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Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing

Author

Listed:
  • Likuan Qin

    (Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, Evanston, Illinois 60208)

  • Vadim Linetsky

    (Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, Evanston, Illinois 60208)

Abstract

This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process, or BRP) and the stochastic discount factor (SDF) is a positive semimartingale multiplicative functional of X . A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman factorization of the Markovian SDF corresponding to the recurrent eigenfunction; extend the Recovery Theorem from discrete time, finite state irreducible Markov chains to recurrent BRPs; and obtain the long-maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient conditions for the existence of a recurrent eigenfunction and provide explicit examples in a number of important financial models, including affine and quadratic diffusion models and an affine model with jumps. These examples show that the recurrence assumption, in addition to fixing uniqueness, rules out unstable economic dynamics, such as the short rate asymptotically going to infinity or to a zero lower bound trap without possibility of escaping.

Suggested Citation

  • Likuan Qin & Vadim Linetsky, 2016. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing," Operations Research, INFORMS, vol. 64(1), pages 99-117, February.
  • Handle: RePEc:inm:oropre:v:64:y:2016:i:1:p:99-117
    DOI: 10.1287/opre.2015.1449
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    Citations

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    Cited by:

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    2. Jensen, Christian Skov & Lando, David & Pedersen, Lasse Heje, 2019. "Generalized recovery," Journal of Financial Economics, Elsevier, vol. 133(1), pages 154-174.
    3. Dillschneider, Yannick & Maurer, Raimond, 2019. "Functional Ross recovery: Theoretical results and empirical tests," Journal of Economic Dynamics and Control, Elsevier, vol. 108(C).
    4. Steven Heston & Kris Jacobs & Hyung Joo Kim, 2023. "The Pricing Kernel in Options," Finance and Economics Discussion Series 2023-053, Board of Governors of the Federal Reserve System (U.S.).
    5. Hyungbin Park, 2021. "Influence of risk tolerance on long-term investments: A Malliavin calculus approach," Papers 2104.00911, arXiv.org.
    6. Hyungbin Park & Heejun Yeo, 2022. "Dynamic and static fund separations and their stability for long-term optimal investments," Papers 2212.00391, arXiv.org, revised Mar 2023.
    7. Hyungbin Park, 2019. "Convergence rates of large-time sensitivities with the Hansen--Scheinkman decomposition," Papers 1912.03404, arXiv.org, revised Jan 2021.
    8. Jackwerth, Jens Carsten & Menner, Marco, 2020. "Does the Ross recovery theorem work empirically?," Journal of Financial Economics, Elsevier, vol. 137(3), pages 723-739.
    9. Tim Leung & Hyungbin Park, 2017. "LONG-TERM GROWTH RATE OF EXPECTED UTILITY FOR LEVERAGED ETFs: MARTINGALE EXTRACTION APPROACH," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-33, September.
    10. Johan Walden, 2017. "Recovery with Unbounded Diffusion Processes," Review of Finance, European Finance Association, vol. 21(4), pages 1403-1444.
    11. Likuan Qin & Vadim Linetsky, 2018. "Long-term factorization in Heath–Jarrow–Morton models," Finance and Stochastics, Springer, vol. 22(3), pages 621-641, July.
    12. Shinmi Ahn & Hyungbin Park, 2020. "Examining the Feasibility of the Sturm–Liouville Theory for Ross Recovery," Mathematics, MDPI, vol. 8(4), pages 1-16, April.
    13. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2019. "Gauge transformations in the dual space, and pricing and estimation in the long run in affine jump-diffusion models," Papers 1912.06948, arXiv.org, revised Dec 2019.
    14. Chattopadhyay, Subir, 2018. "The unit root property and optimality with a continuum of states—Pure exchange," Journal of Mathematical Economics, Elsevier, vol. 78(C), pages 105-118.
    15. Hyungbin Park, 2021. "Modified Mean-Variance Risk Measures for Long-Term Portfolios," Mathematics, MDPI, vol. 9(2), pages 1-23, January.

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