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Option pricing and ARCH processes

Listed author(s):
  • Zumbach, Gilles
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    Recent progresses in option pricing using ARCH processes for the underlying are summarized. The stylized facts are multiscale heteroscedasticity, fat-tailed distributions, time reversal asymmetry, and leverage. The process equations are based on a finite time increment, relative returns, fat-tailed innovations, and multiscale ARCH volatility. The European option price is the expected payoff in the physical measure P weighted by the change of measure dQ/dP, and an expansion in the process increment δt allows for numerical evaluations. A cross-product decomposition of the implied volatility surface allows to compute efficiently option prices, Greeks, replication cost, replication risk, and real option prices. The theoretical implied volatility surface and the empirical mean surface for options on the SP500 index are in excellent agreement.

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    File URL: http://www.sciencedirect.com/science/article/pii/S154461231200013X
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    Article provided by Elsevier in its journal Finance Research Letters.

    Volume (Year): 9 (2012)
    Issue (Month): 3 ()
    Pages: 144-156

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    Handle: RePEc:eee:finlet:v:9:y:2012:i:3:p:144-156
    DOI: 10.1016/j.frl.2012.01.002
    Contact details of provider: Web page: http://www.elsevier.com/locate/frl

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    1. Gilles Zumbach, 2009. "Time reversal invariance in finance," Quantitative Finance, Taylor & Francis Journals, vol. 9(5), pages 505-515.
    2. Peter Christoffersen & Redouane Elkamhi & Bruno Feunou & Kris Jacobs, 2010. "Option Valuation with Conditional Heteroskedasticity and Nonnormality," Review of Financial Studies, Society for Financial Studies, vol. 23(5), pages 2139-2183.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    4. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. " On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    5. Gilles Zumbach, 2011. "Characterizing heteroskedasticity," Quantitative Finance, Taylor & Francis Journals, vol. 11(9), pages 1357-1369, October.
    6. Paul Lynch & Gilles Zumbach, 2003. "Market heterogeneities and the causal structure of volatility," Quantitative Finance, Taylor & Francis Journals, vol. 3(4), pages 320-331.
    7. Duan, Jin-Chuan & Zhang, Hua, 2001. "Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1989-2014, November.
    8. 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.
    9. Jin-Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32.
    10. Gilles Zumbach, 2004. "Volatility processes and volatility forecast with long memory," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 70-86.
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