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Speed and biases of Fourier-based pricing choices: Analysis of the Bates and Asymmetric Variance Gamma models

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  • Ricardo Crisóstomo

Abstract

This paper compares the CPU effort and numerical biases of six Fourier-based implementations. Our analyses focus on two jump models that can consistently price options with different strikes and maturities: (i) the Bates jump-diffusion model, which combines jumps with stochastic volatility and (ii) the Asymmetric Variance Gamma (AVG) model, a pure-jump process where an infinite number of jumps can occur in any interval of time. We show that both truncation and discretization errors significantly increase as we move away from the diffusive Black-Scholes-Merton dynamics. While most pricing choices converge to the Bates reference values, Attari’s formula is the only Fourier-based method that does not completely blow up in any AVG problematic region. In terms of CPU speed, the strike vector computations proposed by Zhu (2010) significantly improve the computational burden, rendering the use of fast Fourier transforms and plain delta-probability decompositions inefficient.

Suggested Citation

  • Ricardo Crisóstomo, 2017. "Speed and biases of Fourier-based pricing choices: Analysis of the Bates and Asymmetric Variance Gamma models," CNMV Working Papers CNMV Working Papers no. 6, CNMV- Spanish Securities Markets Commission - Research and Statistics Department.
    • Handle: RePEc:cnv:wpaper:dt_64en
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    References listed on IDEAS

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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    3. Göncü, Ahmet & Karahan, Mehmet Oğuz & Kuzubaş, Tolga Umut, 2016. "A comparative goodness-of-fit analysis of distributions of some Lévy processes and Heston model to stock index returns," The North American Journal of Economics and Finance, Elsevier, vol. 36(C), pages 69-83.
    4. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    5. Bakshi, Gurdip & Cao, Charles & Chen, Zhiwu, 1997. "Empirical Performance of Alternative Option Pricing Models," Journal of Finance, American Finance Association, vol. 52(5), pages 2003-2049, December.
    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. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    8. Ricardo Crisóstomo, 2014. "An analisys of the Heston Stochastic Volatility Model: Implementation and Calibration using Matlab," CNMV Working Papers CNMV Working Papers no 58, CNMV- Spanish Securities Markets Commission - Research and Statistics Department.
    9. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, 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. Kilin, Fiodar, 2006. "Accelerating the calibration of stochastic volatility models," MPRA Paper 2975, University Library of Munich, Germany, revised 22 Apr 2007.
    12. Jones, Christopher S., 2003. "The dynamics of stochastic volatility: evidence from underlying and options markets," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 181-224.
    13. 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.
    14. Fiorani, Filo, 2004. "Option Pricing Under the Variance Gamma Process," MPRA Paper 15395, University Library of Munich, Germany.
    15. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    16. 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.
    17. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    18. Bakshi, Gurdip & Madan, Dilip, 2000. "Spanning and derivative-security valuation," Journal of Financial Economics, Elsevier, vol. 55(2), pages 205-238, February.
    19. 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.
    20. Andrey Itkin, 2005. "Pricing options with VG model using FFT," Papers physics/0503137, arXiv.org, revised Jan 2010.
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    More about this item

    Keywords

    Jump processes; Bates model; Variance Gamma; Fourier transforms; pricing errors; speed comparisons;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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