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P-Bubbles, Q-Bubbles, and Risk Premia

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  • Robert A. Jarrow
  • Simon S. Kwok

Abstract

We develop a unified modeling framework that connects two distinct types of bubbles defined in the literature: the rational bubbles (aka P-bubbles), and the local martingale bubbles (aka Q-bubbles). We show that the local martingale bubble model includes the classical rational bubble as a special case. We relate both types of bubbles to an equity's risk premium via a novel decomposition.

Suggested Citation

  • Robert A. Jarrow & Simon S. Kwok, 2026. "P-Bubbles, Q-Bubbles, and Risk Premia," Papers 2608.01554, arXiv.org.
    • Handle: RePEc:arx:papers:2608.01554
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    References listed on IDEAS

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    1. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    2. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Limit Theory Of Real‐Time Detectors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56, pages 1079-1134, November.
    3. Robert A. Jarrow & Simon S. Kwok, 2021. "Inferring financial bubbles from option data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 36(7), pages 1013-1046, November.
    4. Fernando M. Duarte & Carlo Rosa, 2015. "The equity risk premium: a review of models," Economic Policy Review, Federal Reserve Bank of New York, issue 2, pages 39-57.
    5. Alberto Martin & Jaume Ventura, 2018. "The Macroeconomics of Rational Bubbles: A User's Guide," Annual Review of Economics, Annual Reviews, vol. 10(1), pages 505-539, August.
    6. Flood, Robert P & Garber, Peter M, 1980. "Market Fundamentals versus Price-Level Bubbles: The First Tests," Journal of Political Economy, University of Chicago Press, vol. 88(4), pages 745-770, August.
    7. John Y. Campbell & Samuel B. Thompson, 2008. "Predicting Excess Stock Returns Out of Sample: Can Anything Beat the Historical Average?," The Review of Financial Studies, Society for Financial Studies, vol. 21(4), pages 1509-1531, July.
    8. Ian Martin, 2017. "What is the Expected Return on the Market?," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 132(1), pages 367-433.
    9. Blanchard, Olivier Jean, 1979. "Speculative bubbles, crashes and rational expectations," Economics Letters, Elsevier, vol. 3(4), pages 387-389.
    10. Loewenstein, Mark & Willard, Gregory A., 2000. "Rational Equilibrium Asset-Pricing Bubbles in Continuous Trading Models," Journal of Economic Theory, Elsevier, vol. 91(1), pages 17-58, March.
    11. Michael Heinrich Baumann & Anja Janischewski, 2025. "What are Asset Price Bubbles? A Survey on Definitions of Financial Bubbles," Papers 2502.10101, arXiv.org.
    12. Robert A. Jarrow & Simon S. Kwok, 2024. "A study on asset price bubble dynamics: explosive trend or quadratic variation?," Quantitative Finance, Taylor & Francis Journals, vol. 24(5), pages 613-626, May.
    13. Ian W. R. Martin & Christian Wagner, 2019. "What Is the Expected Return on a Stock?," Journal of Finance, American Finance Association, vol. 74(4), pages 1887-1929, August.
    14. Diba, Behzad T & Grossman, Herschel I, 1988. "Explosive Rational Bubbles in Stock Prices?," American Economic Review, American Economic Association, vol. 78(3), pages 520-530, June.
    15. Evans, George W, 1991. "Pitfalls in Testing for Explosive Bubbles in Asset Prices," American Economic Review, American Economic Association, vol. 81(4), pages 922-930, September.
    16. Peter C. B. Phillips & Yangru Wu & Jun Yu, 2011. "EXPLOSIVE BEHAVIOR IN THE 1990s NASDAQ: WHEN DID EXUBERANCE ESCALATE ASSET VALUES?," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 52(1), pages 201-226, February.
    17. Baumann, Michael Heinrich & Janischewski, Anja, 2025. "What are asset price bubbles? A survey on definitions of financial bubbles," MPRA Paper 123676, University Library of Munich, Germany.
    18. Anja Janischewski & Michael Heinrich Baumann, 2025. "What are Asset Price Bubbles? A Survey on Definitions of Financial Bubbles," Chemnitz Economic Papers 065, Department of Economics, Chemnitz University of Technology.
    19. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Limit Theory Of Real‐Time Detectors," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56(4), pages 1079-1134, November.
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