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Affine processes under parameter uncertainty

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  • Tolulope Fadina
  • Ariel Neufeld
  • Thorsten Schmidt

Abstract

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non-linear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in the parameter set. This non-linear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity. We then develop an appropriate Ito-formula, the respective term-structure equations and study the non-linear versions of the Vasicek and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear Vasicek-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence the approach solves this modelling issue arising with negative interest rates.

Suggested Citation

  • Tolulope Fadina & Ariel Neufeld & Thorsten Schmidt, 2018. "Affine processes under parameter uncertainty," Papers 1806.02912, arXiv.org, revised Mar 2019.
    • Handle: RePEc:arx:papers:1806.02912
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    References listed on IDEAS

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    1. Larry G. Epstein & Shaolin Ji, 2013. "Ambiguous Volatility and Asset Pricing in Continuous Time," The Review of Financial Studies, Society for Financial Studies, vol. 26(7), pages 1740-1786.
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    7. B. Acciaio & M. Beiglböck & F. Penkner & W. Schachermayer, 2016. "A Model-Free Version Of The Fundamental Theorem Of Asset Pricing And The Super-Replication Theorem," Mathematical Finance, Wiley Blackwell, vol. 26(2), pages 233-251, April.
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    Citations

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    Cited by:

    1. Sandrine Gumbel & Thorsten Schmidt, 2020. "Machine learning for multiple yield curve markets: fast calibration in the Gaussian affine framework," Papers 2004.07736, arXiv.org, revised Apr 2020.
    2. Sandrine Gümbel & Thorsten Schmidt, 2020. "Machine Learning for Multiple Yield Curve Markets: Fast Calibration in the Gaussian Affine Framework," Risks, MDPI, vol. 8(2), pages 1-18, May.
    3. Hölzermann, Julian, 2020. "Pricing Interest Rate Derivatives under Volatility Uncertainty," Center for Mathematical Economics Working Papers 633, Center for Mathematical Economics, Bielefeld University.
    4. Benedikt Geuchen & Katharina Oberpriller & Thorsten Schmidt, 2022. "Affine models with path-dependence under parameter uncertainty and their application in finance," Papers 2207.13350, arXiv.org, revised Jun 2024.
    5. Blessing, Jonas & Kupper, Michael & Nendel, Max, 2023. "Convergence of Infintesimal Generators and Stability of Convex Montone Semigroups," Center for Mathematical Economics Working Papers 680, Center for Mathematical Economics, Bielefeld University.
    6. Schmidt, Thorsten & Tappe, Stefan & Yu, Weijun, 2020. "Infinite dimensional affine processes," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7131-7169.
    7. Akhtari, Bahar & Biagini, Francesca & Mazzon, Andrea & Oberpriller, Katharina, 2023. "Generalized Feynman–Kac formula under volatility uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    8. Meriam El Mansour & Emmanuel Lepinette, 2023. "Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty," Papers 2311.08847, arXiv.org.
    9. Biagini, Francesca & Mazzon, Andrea & Oberpriller, Katharina, 2023. "Reduced-form framework for multiple ordered default times under model uncertainty," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 1-43.
    10. Tolulope Fadina & Thorsten Schmidt, 2024. "The Unfairness of $\varepsilon$-Fairness," Papers 2405.09360, arXiv.org, revised Jun 2024.
    11. Changhong Guo & Shaomei Fang & Yong He, 2023. "A Generalized Stochastic Process: Fractional G-Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-34, March.
    12. Eva Lutkebohmert & Thorsten Schmidt & Julian Sester, 2021. "Robust deep hedging," Papers 2106.10024, arXiv.org, revised Nov 2021.
    13. Criens, David & Niemann, Lars, 2024. "A class of multidimensional nonlinear diffusions with the Feller property," Statistics & Probability Letters, Elsevier, vol. 208(C).
    14. Criens, David & Niemann, Lars, 2024. "Markov selections and Feller properties of nonlinear diffusions," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    15. Tolulope Fadina & Thorsten Schmidt, 2019. "Default Ambiguity," Risks, MDPI, vol. 7(2), pages 1-17, June.
    16. David Criens & Lars Niemann, 2022. "Robust utility maximization with nonlinear continuous semimartingales," Papers 2206.14015, arXiv.org, revised Aug 2023.
    17. David Criens & Lars Niemann, 2023. "Robust utility maximization with nonlinear continuous semimartingales," Mathematics and Financial Economics, Springer, volume 17, number 5, December.
    18. Bahar Akhtari & Francesca Biagini & Andrea Mazzon & Katharina Oberpriller, 2020. "Generalized Feynman-Kac Formula under volatility uncertainty," Papers 2012.08163, arXiv.org, revised Nov 2022.
    19. Francesca Biagini & Georg Bollweg & Katharina Oberpriller, 2022. "Non-linear Affine Processes with Jumps," Papers 2207.03710, arXiv.org, revised Jul 2022.
    20. Fuhrmann, Sven & Kupper, Michael & Nendel, Max, 2021. "Wasserstein Perturbations of Markovian Transition Semigroups," Center for Mathematical Economics Working Papers 649, Center for Mathematical Economics, Bielefeld University.

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