Updating pricing rules

  • Aloisio Araujo
  • Alain Chateauneuf
  • José Heleno Faro
  • Bruno Holanda
Research Article
  • 28 Downloads

Abstract

This paper studies the problem of updating the super-replication prices of arbitrage-free finite financial markets with a frictionless bond. Any super-replication price is a pricing rule represented as the support function of some polytope of probabilities containing at least one strict positive probability, which captures the closure of the set of risk-neutral probabilities of any underlying market consistent with the given pricing rule. We show that a weak form of dynamic consistency characterizes the full (prior-by-prior) Bayesian updating of pricing rules. In order to study the problem of updating pricing rules revealing incomplete markets without frictions on all tradable securities, we first show that the corresponding polytope of probabilities must be non-expansible. We find that the full Bayesian updating does not preserve non-expansibility, unless a condition of non-trivial updating is satisfied. Finally, we show that the full Bayesian updating of pricing rules of efficient complete markets is completely stable. We also show that efficient complete markets with uniform bid–ask spreads are stable under full Bayesian updating, while efficient complete markets that fulfill the put–call parity are stable only under a Choquet pricing rule computed with respect to a regular concave nonadditive risk-neutral probability.

Keywords

Pricing rules Full Bayesian update Incomplete markets Efficient complete markets Bid–ask spreads 

JEL Classification

D52 D53 

References

  1. Acciaio, B., Föllmer, H., Penner, I.: Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles. Finance Stoch. 16(4), 669–709 (2012)CrossRefGoogle Scholar
  2. Al-Najjar, N.I., Weinstein, J.: The ambiguity aversion literature: a critical assessment. Econ. Philos. 25(03), 249–284 (2009)CrossRefGoogle Scholar
  3. Araujo, A., Chateauneuf, A., Faro, J.: Pricing rules and Arrow–Debreu ambiguous valuation. Econ. Theory 49, 1–35 (2012).  https://doi.org/10.1007/s00199-011-0660-4 CrossRefGoogle Scholar
  4. Araujo, A., Chateauneuf, A., Faro, J.H.: Financial market structures revealed by pricing rules: efficient complete markets are prevalent. J. Econ. Theory (2017).  https://doi.org/10.1016/j.jet.2017.11.002
  5. Castagnoli, E., Maccheroni, F., Marinacci, M.: Insurance premia consistent with the market. Insur. Math. Econ. 31(2), 267–284 (2002).  https://doi.org/10.1016/S0167-6687(02)00155-5 CrossRefGoogle Scholar
  6. Castagnoli, E., Favero, G., Maccheroni, F.: A problem in sublinear pricing along time. Paper presented at VIII Workshop on Quantitative Finance, Università Ca’ Foscari di Venezia, 30123 Venezia, Italy. http://virgo.unive.it/quantitativefinance2007/viewabstract.php?id=129 (2006)
  7. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M.: Put-call parity and market frictions. J. Econ. Theory 157, 730–762 (2015).  https://doi.org/10.1016/j.jet.2014.12.011 CrossRefGoogle Scholar
  8. Chateauneuf, A., Jaffray, J.Y.: Local möbius transforms of monotone capacities. In: Symbolic and Quantitative Approaches to Reasoning and Uncertainty, pp. 115–124. Springer, Berlin (1995).  https://doi.org/10.1007/3-540-60112-0_14
  9. Chateauneuf, A., Kast, R., Lapied, A.: Choquet pricing for financial markets with frictions. Math. Finance 6, 323–330 (1996)CrossRefGoogle Scholar
  10. Chateauneuf, A., Gajdos, T., Jaffray, J.Y.: Regular updating. Theory Decis. 71, 111–128 (2011)CrossRefGoogle Scholar
  11. Cox, J.C., Ross, S.A.: The valuation of options for alternative stochastic processes. J. Financ. Econ. 3, 145–166 (1976)CrossRefGoogle Scholar
  12. Detlefsen, K., Scandolo, G.: Conditional and dynamic convex risk measures. Finance Stoch. 9(4), 539–561 (2005)CrossRefGoogle Scholar
  13. Epstein, L., Le Breton, M.: Dynamically consistent beliefs must be Bayesian. J. Econ. Theory 61(1), 1–22 (1993)CrossRefGoogle Scholar
  14. Epstein, L.G., Schneider, M.: Recursive multiple-priors. J. Econ. Theory 113, 1–31 (2003).  https://doi.org/10.1016/S0022-0531(03)00097-8 CrossRefGoogle Scholar
  15. Fagin, R., Halpern, J.Y.: A new approach to updating beliefs. In: Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence (1990)Google Scholar
  16. Faro, J.H., Lefort, J.P.: Dynamic objective and subjective rationality. Insper Working Papers wpe312, Insper Working Paper, Insper Instituto de Ensino e Pesquisa. https://ideas.repec.org/p/ibm/ibmecp/wpe_312.html (2013)
  17. Florenzano, M., Gourdel, P., Van, C.: Finite Dimensional Convexity and Optimization. Studies in Economic Theory. Springer, Berlin (2001)CrossRefGoogle Scholar
  18. Galanis, S.: Dynamic consistency and subjective beliefs. Tech. rep., mimeo.  https://doi.org/10.2139/ssrn.2756612. https://ssrn.com/abstract=2756612 (2017). Accessed 1 Nov 2017
  19. Ghirardato, P.: Revisiting savage in a conditional world. Econ. Theory 20(1), 83–92 (2002).  https://doi.org/10.1007/s001990100188 CrossRefGoogle Scholar
  20. Ghirardato, P., Maccheroni, F., Marinacci, M.: Revealed Ambiguity and Its Consequences: Updating, Chapter 1, pp. 3–18. Springer, Berlin (2008).  https://doi.org/10.1007/978-3-540-68437-4_1 Google Scholar
  21. Hansen, L., Jagannathan, R.: Implications of security market data for models of dynamic economies. J. Polit. Econ. 99(2), 225–262 (1991)CrossRefGoogle Scholar
  22. Harrison, J.M., Kreps, D.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20(3), 381–408 (1979)CrossRefGoogle Scholar
  23. Huber, P.J.: Robust Statistics. Wiley Series in Probabilities and Mathematical Statistics. Wiley, New York (1981)Google Scholar
  24. Jaffray, J.Y.: Bayesian updating and belief functions. IEEE Trans. Syst. Man Cybern. 22, 1144–1152 (1992)CrossRefGoogle Scholar
  25. Jouini, E.: Price functionals with bidask spreads: an axiomatic approach. J. Math. Econ. 34(4), 547–558 (2000).  https://doi.org/10.1016/S0304-4068(99)00023-3 CrossRefGoogle Scholar
  26. Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transaction costs. J. Econ. Theory 66, 178–197 (1995)CrossRefGoogle Scholar
  27. Jouini, E., Kallal, H.: Efficient trading strategies in the presence of market frictions. Rev. Financ. Stud. 14(2), 343–369 (2001)CrossRefGoogle Scholar
  28. Luttmer, E.G.J.: Asset pricing in economies with frictions. Econometrica 64(6), 1439–1467 (1996)CrossRefGoogle Scholar
  29. Pires, C.P.: A rule for updating ambiguous beliefs. Theory Decis. 53, 137–152 (2002)CrossRefGoogle Scholar
  30. Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1997)Google Scholar
  31. Ross, S.A.: The arbitrage theory of capital asset pricing. J. Econ. Theory 13, 341–360 (1976)CrossRefGoogle Scholar
  32. Ross, S.A.: A simple approach to the valuation of risky streams. J. Bus. 51, 453–475 (1978)CrossRefGoogle Scholar
  33. Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)CrossRefGoogle Scholar
  34. Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  35. Siniscalchi, M.: Dynamic choice under ambiguity. Theor. Econ. 6(3). https://EconPapers.repec.org/RePEc:the:publsh:571 (2011). Accessed 15 Aug 2017
  36. Tutsch, S.: Update rules for convex risk measures. Quant. Finance 8(8), 833–843 (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.Brazilian School of Economics and FinanceFGV EPGERio de JaneiroBrazil
  3. 3.Research Department in EconomicsIPAG Business SchoolParis Cedex 05France
  4. 4.Paris School of EconomicsUniversité de Paris IParis Cedex 13France
  5. 5.InsperVila OlímpiaBrazil
  6. 6.Faculdade de Administraçño, Contabilidade e EconomiaUniversidade Federal de GoiásGoiâniaBrazil

Personalised recommendations