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Chebyshev interpolation for parametric option pricing

Author

Listed:
  • Maximilian Gaß

    (Technical University of Munich)

  • Kathrin Glau

    (Technical University of Munich
    Queen Mary University of London)

  • Mirco Mahlstedt

    (Technical University of Munich)

  • Maximilian Mair

    (Technical University of Munich)

Abstract

Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real time. We concentrate on parametric option pricing (POP) as a generic instance of parametric conditional expectations and show that polynomial interpolation in the parameter space promises to considerably reduce run-times while maintaining accuracy. The attractive properties of Chebyshev interpolation and its tensorized extension enable us to identify broadly applicable criteria for (sub)exponential convergence and explicit error bounds. The method is most promising when the computation of the prices is most challenging. We therefore investigate its combination with Monte Carlo simulation and analyze the effect of (stochastic) approximations of the interpolation. For a wide and important range of problems, the Chebyshev method turns out to be more efficient than parametric multilevel Monte Carlo. We conclude with a numerical efficiency study.

Suggested Citation

  • Maximilian Gaß & Kathrin Glau & Mirco Mahlstedt & Maximilian Mair, 2018. "Chebyshev interpolation for parametric option pricing," Finance and Stochastics, Springer, vol. 22(3), pages 701-731, July.
    • Handle: RePEc:spr:finsto:v:22:y:2018:i:3:d:10.1007_s00780-018-0361-y
    DOI: 10.1007/s00780-018-0361-y
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    References listed on IDEAS

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    1. Brennan, Michael J & Schwartz, Eduardo S, 1977. "The Valuation of American Put Options," Journal of Finance, American Finance Association, vol. 32(2), pages 449-462, May.
    2. Rama Cont & Nicolas Lantos & Olivier Pironneau, 2011. "A reduced basis for option pricing," Post-Print hal-00522410, HAL.
    3. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    4. 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..
    5. Lord, Roger & Fang, Fang & Bervoets, Frank & Oosterlee, Kees, 2007. "A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes," MPRA Paper 1952, University Library of Munich, Germany.
    6. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
    7. 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.
    8. 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.
    9. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.
    10. Martijn Pistorius & Johannes Stolte, 2012. "Fast Computation Of Vanilla Prices In Time-Changed Models And Implied Volatilities Using Rational Approximations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(04), pages 1-34.
    11. Ernst Eberlein & Kathrin Glau & Antonis Papapantoleon, 2010. "Analysis of Fourier Transform Valuation Formulas and Applications," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(3), pages 211-240.
    12. Ernst Eberlein & Fehmi Özkan, 2005. "The Lévy LIBOR model," Finance and Stochastics, Springer, vol. 9(3), pages 327-348, July.
    13. Martijn Pistorius & Johannes Stolte, 2012. "Fast computation of vanilla prices in time-changed models and implied volatilities using rational approximations," Papers 1203.6899, arXiv.org.
    14. Damir Filipović & Martin Larsson & Anders B. Trolle, 2017. "Linear-Rational Term Structure Models," Journal of Finance, American Finance Association, vol. 72(2), pages 655-704, April.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Kathrin Glau & Mirco Mahlstedt & Christian Potz, 2018. "A new approach for American option pricing: The Dynamic Chebyshev method," Papers 1806.05579, arXiv.org.
    2. Griselda Deelstra & Lech A. Grzelak & Felix L. Wolf, 2022. "Accelerated Computations of Sensitivities for xVA," Papers 2211.17026, arXiv.org, revised Jan 2024.
    3. Kathrin Glau & Ricardo Pachon & Christian Potz, 2019. "Speed-up credit exposure calculations for pricing and risk management," Papers 1912.01280, arXiv.org.
    4. Mariano Zeron & Ignacio Ruiz, 2020. "Dynamic sensitivities and Initial Margin via Chebyshev Tensors," Papers 2011.04544, arXiv.org.
    5. M. Khasi & J. Rashidinia, 2024. "A Bilinear Pseudo-spectral Method for Solving Two-asset European and American Pricing Options," Computational Economics, Springer;Society for Computational Economics, vol. 63(2), pages 893-918, February.
    6. Leonardo Perotti & Lech A. Grzelak, 2022. "On Pricing of Discrete Asian and Lookback Options under the Heston Model," Papers 2211.03638, arXiv.org, revised Feb 2024.
    7. Grzelak, Lech A., 2022. "Sparse grid method for highly efficient computation of exposures for xVA," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    8. Mariano Zeron-Medina Laris & Ignacio Ruiz, 2019. "Denting the FRTB IMA computational challenge via Orthogonal Chebyshev Sliding Technique," Papers 1911.10948, arXiv.org, revised Dec 2020.
    9. Kathrin Glau & Ricardo Pachon & Christian Potz, 2019. "Fast Calculation of Credit Exposures for Barrier and Bermudan options using Chebyshev interpolation," Papers 1905.00238, arXiv.org.
    10. Andrea Maran & Andrea Pallavicini & Stefano Scoleri, 2021. "Chebyshev Greeks: Smoothing Gamma without Bias," Papers 2106.12431, arXiv.org.
    11. Shuaiqiang Liu & Lech A. Grzelak & Cornelis W. Oosterlee, 2022. "The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations," Risks, MDPI, vol. 10(3), pages 1-27, February.
    12. Mariano Zeron & Ignacio Ruiz, 2020. "Tensoring volatility calibration," Papers 2012.07440, arXiv.org, revised Dec 2020.
    13. Tat Lung Chan & Nicholas Hale, 2018. "Hedging and Pricing European-type, Early-Exercise and Discrete Barrier Options using Algorithm for the Convolution of Legendre Series," Papers 1811.09257, arXiv.org, revised May 2019.
    14. Damien Ackerer & Damir Filipović, 2020. "Linear credit risk models," Finance and Stochastics, Springer, vol. 24(1), pages 169-214, January.
    15. Kathrin Glau & Daniel Kressner & Francesco Statti, 2019. "Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing," Papers 1902.04367, arXiv.org.
    16. Lech A. Grzelak, 2021. "Sparse Grid Method for Highly Efficient Computation of Exposures for xVA," Papers 2104.14319, arXiv.org, revised May 2022.

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    More about this item

    Keywords

    Multivariate option pricing; Complexity reduction; (Tensorized) Chebyshev polynomials; Polynomial interpolation; Fourier transform methods; Monte Carlo; Parametric Monte Carlo; Online–offline decomposition;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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