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Option Pricing with Normal Mixture Returns: Modelling Excess Kurtosis and Uncertanity in Volatility

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  • Carol Alexander

    (ICMA Centre, University of Reading)

  • Sujit Narayanan

    (ICMA Centre, University of Reading)

Abstract

his paper addresses the problem of uncertainty in volatility, and how this affects option prices. The volatility uncertainty adjustment to Black-Scholes option prices is quantified in this paper using a normal mixture model for the distribution of underlying returns, or equivalently, assuming a mixture of lognormal densities for the density of the asset price. The use of a lognormal mixture price process for pricing options is not new (Ritchey, 1990) but the local volatility that should be used in the lognormal mixture price process has only recently been established (Brigo and Mercurio, 2000a, 2001).Â

Suggested Citation

  • Carol Alexander & Sujit Narayanan, 2001. "Option Pricing with Normal Mixture Returns: Modelling Excess Kurtosis and Uncertanity in Volatility," ICMA Centre Discussion Papers in Finance icma-dp2001-10, Henley Business School, University of Reading, revised Dec 2001.
    • Handle: RePEc:rdg:icmadp:icma-dp2001-10
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    File URL: http://www.icmacentre.ac.uk/pdf/discussion/DP2001-10.pdf
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    References listed on IDEAS

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    1. 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.
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    6. 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.
    7. Hsieh, David A., 1988. "The statistical properties of daily foreign exchange rates: 1974-1983," Journal of International Economics, Elsevier, vol. 24(1-2), pages 129-145, February.
    8. Melick, William R. & Thomas, Charles P., 1997. "Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 32(1), pages 91-115, March.
    9. Muller, Ulrich A. & Dacorogna, Michel M. & Olsen, Richard B. & Pictet, Olivier V. & Schwarz, Matthias & Morgenegg, Claude, 1990. "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis," Journal of Banking & Finance, Elsevier, vol. 14(6), pages 1189-1208, December.
    10. 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..
    11. McNeil, Alexander J., 1997. "Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory," ASTIN Bulletin, Cambridge University Press, vol. 27(1), pages 117-137, May.
    12. 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.
    13. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
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    Cited by:

    1. Kole, H.J.W.G. & Koedijk, C.G. & Verbeek, M.J.C.M., 2004. "The effects of systemic crises when investors can be crisis ignorant," ERIM Report Series Research in Management ERS-2004-027-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.
    2. En-Der Su & Feng-Jeng Lin, 2012. "Two-State Volatility Transition Pricing and Hedging of TXO Options," Computational Economics, Springer;Society for Computational Economics, vol. 39(3), pages 259-287, March.
    3. Halperin, Igor, 2022. "Non-equilibrium skewness, market crises, and option pricing: Non-linear Langevin model of markets with supersymmetry," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 594(C).
    4. Damiano Brigo, 2008. "The general mixture-diffusion SDE and its relationship with an uncertain-volatility option model with volatility-asset decorrelation," Papers 0812.4052, arXiv.org.
    5. Fleming, Jeff & Paye, Bradley S., 2011. "High-frequency returns, jumps and the mixture of normals hypothesis," Journal of Econometrics, Elsevier, vol. 160(1), pages 119-128, January.
    6. Igor Halperin, 2020. "Non-Equilibrium Skewness, Market Crises, and Option Pricing: Non-Linear Langevin Model of Markets with Supersymmetry," Papers 2011.01417, arXiv.org, revised Dec 2021.

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