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Option Pricing with Orthogonal Polynomial Expansions

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  • Damien Ackerer
  • Damir Filipovic

Abstract

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.

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  • Damien Ackerer & Damir Filipovic, 2017. "Option Pricing with Orthogonal Polynomial Expansions," Papers 1711.09193, arXiv.org, revised May 2019.
    • Handle: RePEc:arx:papers:1711.09193
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    References listed on IDEAS

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    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Filipović, Damir & Gourier, Elise & Mancini, Loriano, 2016. "Quadratic variance swap models," Journal of Financial Economics, Elsevier, vol. 119(1), pages 44-68.
    3. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    4. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    5. Rainer Schöbel & Jianwei Zhu, 1999. "Stochastic Volatility With an Ornstein–Uhlenbeck Process: An Extension," Review of Finance, European Finance Association, vol. 3(1), pages 23-46.
    6. 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..
    7. T. A. McWalter & R. Rudd & J. Kienitz & E. Platen, 2018. "Recursive marginal quantization of higher-order schemes," Quantitative Finance, Taylor & Francis Journals, vol. 18(4), pages 693-706, April.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    9. Gilles Pagès & Abass Sagna, 2015. "Recursive Marginal Quantization of the Euler Scheme of a Diffusion Process," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(5), pages 463-498, November.
    10. 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.
    11. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, vol. 176(2), pages 93-111.
    12. Drost, Feike C. & Werker, Bas J. M., 1996. "Closing the GARCH gap: Continuous time GARCH modeling," Journal of Econometrics, Elsevier, vol. 74(1), pages 31-57, September.
    13. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    14. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    15. Paul Glasserman & Kyoung-Kuk Kim, 2011. "Gamma expansion of the Heston stochastic volatility model," Finance and Stochastics, Springer, vol. 15(2), pages 267-296, June.
    16. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    17. Barone-Adesi, Giovanni & Rasmussen, Henrik & Ravanelli, Claudia, 2005. "An option pricing formula for the GARCH diffusion model," Computational Statistics & Data Analysis, Elsevier, vol. 49(2), pages 287-310, April.
    18. Damir Filipović & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Swiss Finance Institute Research Paper Series 17-60, Swiss Finance Institute.
    19. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
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    Cited by:

    1. Peter Carr & Sander Willems, 2019. "A lognormal type stochastic volatility model with quadratic drift," Papers 1908.07417, arXiv.org.
    2. Sander Willems, 2018. "Asian Option Pricing with Orthogonal Polynomials," Papers 1802.01307, arXiv.org, revised Sep 2018.
    3. Pierre-Edouard Arrouy & Sophian Mehalla & Bernard Lapeyre & Alexandre Boumezoued, 2020. "Jacobi Stochastic Volatility factor for the Libor Market Model," Working Papers hal-02468583, HAL.
    4. Damir Filipovi'c & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Papers 1711.08043, arXiv.org, revised Jul 2019.
    5. Damien Ackerer & Damir Filipović & Sergio Pulido, 2018. "The Jacobi stochastic volatility model," Finance and Stochastics, Springer, vol. 22(3), pages 667-700, July.

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