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Polynomial approximation of discounted moments

Author

Listed:
  • Chenyu Zhao
  • Misha Beek
  • Peter Spreij

    (University of Amsterdam)

  • Makhtar Ba

Abstract

We introduce an approximation strategy for the discounted moments of a stochastic process that can approximate the true moments for a large class of problems. These moments appear in pricing formulas of financial products such as bonds and credit derivatives. The approximation relies on a high-order power series expansion of the infinitesimal generator and draws parallels with the theory of polynomial processes. We demonstrate applications to bond pricing and credit derivatives. In the special cases that allow an analytical solution, the approximation error decreases to around 10 to 100 times machine precision for higher orders. When no analytical solution exists, we numerically compare the approximation with existing numerical techniques.

Suggested Citation

  • Chenyu Zhao & Misha Beek & Peter Spreij & Makhtar Ba, 2025. "Polynomial approximation of discounted moments," Finance and Stochastics, Springer, vol. 29(1), pages 63-95, January.
    • Handle: RePEc:spr:finsto:v:29:y:2025:i:1:d:10.1007_s00780-024-00550-4
    DOI: 10.1007/s00780-024-00550-4
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    References listed on IDEAS

    as
    1. Angelo Arvanitis & Jonathan Gregory & Jean-Paul Laurent, 1999. "Building Models for Credit Spreads," Post-Print hal-03679527, HAL.
    2. Jean-Pierre Chateau & Daniel Dufresne, 2017. "Gram-Charlier Processes and Applications to Option Pricing," Journal of Probability and Statistics, Hindawi, vol. 2017, pages 1-19, February.
    3. Luca Capriotti, 2006. "The Exponent Expansion: An Effective Approximation Of Transition Probabilities Of Diffusion Processes And Pricing Kernels Of Financial Derivatives," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(07), pages 1179-1199.
    4. Hao Zhou, 2003. "Itô Conditional Moment Generator and the Estimation of Short-Rate Processes," Journal of Financial Econometrics, Oxford University Press, vol. 1(2), pages 250-271.
    5. 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.
    6. Luca Capriotti, 2006. "The Exponent Expansion: An Effective Approximation of Transition Probabilities of Diffusion Processes and Pricing Kernels of Financial Derivatives," Papers physics/0602107, arXiv.org.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Markov processes; Pricing; Hedging; Short-rate models; Credit models; Generator; Resolvent;
    All these keywords.

    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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