Abstract
This paper studies arbitrage-free financial markets with bid-ask spreads whose super-hedging prices are submodular. The submodular assumption on the super-hedging price, or the supermodularity usually assumed on utility functions, is the formal expression of perfect complementarity, which dates back to Fisher, Pareto, and Edgeworth, according to Samuelson (J Econ Lit 12:1255–1289, 1974). Our main contribution provides several characterizations of financial markets with frictions that are submodular as a consequence of a more general study of submodular pricing rules. First, a market is submodular if and only if its super-hedging price is a Choquet integral and if and only if its set of risk-neutral probabilities is representable as the core of a submodular non-additive probability that is uniquely defined, called risk-neutral capacity. Second, a market is representable by its risk neutral capacity if and only if it is equivalent to a market, only composed of bid-ask event securities.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
08 April 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00199-022-01430-8
References
Amihud, Y., Mendelson, H.: Asset pricing and the bid-ask spread. J. Financ. Econ. 17, 223–249 (1986)
Angeloni, L., Cornet, B.: Existence of financial equilibria in a multi-period stochastic economy. Adv. Math. Econ. 8, 1–31 (2006)
Araujo, A., Chateauneuf, A., Faro, J.H.: Pricing rules and Arrow–Debreu ambiguous valuation. Econ. Theory 49, 1–35 (2012). https://doi.org/10.1007/s00199-011-0660-4
Araujo, A., Chateauneuf, A., Faro, J.H.: Financial market structures revealed by pricing rules: efficient complete markets are prevalent. J. Econ. Theory 173, 257–288 (2018)
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)
Aouani, Z., Cornet, B.: Existence of financial equilibria with restricted participation. J. Math. Econ. 45, 772–786 (2009)
Aouani, Z., Cornet, B.: Reduced equivalent form of a financial structure. J. Math. Econ. 47, 318–327 (2011)
Aouani, Z., Cornet, B.: Characterizing useless-free financial structures. Set Valued Var. Anal. 24, 149–166 (2016)
Aouani, Z., Cornet, B.: Eliminating useless portfolios in constrained financial economies. Econ. Theory 63(1), 159–190 (2017). https://doi.org/10.1007/s00199-016-1000-5
Bejan, C.: Investment and financing in incomplete markets. Econ. Theory 69, 149–182 (2020). https://doi.org/10.1007/s00199-018-1160-6
Bensaid, B., Lesne, J.-P., Pagés, H., Scheinkman, J.: Derivative asset pricing with transaction costs. Math. Finance 2, 63–86 (1992)
Berger, J.O.: Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer, Berlin (1985)
Cass, D.: Competitive equilibrium with incomplete financial markets. J. Math. Econ. 42, 384–405 (2006)
Castagnoli, E., Maccheroni, F., Marinacci, M.: Insurance premia consistent with the market. Insur. Math. Econ. 31, 267–284 (2002)
Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M.: Put–Call parity and market frictions. J. Econ. Theory 157, 730–762 (2015)
Chateauneuf, A., Cornet, B.: Choquet representability of submodular functions. Math. Program. 168(1–2), 615–629 (2018)
Chateauneuf, A., Cornet, B.: The risk neutral non-additive probability with market frictions. Econ. Theory Bull. (2022). https://doi.org/10.1007/s40505-022-00216-4
Chateauneuf, A., Kast, R., Lapied, A.: Choquet pricing for financial markets with frictions. Math. Finance 6, 323–330 (1996)
Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)
Cornet, B., Gopalan, R.: Arbitrage and equilibrium with portfolio constraints. Econ. Theory 45, 227–252 (2010). https://doi.org/10.1007/s00199-009-0506-5
Cornet, B., Ranjan, A.: A remark on the set of arbitrage-free prices in a multi-period economy. Int. J. Econ. Theory 9, 35–43 (2013)
Cox, J.C., Ross, S.A.: A survey of some new results in financial option pricing theory. J. Finance 31, 383–402 (1976)
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)
Denneberg, D.: Non-additive Measure and Integral. Kluwer, Dordrecht (1994)
Duffie, D., Huang, C.: Multiperiod security markets with differential information. J. Math. Econ. 15, 283–303 (1986)
Flam, S.D.: Emergence of price-taking behavior. Econ. Theory 70, 847–870 (2020). https://doi.org/10.1007/s00199-019-01232-5
Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econ. 16, 65–88 (1987)
Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18, 141–153 (1989)
Harrison, M.J., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20, 381–408 (1979)
Huber, P.J.: Robust Statistics. Wiley Series in Probabilities and Mathematical Statistics. Wiley, Hoboken (1981)
Jouini, E.: Price functionals with bid-ask spreads: an axiomatic approach. J. Math. Econ. 34, 547–558 (2000)
Jouini, E., Kallal, H.: Martingales and arbitrage in securities markets with transactions costs. J. Econ. Theory 66, 178–197 (1995)
Jouini, E., Kallal, H.: Efficient trading strategies in the presence of market frictions. Rev. Financ. Stud. 14, 343–369 (2001)
Luttmer, E.G.: Asset pricing in economies with frictions. Econometrica 64, 1439–1467 (1996)
Marinacci, M., Montrucchio, L.: Introduction to the Mathematics of Ambiguity. Uncertainty in Economic Theory, Routledge (2004)
Marinacci, M., Montrucchio, L.: Ultramodular functions. Math. Oper. Res. 30, 311–332 (2005)
Markeprand, T.: On financial equilibrium with intermediation costs. J. Math. Econ. 44, 148–156 (2008)
Martins-da-Rocha, V.F., Vailakis, Y.: Financial markets with endogenous transaction costs. Econ. Theory 45, 65–97 (2010). https://doi.org/10.1007/s00199-009-0498-1
Milgrom, P., Shannon, C.: Monotone comparative statics. Econometrica 62, 157–180 (1994)
Milgrom, P., Roberts, J.: Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58, 1255–78 (1990)
Moulin, H.: An application of the Shapley value to fair division with money. Econometrica 60, 1331–1349 (1992)
Müller, A., Scarsini, M.: Fear of loss, inframodularity, and transfers. J. Econ. Theory 147(4), 41–48 (2012)
Prisman, E.: Valuation of risky assets in arbitrage-free economies with frictions. J. Finance 56, 545–557 (1986)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Ross, S.A.: Risk, return, and arbitrage. In: Friend, I., Bicksler, J. (eds.) Risk and Return in Finance. Ballinger, Cambridge (1976)
Ross, S.A.: A simple approach to the valuation of risky streams. J. Bus. 51, 453–485 (1978)
Ross, S.A.: Arbitrage and martingales with taxation. J. Polit. Econ. 95, 371–393 (1987)
Samuelson, P.: Complementarity: an essay on the 40th anniversary of the Hicks–Allen revolution in demand theory. J. Econ. Lit. 12, 1255–1289 (1974)
Sharkey, W.W., Telser, L.G.: Supportable cost functions for the multiproduct firm. J. Econ. Theory 18, 23–37 (1978)
Schmeidler, D.: Integral representation without additivity. Proc. Am. Math. Soc. 97, 255–261 (1986)
Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57, 571–587 (1989)
Shapley, L.S.: Cores of convex games. Int. J. Game Theory 1, 12–26 (1971)
Siconolfi, P.: Equilibrium with asymmetric constraints on portfolio holdings and incomplete financial markets. In: Galeotti, M., et al. (eds.) Non-linear Dynamics in Economics and Social Sciences, pp. 271–292. Societa’Pitagora, Bergamo (1989)
Topkis, D.: Minimizing a submodular function on a lattice. Oper. Res. 26, 305–321 (1978)
Topkis, D.: Supermodularity and Complementarity. Princeton University Press, Princeton (1998)
Vives, X.: Nash equilibrium with strategic complementarities. J. Math. Econ. 19, 305–321 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors have benefited from valuable comments and suggestions from the participants of the SAET Meetings at Paris, Warwick, Tokyo, and Cambridge, the RUD meeting, the D-TEA Conference, the European Workshops on General Equilibrium Theory, the Kansas Workshop in Economic Theory, and seminars in Bielefeld, Detroit, Glasgow, Iowa, Louvain, New York, and San Diego, where the paper has been presented. The first author thanks the University of Kansas for its generous support and its hospitality for research stays made during several consecutive years.
The original online version of this article was revised due to a retrospective Open Access order.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chateauneuf, A., Cornet, B. Submodular financial markets with frictions. Econ Theory 73, 721–744 (2022). https://doi.org/10.1007/s00199-022-01415-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-022-01415-7
Keywords
- Submodularity
- financial markets
- Frictions
- Bid-ask
- Arbitrage
- Multi-prior model
- Super-hedging price
- Super-replication
- Risk measure
- Pricing rules
- Choquet integral
- Event securities