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American Call Options on Jump-Diffusion Processes: A Fourier Transform Approach

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Abstract

This paper considers the Fourier transform approach to derive the implicit integral equation for the price of an American call option in the case where the underlying asset follows a jump-diffusion process. Using the method of Jamshidian (1992), we demonstrate that the call option price is given by the solution to an inhomogeneous integro-partial differential equation in an unbounded domain, and subsequently derive the solution using Fourier transforms. We also extend McKean’s incomplete Fourier transform approach to solve the free boundary problem under Merton’s framework, for a general jump size distribution. We show how the two methods are related to each other, and also to the Geske-Johnson compound option approach used by Gukhal (2001). The paper also derives results concerning the limit for the free boundary at expiry, and presents a numerical algorithm for solving the linked integral equation system for the American call price, delta and early exercise boundary. This scheme is applied to Merton’s jump-diffusion model, where the jumps are log-normally distributed.

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  • Carl Chiarella & Andrew Ziogas, 2006. "American Call Options on Jump-Diffusion Processes: A Fourier Transform Approach," Research Paper Series 174, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:174
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-02/QFR-rp174.pdf
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    1. Leif Andersen & Jesper Andreasen, 2000. "Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing," Review of Derivatives Research, Springer, vol. 4(3), pages 231-262, October.
    2. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-351, July.
    3. Barone-Adesi, Giovanni, 2005. "The saga of the American put," Journal of Banking & Finance, Elsevier, vol. 29(11), pages 2909-2918, November.
    4. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    5. Siim Kallast & Andi Kivinukk, 2003. "Pricing and Hedging American Options Using Approximations by Kim Integral Equations," Review of Finance, Springer, vol. 7(3), pages 361-383.
    6. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
    7. Mark Broadie & Yusaku Yamamoto, 2003. "Application of the Fast Gauss Transform to Option Pricing," Management Science, INFORMS, vol. 49(8), pages 1071-1088, August.
    8. Ball, Clifford A & Torous, Walter N, 1985. " On Jumps in Common Stock Prices and Their Impact on Call Option Pricing," Journal of Finance, American Finance Association, vol. 40(1), pages 155-173, March.
    9. Amin, Kaushik I, 1993. " Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-1863, December.
    10. Carl Chiarella & Adam Kucera & Andrew Ziogas, 2004. "A Survey of the Integral Representation of American Option Prices," Research Paper Series 118, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv.
    12. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
    13. Geske, Robert & Johnson, Herb E, 1984. " The American Put Option Valued Analytically," Journal of Finance, American Finance Association, vol. 39(5), pages 1511-1524, December.
    14. Mulinacci, Sabrina, 1996. "An approximation of American option prices in a jump-diffusion model," Stochastic Processes and their Applications, Elsevier, vol. 62(1), pages 1-17, March.
    15. Chandrasekhar Reddy Gukhal, 2001. "Analytical Valuation of American Options on Jump-Diffusion Processes," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 97-115.
    16. Kim, In Joon, 1990. "The Analytic Valuation of American Options," Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 547-572.
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    Citations

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    Cited by:

    1. Carl Chiarella & Boda Kang & Gunter H. Meyer & Andrew Ziogas, 2009. "The Evaluation Of American Option Prices Under Stochastic Volatility And Jump-Diffusion Dynamics Using The Method Of Lines," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 393-425.
    2. Gerald Cheang & Carl Chiarella & Andrew Ziogas, 2009. "An Analysis of American Options Under Heston Stochastic Volatility and Jump-Diffusion Dynamics," Research Paper Series 256, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Simonato, Jean-Guy, 2011. "Computing American option prices in the lognormal jump–diffusion framework with a Markov chain," Finance Research Letters, Elsevier, vol. 8(4), pages 220-226.

    More about this item

    Keywords

    American options; jump-diffusion; Volterra integral equation; free boundary problem;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory

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