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Liquidity risk theory and coherent measures of risk


  • Carlo Acerbi
  • Giacomo Scandolo


We discuss liquidity risk from a pure risk-theoretical point of view in the axiomatic context of coherent measures of risk. We propose a formalism for liquidity risk that is compatible with the axioms of coherency. We emphasize the difference between 'coherent risk measures' (CRM) ρ(X ) defined on portfolio values X as opposed to 'coherent portfolio risk measures' (CPRM) ρ(p) defined on the vector space of portfolios p, and we observe that in the presence of liquidity risk the value function on the space of portfolios is no longer necessarily linear. We propose a new nonlinear 'Value' function VL(p) that depends on a new notion of 'liquidity policy' L. The function VL(p) naturally arises from a general description of the impact that the microstructure of illiquid markets has when marking a portfolio to market. We discuss the consequences of the introduction of the function VL(p) in the coherency axioms and we study the properties induced on CPRMs. We show in particular that CPRMs are convex, finding a result that was proposed as a new axiom in the literature of so called 'convex measures of risk'. The framework we propose is not a model but rather a new formalism, in the sense that it is completely free from hypotheses on the dynamics of the market. We provide interpretation and characterization of the formalism as well as some stylized examples.

Suggested Citation

  • Carlo Acerbi & Giacomo Scandolo, 2008. "Liquidity risk theory and coherent measures of risk," Quantitative Finance, Taylor & Francis Journals, vol. 8(7), pages 681-692.
  • Handle: RePEc:taf:quantf:v:8:y:2008:i:7:p:681-692
    DOI: 10.1080/14697680802373975

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    Cited by:

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    2. Bianchetti, Marco & Carlicchi, Mattia, 2012. "Markets Evolution After the Credit Crunch," MPRA Paper 44023, University Library of Munich, Germany.
    3. Csóka, Péter & Herings, P. Jean-Jacques, 2014. "Risk allocation under liquidity constraints," Journal of Banking & Finance, Elsevier, vol. 49(C), pages 1-9.
    4. Havran, Dániel & Erb, Tamás, 2015. "Mit veszítünk a piaci súrlódásokkal?. A pénzügyi piacok mikrostruktúrája
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      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(3), pages 229-262.
    5. Marco Bianchetti & Mattia Carlicchi, 2013. "Markets Evolution After the Credit Crunch," Papers 1301.7078,
    6. Csóka Péter & Pintér Miklós, 2016. "On the Impossibility of Fair Risk Allocation," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 16(1), pages 143-158, January.
    7. Stange, Sebastian & Kaserer, Christoph, 2008. "Why and how to integrate liquidity risk into a VaR-framework," CEFS Working Paper Series 2008-10, Technische Universität München (TUM), Center for Entrepreneurial and Financial Studies (CEFS).
    8. Sebestyén, Tamás, 2016. "Doktoranduszhallgatók IV. Nyári Műhelye. MKE-PTE KTK, Pécs, 2016. június 3
      [The 4th Summer Workshop of Doctoral Students. Hungarian Economics Association/Pécs University, Faculty of Economics]
      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(9), pages 1011-1018.
    9. repec:eee:finlet:v:21:y:2017:i:c:p:228-234 is not listed on IDEAS
    10. Fabio Caccioli & Imre Kondor & Matteo Marsili & Susanne Still, 2016. "Liquidity Risk And Instabilities In Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-28, August.
    11. Tomasz R. Bielecki & Igor Cialenco & Tao Chen, 2014. "Dynamic Conic Finance via Backward Stochastic Difference Equations," Papers 1412.6459,, revised Dec 2014.
    12. repec:ksa:szemle:1702 is not listed on IDEAS
    13. Marco, Bianchetti & Mattia, Carlicchi, 2012. "Interest Rates After The Credit Crunch: Multiple-Curve Vanilla Derivatives and SABR," MPRA Paper 42248, University Library of Munich, Germany.
    14. Damiano Brigo & Mirela Predescu & Agostino Capponi, 2010. "Credit Default Swaps Liquidity modeling: A survey," Papers 1003.0889,, revised Mar 2010.
    15. Csóka, Péter & Bátyi, Tamás László & Pintér, Miklós & Balog, Dóra, 2011. "Tőkeallokációs módszerek és tulajdonságaik a gyakorlatban
      [Methods of capital allocation and their characteristics in practice]
      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(7), pages 619-632.
    16. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2018. "An improved Least Squares Monte Carlo method for portfolio optimization with high dimensional control," Papers 1803.11467,
    17. Erindi Allaj, 2014. "Risk measuring under liquidity risk," Papers 1412.6745,
    18. Fabio Caccioli & Jean-Philippe Bouchaud & J. Doyne Farmer, 2012. "A proposal for impact-adjusted valuation: Critical leverage and execution risk," Papers 1204.0922,, revised Aug 2012.
    19. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2016. "Dynamic Portfolio Optimization with Liquidity Cost and Market Impact: A Simulation-and-Regression Approach," Papers 1610.07694,, revised Oct 2017.
    20. Kountzakis, C. & Polyrakis, I.A., 2013. "Coherent risk measures in general economic models and price bubbles," Journal of Mathematical Economics, Elsevier, vol. 49(3), pages 201-209.
    21. Balog, Dóra & Bátyi, Tamás László & Csóka, Péter & Pintér, Miklós, 2017. "Properties and comparison of risk capital allocation methods," European Journal of Operational Research, Elsevier, vol. 259(2), pages 614-625.
    22. Csóka, Péter & Havran, Dániel & Váradi, Kata, 2013. "Konferencia a pénzügyi piacok likviditásáról. Third Annual Financial Market Liquidity Conference BCE Befektetések és Vállalati Pénzügy Tanszék-MTA KRTK KTI Játékelméleti Kutatócsoport-Nemzetközi Banká," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(4), pages 477-485.
    23. Csóka, Péter, 2017. "Fair risk allocation in illiquid markets," Finance Research Letters, Elsevier, vol. 21(C), pages 228-234.
    24. Daniel Havran & Kata Varadi, 2015. "Price Impact and the Recovery of the Limit Order Book: Why Should We Care About Informed Liquidity Providers?," IEHAS Discussion Papers 1540, Institute of Economics, Centre for Economic and Regional Studies, Hungarian Academy of Sciences.
    25. Daniel Lacker, 2015. "Liquidity, risk measures, and concentration of measure," Papers 1510.07033,, revised Oct 2015.


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