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Filtration shrinkage, the structure of deflators, and failure of market completeness

Author

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  • Constantinos Kardaras

    (London School of Economics and Political Science)

  • Johannes Ruf

    (London School of Economics and Political Science)

Abstract

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale parts in the multiplicative Doob–Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.

Suggested Citation

  • Constantinos Kardaras & Johannes Ruf, 2020. "Filtration shrinkage, the structure of deflators, and failure of market completeness," Finance and Stochastics, Springer, vol. 24(4), pages 871-901, October.
    • Handle: RePEc:spr:finsto:v:24:y:2020:i:4:d:10.1007_s00780-020-00435-2
    DOI: 10.1007/s00780-020-00435-2
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    References listed on IDEAS

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    1. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1761-1784.
    2. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," LSE Research Online Documents on Economics 65150, London School of Economics and Political Science, LSE Library.
    3. Jeanblanc, Monique & Song, Shiqi, 2015. "Martingale representation property in progressively enlarged filtrations," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4242-4271.
    4. Claudio Fontana & Monique Jeanblanc & Shiqi Song, 2014. "On arbitrages arising with honest times," Finance and Stochastics, Springer, vol. 18(3), pages 515-543, July.
    5. Prokaj, Vilmos, 2009. "Unfolding the Skorohod reflection of a semimartingale," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 534-536, February.
    6. Anna Aksamit & Tahir Choulli & Jun Deng & Monique Jeanblanc, 2017. "No-arbitrage up to random horizon for quasi-left-continuous models," Finance and Stochastics, Springer, vol. 21(4), pages 1103-1139, October.
    7. Huy N. Chau & Andrea Cosso & Claudio Fontana, 2020. "The value of informational arbitrage," Finance and Stochastics, Springer, vol. 24(2), pages 277-307, April.
    8. Ioannis Karatzas & Constantinos Kardaras, 2007. "The numéraire portfolio in semimartingale financial models," Finance and Stochastics, Springer, vol. 11(4), pages 447-493, October.
    9. Song, Shiqi, 2016. "Drift operator in a viable expansion of information flow," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2297-2322.
    10. Anna Aksamit & Tahir Choulli & Jun Deng & Monique Jeanblanc, 2018. "No-arbitrage under a class of honest times," Finance and Stochastics, Springer, vol. 22(1), pages 127-159, January.
    11. Delia Coculescu & Monique Jeanblanc & Ashkan Nikeghbali, 2012. "Default times, no-arbitrage conditions and changes of probability measures," Finance and Stochastics, Springer, vol. 16(3), pages 513-535, July.
    12. Fontana, Claudio, 2018. "The strong predictable representation property in initially enlarged filtrations under the density hypothesis," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 1007-1033.
    13. Kardaras, Constantinos & Ruf, Johannes, 2019. "Projections of scaled bessel processes," LSE Research Online Documents on Economics 100939, London School of Economics and Political Science, LSE Library.
    14. Francesca Biagini & Andrea Mazzon & Ari-Pekka Perkkio, 2020. "Optional projection under equivalent local martingale measures," Papers 2003.09940, arXiv.org.
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    Cited by:

    1. David Criens & Mikhail Urusov, 2023. "Criteria for NUPBR, NFLVR and the existence of EMMs in integrated diffusion markets," Papers 2306.11470, arXiv.org.
    2. Francesca Biagini & Andrea Mazzon & Ari-Pekka Perkkiö, 2023. "Optional projection under equivalent local martingale measures," Finance and Stochastics, Springer, vol. 27(2), pages 435-465, April.
    3. Černý, Aleš & Ruf, Johannes, 2023. "Simplified calculus for semimartingales: Multiplicative compensators and changes of measure," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 572-602.
    4. Çetin, Umut & Larsen, Kasper, 2023. "Uniqueness in cauchy problems for diffusive real-valued strict local martingales," LSE Research Online Documents on Economics 118743, London School of Economics and Political Science, LSE Library.
    5. Umut Cetin & Kasper Larsen, 2020. "Uniqueness in Cauchy problems for diffusive real-valued strict local martingales," Papers 2007.15041, arXiv.org, revised May 2022.

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    More about this item

    Keywords

    Bayes rule; Brownian motion; Deflator; Lévy transform; Local martingale; Market completeness; Predictable representation property;
    All these keywords.

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading

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