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The Evaluation Of American Option Prices Under Stochastic Volatility And Jump-Diffusion Dynamics Using The Method Of Lines

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Author Info

  • CARL CHIARELLA

    ()
    (School of Finance and Economics, University of Technology, Sydney, PO Box 123, Broadway, NSW 2007, Australia)

  • BODA KANG

    ()
    (School of Finance and Economics, University of Technology, Sydney, Australia)

  • GUNTER H. MEYER

    ()
    (School of Mathematics, Georgia Institute of Technology, Atlanta, USA)

  • ANDREW ZIOGAS

    ()
    (Integral Energy, Australia)

Abstract

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston [18], and by a Poisson jump process of the type originally introduced by Merton [25]. We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer [26] for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen and Toivanen [21]. The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation and which is taken as the benchmark for the price. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.

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Bibliographic Info

Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

Volume (Year): 12 (2009)
Issue (Month): 03 ()
Pages: 393-425

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Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:03:p:393-425

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Related research

Keywords: American options; stochastic volatility; jump-diffusion processes; Volterra integral equations; free boundary problem; method of lines;

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References

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  1. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
  2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
  3. 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.
  4. Nigel Clarke & Kevin Parrott, 1999. "Multigrid for American option pricing with stochastic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 177-195.
  5. Elias Tzavalis & Shijun Wang, 2003. "Pricing American Options under Stochastic Volatility: A New Method Using Chebyshev Polynomials to Approximate the Early Exercise Boundary," Working Papers 488, Queen Mary, University of London, School of Economics and Finance.
  6. Amin, Kaushik I, 1993. " Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-63, December.
  7. Thomas Adolfsson & Carl Chiarella & Andrew Ziogas & Jonathan Ziveyi, 2013. "Representation and Numerical Approximation of American Option Prices under Heston Stochastic Volatility Dynamics," Research Paper Series 327, Quantitative Finance Research Centre, University of Technology, Sydney.
  8. 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.
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Cited by:
  1. 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.
  2. Ballestra, Luca Vincenzo & Pacelli, Graziella, 2013. "Pricing European and American options with two stochastic factors: A highly efficient radial basis function approach," Journal of Economic Dynamics and Control, Elsevier, vol. 37(6), pages 1142-1167.
  3. Chen Xiaoshan & Song Qingshuo, 2013. "American option of stochastic volatility model with negative Fichera function on degenerate boundary," Science & Finance (CFM) working paper archive 1306.0345, Science & Finance, Capital Fund Management.
  4. Carl Chiarella & Jonathan Ziveyi, 2011. "Two Stochastic Volatility Processes - American Option Pricing," Research Paper Series 292, Quantitative Finance Research Centre, University of Technology, Sydney.
  5. 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.
  6. Christoph Reisinger & Jan Hendrik Witte, 2010. "On the Use of Policy Iteration as an Easy Way of Pricing American Options," Science & Finance (CFM) working paper archive 1012.4976, Science & Finance, Capital Fund Management, revised Sep 2011.
  7. Jun Cheng & Jin Zhang, 2012. "Analytical pricing of American options," Review of Derivatives Research, Springer, vol. 15(2), pages 157-192, July.
  8. Mahayni, Antje & Schoenmakers, John G.M., 2011. "Minimum return guarantees with fund switching rights—An optimal stopping problem," Journal of Economic Dynamics and Control, Elsevier, vol. 35(11), pages 1880-1897.
  9. Ballestra, Luca Vincenzo & Ottaviani, Massimiliano & Pacelli, Graziella, 2012. "An operator splitting harmonic differential quadrature approach to solve Young’s model for life insurance risk," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 442-448.
  10. Andrey Itkin, 2014. "High-Order Splitting Methods for Forward PDEs and PIDEs," Science & Finance (CFM) working paper archive 1403.1804, Science & Finance, Capital Fund Management.

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