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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.

Suggested Citation

  • 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, 2006. "American Call Options on Jump-Diffusion Processes: A Fourier Transform Approach," Research Paper Series 174, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. 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.
    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. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    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. 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.
    7. 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.
    8. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    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. 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.
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    1. Gerald H. L. Cheang & Carl Chiarella & Andrew Ziogas, 2013. "The representation of American options prices under stochastic volatility and jump-diffusion dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 13(2), pages 241-253, January.
    2. Ming Xi Huang, 2010. "Modelling Default Correlations in a Two-Firm Model with Dynamic Leverage Ratios," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 15.
    3. 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.
    4. Carl Chiarella & Christina Nikitopoulos-Sklibosios & Erik Schlogl & Hongang Yang, 2016. "Pricing American Options under Regime Switching Using Method of Lines," Research Paper Series 368, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. repec:eee:insuma:v:79:y:2018:i:c:p:43-56 is not listed on IDEAS
    6. Carl Chiarella & Jonathan Ziveyi, 2014. "Pricing American options written on two underlying assets," Quantitative Finance, Taylor & Francis Journals, vol. 14(3), pages 409-426, March.
    7. 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.
    8. Robert J. Elliott & Katsumasa Nishide & Carlton‐James U. Osakwe, 2016. "Heston‐Type Stochastic Volatility with a Markov Switching Regime," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 36(9), pages 902-919, September.
    9. Carl Chiarella & Jonathan Ziveyi, 2011. "Two Stochastic Volatility Processes - American Option Pricing," Research Paper Series 292, Quantitative Finance Research Centre, University of Technology, Sydney.
    10. Jérôme Detemple, 2014. "Optimal Exercise for Derivative Securities," Annual Review of Financial Economics, Annual Reviews, vol. 6(1), pages 459-487, December.
    11. Jonathan Ziveyi, 2011. "The Evaluation of Early Exercise Exotic Options," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 12.
    12. Christoph Reisinger & Jan Hendrik Witte, 2010. "On the Use of Policy Iteration as an Easy Way of Pricing American Options," Papers 1012.4976, arXiv.org, revised Sep 2011.
    13. Zhylyevskyy, Oleksandr, 2012. "Joint Characteristic Function of Stock Log-Price and Squared Volatility in the Bates Model and Its Asset Pricing Applications," Staff General Research Papers Archive 35559, Iowa State University, Department of Economics.
    14. repec:wsi:ijtafx:v:18:y:2015:i:05:n:s0219024915500314 is not listed on IDEAS
    15. Andrey Itkin, 2015. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-24.
    16. 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.
    17. Jamal Amani Rad & Kourosh Parand, 2014. "Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method," Papers 1412.6064, arXiv.org.
    18. Chen Xiaoshan & Song Qingshuo, 2013. "American option of stochastic volatility model with negative Fichera function on degenerate boundary," Papers 1306.0345, arXiv.org.
    19. Rafael Company & Vera Egorova & Lucas J'odar & Fazlollah Soleymani, 2017. "Computing stable numerical solutions for multidimensional American option pricing problems: a semi-discretization approach," Papers 1701.08545, arXiv.org.
    20. Carl Chiarella & Boda Kang & Gunter H. Meyer, 2010. "The Evaluation Of Barrier Option Prices Under Stochastic Volatility," Research Paper Series 266, Quantitative Finance Research Centre, University of Technology, Sydney.
    21. Maya Briani & Lucia Caramellino & Giulia Terenzi & Antonino Zanette, 2016. "On a hybrid method using trees and finite-differences for pricing options in complex models," Papers 1603.07225, arXiv.org, revised May 2017.
    22. Andrey Itkin, 2015. "LSV models with stochastic interest rates and correlated jumps," Papers 1511.01460, arXiv.org, revised Nov 2016.

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    Keywords

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

    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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