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The evaluation of European compound option prices under stochastic volatility using Fourier transform techniques

  • Susanne Griebsch

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    Compound options are not only sensitive to future movements of the underlying asset price, but also to future changes in volatility levels. Because the Black–Scholes analytical valuation formula for compound options is not able to incorporate the sensitivity to volatility, the aim of this paper is to develop a numerical pricing procedure for this type of option in stochastic volatility models, specifically focusing on the model of Heston. For this, the compound option value is represented as the difference of its exercise probabilities, which depend on three random variables through a complex functional form. Then the joint distribution of these random variables is uniquely determined by their characteristic function and therefore the probabilities can each be expressed as a multiple inverse Fourier transform. Solving the inverse Fourier transform with respect to volatility, we can reduce the pricing problem from three to two dimensions. This reduced dimensionality simplifies the application of the fast Fourier transform (FFT) method developed by Dempster and Hong when transferred to our stochastic volatility framework. After combining their approach with a new extension of the fractional FFT technique for option pricing to the two-dimensional case, it is possible to obtain good approximations to the exercise probabilities. The resulting upper and lower bounds are then compared with other numerical methods such as Monte Carlo simulations and show promising results. Copyright Springer Science+Business Media New York 2013

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    File URL: http://hdl.handle.net/10.1007/s11147-012-9083-z
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    Article provided by Springer in its journal Review of Derivatives Research.

    Volume (Year): 16 (2013)
    Issue (Month): 2 (July)
    Pages: 135-165

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    Handle: RePEc:kap:revdev:v:16:y:2013:i:2:p:135-165
    Contact details of provider: Web page: http://www.springerlink.com/link.asp?id=102989

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    1. Geske, Robert & Johnson, Herb E, 1984. " The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-24, December.
    2. Susanne Griebsch & Uwe Wystup, 2011. "On the valuation of fader and discrete barrier options in Heston's stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 693-709.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
    4. Geske, Robert, 1979. "The valuation of compound options," Journal of Financial Economics, Elsevier, vol. 7(1), pages 63-81, March.
    5. Oleksandr Zhylyevskyy, 2010. "A fast Fourier transform technique for pricing American options under stochastic volatility," Review of Derivatives Research, Springer, vol. 13(1), pages 1-24, April.
    6. Carl Chiarella & Boda Kang, 2009. "The Evaluation of American Compound Option Prices Under Stochastic Volatility Using the Sparse Grid Approach," Research Paper Series 245, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Carr, Peter P, 1988. " The Valuation of Sequential Exchange Opportunities," Journal of Finance, American Finance Association, vol. 43(5), pages 1235-56, December.
    8. Brenner, Menachem & Ou, Ernest Y. & Zhang, Jin E., 2006. "Hedging volatility risk," Journal of Banking & Finance, Elsevier, vol. 30(3), pages 811-821, March.
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