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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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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 (1993), and by a Poisson jump process of the type originally introduced by Merton (1976). 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 (1998) 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 & Toivanen (2007). The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation which is taken as the benchmark. 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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  • Carl Chiarella & Boda Kang & Gunter H. Meyer & Andrew Ziogas, 2008. "The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines," Research Paper Series 219, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:219
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    1. Carl Chiarella & Andrew Ziogas, 2009. "American Call Options Under Jump-Diffusion Processes - A Fourier Transform Approach," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(1), pages 37-79.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. ,, 2004. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 20(2), pages 427-429, April.
    4. 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.
    5. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    6. Amin, Kaushik I, 1993. "Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-1863, December.
    7. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    8. 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.
    9. 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.
    10. ,, 2004. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 20(1), pages 223-229, February.
    11. 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.
    12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    13. Louis O. Scott, 1997. "Pricing Stock Options in a Jump‐Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier Inversion Methods," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 413-426, October.
    14. 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.
    15. Samuli Ikonen & Jari Toivanen, 2007. "Componentwise Splitting Methods For Pricing American Options Under Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(02), pages 331-361.
    16. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," The Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
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    Keywords

    American options; stochastic volatility; jump-diffusion processes; Volterra integral equations; free boundary problem; method of lines;
    All these keywords.

    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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