Implied liquidity risk premia in option markets


The theory of conic finance replaces the classical one-price model by a two-price model by determining bid and ask prices for future terminal cash flows in a consistent manner. In this framework, we derive closed-form solutions for bid and ask prices of plain vanilla European options, when the density of the log-returns is log-concave. Assuming that log-returns are normally or Laplace distributed, we apply the results to a time-series of real market data and compute an implied liquidity risk premium to describe the bid–ask spread. We compare this approach to the classical attempt of describing the spread by quoting Black–Scholes implied bid and ask volatilities and demonstrate that the new approach characterize liquidity over time significantly better.

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  1. Albrecher, H., Guillaume, F., Schoutens, W.: Implied liquidity: model sensitivity. J Empir Finance 23, 48–67 (2013)

    Article  Google Scholar 

  2. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math Finance 9(3), 203–228 (1999)

    Article  Google Scholar 

  3. Balbás, A., Garrido, J., Mayoral, S.: Properties of distortion risk measures. Methodol Comput Appl Probab 11, 385–399 (2009)

    Article  Google Scholar 

  4. Bernardo, A.E., Ledoit, O.: Gain, loss, and asset pricing. J Polit Econ 108(1), 144–172 (2000)

    Article  Google Scholar 

  5. Cherny, A., Madan, D.B.: New measures for performance evaluation. Rev Financ Stud 22(7), 2571–2606 (2009)

    Article  Google Scholar 

  6. Corcuera, J.M., Guillaume, F., Madan, D.B., Schoutens, W.: Implied liquidity: towards stochastic liquidity modelling and liquidity trading. Int J Portfolio Anal Manag 1(1), 80–91 (2012)

    Article  Google Scholar 

  7. Corcuera, J.M., Guillaume, F., Leoni, P., Schoutens, W.: Implied Lévy volatility. Quant Finance 9(4), 383–393 (2009)

    Article  Google Scholar 

  8. Dhaene, J., Dony, J., Forys, M. B., Linders, D., Schoutens, W.: FIX: the fear index—measuring market fear. In: Topics in Numerical Methods for Finance, pp 37-55. Springer, Boston, MA (2012)

  9. Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter, Berlin/New York (2011)

    Google Scholar 

  10. Kijima, M.: A multivariate extension of equilibrium pricing transforms: the multivariate Esscher and Wang transforms for pricing financial and insurance risks. ASTIN Bull J IAA 36(1), 269–283 (2006)

    Article  Google Scholar 

  11. Kijima, M., Muromachi, Y.: On the Wang transform with fat-tail distributions. In: Proceedings of Stanford—Tsukuba Workshop (2006)

  12. Kijima, M., Muromachi, Y.: An extension of the Wang transform derived from Bühlmann’s economic premium principle for insurance risk. Insur Math Econ 42(3), 887–896 (2008)

    Article  Google Scholar 

  13. Kusuoka, S.: On law invariant coherent risk measures. In: Advances in Mathematical Economics, pp. 83–95 (2001)

  14. Madan, D.B.: A two price theory of financial equilibrium with risk management implications. Ann Finance 8(4), 489–505 (2012)

    Article  Google Scholar 

  15. Madan, D.B.: Asset pricing theory for two price economies. Ann Finance 11(1), 1–35 (2015)

    Article  Google Scholar 

  16. Madan, D.B.: Adapted hedging. Ann Finance 12(3–4), 305–334 (2016a)

    Article  Google Scholar 

  17. Madan, D.B.: Conic portfolio theory. Int J Theor Appl Finance 19(03), 1650019 (2016b)

    Article  Google Scholar 

  18. Madan, D.B., Cherny, A.: Markets as a counterparty: an introduction to conic finance. Int J Theor Appl Finance 13(08), 1149–1177 (2010)

    Article  Google Scholar 

  19. Madan, D.B., Schoutens, W.: Conic coconuts: the pricing of contingent capital notes using conic finance. Math Financ Econ 4(2), 87–106 (2011)

    Article  Google Scholar 

  20. Madan, D.B., Schoutens, W.: Applied Conic Finance. Cambridge University Press, Cambridge (2016)

    Google Scholar 

  21. Madan, D.B., Schoutens, W.: Conic asset pricing and the costs of price fluctuations. Ann Finance (2018).

  22. Tsukahara, H.: One-parameter families of distortion risk measures. Math Finance 19(4), 691–705 (2009)

    Article  Google Scholar 

  23. Wang, S.S.: A class of distortion operators for pricing financial and insurance risks. J Risk Insur 67, 15–36 (2000)

    Article  Google Scholar 

  24. Wang, S.S.: A universal framework for pricing financial and insurance risks. Astin Bull J IAA 32(02), 213–234 (2002)

    Article  Google Scholar 

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Correspondence to Gero Junike.

Additional information

G. Junike: This research is partially supported by 13th UAB-PIF scholarship.

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Guillaume, F., Junike, G., Leoni, P. et al. Implied liquidity risk premia in option markets. Ann Finance 15, 233–246 (2019).

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  • Conic finance
  • Distortion functions
  • WANG-transform
  • Laplace distortion

JEL Classification

  • C02
  • D53
  • G12
  • G13