Ltd..
    18. Casper De Vries & Jon Danielsson & Casper G, de Vries, 1996. "Tail Index and Quantile Estimation with Very High Frequency Data," CESifo Working Paper Series 116, CESifo.
    19. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    20. repec:bla:jfinan:v:53:y:1998:i:2:p:499-547 is not listed on IDEAS
    21. 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.
    22. Jackwerth, Jens Carsten & Rubinstein, Mark, 1996. "Recovering Probability Distributions from Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1611-1632, December.
    23. Carmela Quintos & Zhenhong Fan & Peter C. B. Phillips, 2001. "Structural Change Tests in Tail Behaviour and the Asian Crisis," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 68(3), pages 633-663.
    24. repec:bla:jfinan:v:59:y:2004:i:1:p:407-446 is not listed on IDEAS
    25. Robert J. Ritchey, 1990. "Call Option Valuation For Discrete Normal Mixtures," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 13(4), pages 285-296, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Marco Rocco, 2011. "Extreme value theory for finance: a survey," Questioni di Economia e Finanza (Occasional Papers) 99, Bank of Italy, Economic Research and International Relations Area.
    2. Rui Menezes & Sonia Bentes, 2016. "Hysteresis and Duration Dependence of Financial Crises in the US: Evidence from 1871-2016," Papers 1610.00259, arXiv.org.
    3. Marcos Massaki Abe & Eui Jung Chang & Benjamin Miranda Tabak, 2007. "Forecasting Exchange Rate Density Using Parametric Models: the Case of Brazil," Brazilian Review of Finance, Brazilian Society of Finance, vol. 5(1), pages 29-39.
    4. Bari, Chintaman Santosh & Chandra, Satish & Dhamaniya, Ashish, 2022. "Service headway distribution analysis of FASTag lanes under mixed traffic conditions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    5. Frank Fabozzi & Radu Tunaru & George Albota, 2009. "Estimating risk-neutral density with parametric models in interest rate markets," Quantitative Finance, Taylor & Francis Journals, vol. 9(1), pages 55-70.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Markose, Sheri M & Alentorn, Amadeo, 2005. "The Generalized Extreme Value (GEV) Distribution, Implied Tail Index and Option Pricing," Economics Discussion Papers 3726, University of Essex, Department of Economics.
    2. Christoffersen, Peter & Jacobs, Kris & Chang, Bo Young, 2013. "Forecasting with Option-Implied Information," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 581-656, Elsevier.
    3. Bondarenko, Oleg, 2003. "Estimation of risk-neutral densities using positive convolution approximation," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 85-112.
    4. Jin Zhang & Yi Xiang, 2008. "The implied volatility smirk," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 263-284.
    5. Shi-jie Jiang & Mujun Lei & Cheng-Huang Chung, 2018. "An Improvement of Gain-Loss Price Bounds on Options Based on Binomial Tree and Market-Implied Risk-Neutral Distribution," Sustainability, MDPI, vol. 10(6), pages 1-17, June.
    6. Lina M. Cortés & Javier Perote & Andrés Mora-Valencia, 2017. "Implicit probability distribution for WTI options: The Black Scholes vs. the semi-nonparametric approach," Documentos de Trabajo de Valor Público 15923, Universidad EAFIT.
    7. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    8. Vahamaa, Sami, 2005. "Option-implied asymmetries in bond market expectations around monetary policy actions of the ECB," Journal of Economics and Business, Elsevier, vol. 57(1), pages 23-38.
    9. José L. Vilar-Zanón & Olivia Peraita-Ezcurra, 2019. "A linear goal programming method to recover risk neutral probabilities from options prices by maximum entropy," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 259-276, June.
    10. Li, Yifan & Nolte, Ingmar & Pham, Manh Cuong, 2024. "Parametric risk-neutral density estimation via finite lognormal-Weibull mixtures," Journal of Econometrics, Elsevier, vol. 241(2).
    11. Wilkens, Sascha & Roder, Klaus, 2006. "The informational content of option-implied distributions: Evidence from the Eurex index and interest rate futures options market," Global Finance Journal, Elsevier, vol. 17(1), pages 50-74, September.
    12. J. A. Jiménez & V. Arunachalam & G. M. Serna, 2015. "Option Pricing Based On A Log–Skew–Normal Mixture," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(08), pages 1-22, December.
    13. Coutant, Sophie & Jondeau, Eric & Rockinger, Michael, 2001. "Reading PIBOR futures options smiles: The 1997 snap election," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1957-1987, November.
    14. Bogdan Negrea & Bertrand Maillet & Emmanuel Jurczenko, 2002. "Revisited Multi-moment Approximate Option," FMG Discussion Papers dp430, Financial Markets Group.
    15. Cortés, Lina M. & Mora-Valencia, Andrés & Perote, Javier, 2020. "Retrieving the implicit risk neutral density of WTI options with a semi-nonparametric approach," The North American Journal of Economics and Finance, Elsevier, vol. 54(C).
    16. Monteiro, Ana Margarida & Tutuncu, Reha H. & Vicente, Luis N., 2008. "Recovering risk-neutral probability density functions from options prices using cubic splines and ensuring nonnegativity," European Journal of Operational Research, Elsevier, vol. 187(2), pages 525-542, June.
    17. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
    18. Brisset, Nicolas, 2017. "On Performativity: Option Theory And The Resistance Of Financial Phenomena," Journal of the History of Economic Thought, Cambridge University Press, vol. 39(4), pages 549-569, December.
    19. León, à ngel & Mencía, Javier & Sentana, Enrique, 2009. "Parametric Properties of Semi-Nonparametric Distributions, with Applications to Option Valuation," Journal of Business & Economic Statistics, American Statistical Association, vol. 27(2), pages 176-192.
    20. Marie Briere, 2006. "Market Reactions to Central Bank Communication Policies :Reading Interest Rate Options Smiles," Working Papers CEB 38, ULB -- Universite Libre de Bruxelles.

    More about this item

    Keywords

    Risk neutral probability density function; Generalized Extreme Value Distribution; Implied Tail Index.;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:sce:scecf5:397. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Christopher F. Baum (email available below). General contact details of provider: https://edirc.repec.org/data/sceeeea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.