Financial economics without probabilistic prior assumptions

The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knighti...

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Veröffentlicht in:	Decisions in economics and finance Jg. 38; H. 1; S. 75 - 91
1. Verfasser:	Riedel, Frank
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Milan Springer Milan 01.04.2015 Springer Nature B.V
Schlagworte:	Arbitrage Asset backed securities Black swan event Econometrics Economic theory Economic Theory/Quantitative Economics/Mathematical Methods Economics Economics and Finance Equilibrium Finance Financial economics Free markets Hedging Linear algebra Linear programming Management Market equilibrium Mathematical analysis Operations Research/Decision Theory Prices Probability Probability distribution Public Finance Random variables Sophistication Studies Uncertainty Volatility D53 Infinite-dimensional linear programming Full support martingale measure Probability-free finance G12 Fundamental theorem of asset pricing Superhedging
ISSN:	1593-8883, 1129-6569
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Abstract	The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knightian uncertainty, we might challenge such an assumption on the probabilistic sophistication of our market model. The present paper shows that one can still develop a sound model of arbitrage pricing under complete Knightian uncertainty as long as certain continuity conditions are met. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no-arbitrage theory to economic equilibrium by establishing a variant of the Harrison–Kreps theorem on viability and no arbitrage. Finally, we consider (super) hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem.
AbstractList	The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knightian uncertainty, we might challenge such an assumption on the probabilistic sophistication of our market model. The present paper shows that one can still develop a sound model of arbitrage pricing under complete Knightian uncertainty as long as certain continuity conditions are met. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no-arbitrage theory to economic equilibrium by establishing a variant of the Harrison-Kreps theorem on viability and no arbitrage. Finally, we consider (super) hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem. The treatment of uncertainty in general equilibrium theory in the style of Arrow and Debreu does not require a prior probability on the state space. Finance models nevertheless treat payoffs as random variables, implicitly or explicitly using a known probability distribution. In the light of Knightian uncertainty, we might challenge such an assumption on the probabilistic sophistication of our market model. The present paper shows that one can still develop a sound model of arbitrage pricing under complete Knightian uncertainty as long as certain continuity conditions are met. The pricing functional given by an arbitrage-free market can be identified with a full support martingale measure (instead of equivalent martingale measure). We relate the no-arbitrage theory to economic equilibrium by establishing a variant of the Harrison–Kreps theorem on viability and no arbitrage. Finally, we consider (super) hedging of contingent claims and embed it in a classical infinite-dimensional linear programming problem.
Author	Riedel, Frank
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Cites_doi	10.1007/BFb0088364 10.1016/0304-4068(81)90010-0 10.1112/plms/s3-39.1.51 10.1090/S0002-9947-1970-0270129-5 10.1016/j.jmateco.2013.09.005 10.1007/BF01450498 10.1080/17442509008833613 10.1016/0304-4149(94)90128-7 10.1007/s00780-008-0074-8 10.1007/BFb0089488 10.1007/978-3-0348-8268-2_2 10.1093/rfs/hht018 10.1007/s002080050220 10.1016/0304-405X(79)90015-1 10.1016/0022-0531(79)90043-7 10.1016/j.bulsci.2011.06.008 10.1111/mafi.12060 10.1515/9783110198065 10.2307/2951565 10.1112/jlms/49.1.109
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Keywords	D53 Infinite-dimensional linear programming Full support martingale measure Probability-free finance G12 Fundamental theorem of asset pricing Superhedging
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References	van CasterenJAStrictly positive functionals on vector latticesProc. Lond. Math. Soc.1979393517210.1112/plms/s3-39.1.51 Vorbrink, Jörg: Financial Markets with Volatility Uncertainty. IMW Working Paper 441, http://www.imw.uni-bielefeld.de/papers/files/imw-wp-441 (2010) MachinaMJSchmeidlerDA more robust definition of subjective probabilityEconometrcia19926074578010.2307/2951565 DalangRCMortonAWillingerWEquivalent martingale measures and no-arbitrage in stochastic securities market modelsStoch. Stoch. Rep.19902918520110.1080/17442509008833613 Sondermann, D.: Introduction to Stochastic Calculus for Finance, volume 579 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2006) FöllmerHDer Zufall in den Wirtschaftswissenschaften: Zur Rolle der Wahrscheinlichkeitstheorie in der Theorie der FinanzmärkteNova Acta Leopold.1999308113125 DelbaenFSchachermayerWA general version of the fundamental theorem of asset pricingMath. Ann.199430046352010.1007/BF01450498 Föllmer, H.: Calcul d’Itô sans probabilités, vol. 850. Séminaire de Probabilités XV 1979/80. Lecture Notes in Mathematics, pp. 143–150 (1981) SummersWHDual spaces of weighted spacesTrans. Am. Math. Soc.197015132333310.1090/S0002-9947-1970-0270129-5 Bouchard, B., Nutz, M.: Arbitrage and Duality in Non-dominated Discrete-Time Models. Ann. Appl. Probab. (forthcoming) Acciaio, B., Beiglbck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Financ. (forthcoming) KrepsDArbitrage and equilibrium in economies with infinitely many commoditiesJ. Math. Econ.19818153510.1016/0304-4068(81)90010-0 AndersonJENashPLinear Programming in Infinite-Dimensional Spaces1987New YorkWiley BenderChristianSottinenTommiValkeilaEskoPricing by hedging and no-arbitrage beyond semimartingalesFinanc. Stochast.20081244146810.1007/s00780-008-0074-8 Peng, S.: arXiv:0711.2834 (2007) Beissner, P.: Viability and No-Arbitrage Under Uncertainty. IMW Working Paper (2012) Föllmer, H.: Probabilistic Aspects of Financial Risk. Plenary Lecture at the Third European Congress of Mathematics. In: Proceedings of the European Congress of Mathematics, Barcelona 2000 (2001) FöllmerHSchiedAStochastic Finance. Studies in Mathematics2002BerlinWalter de Gruyter HarrisonJMKrepsDMartingales and arbitrage in multi-period security marketsJ. Econ. Theory19792038140810.1016/0022-0531(79)90043-7 DelbaenFSchachermayerWThe fundamental theorem of asset pricing for unbounded stochastic processesMath. Ann.199831221525010.1007/s002080050220 Coviello R., Di Girolami, C., Russo, F.: On stochastic calculus related to financial assets without semimartingales. Bull. Sci. Math. 135, 733–774 (2011) van CasterenJAStrictly positive radon measuresJ. Lond. Math. Soc.199449110912310.1112/jlms/49.1.109 Epstein, L., Ji, S.: Ambiguous volatility, possibility and utility in continuous time. J. Math. Econ. 50, 269–282 (2014) Epstein, L., Ji, S.: Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26, 1740–1786 (2013) Yan, J.A.: Charactérisation d’une classe d’ensembles convexes de l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^1$$\end{document} ou h1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^1$$\end{document}. In: Séminaire de Probabilité XIV, volume 469 of Lecture Notes in Mathematics. Springer, Berlin, pp. 220–222 (1980) BickAWillingerWDynamic spanning without probabilitiesStoch. Process. Their Appl.19945034937410.1016/0304-4149(94)90128-7 Cox, John C., Ross, Stephen A., Rubinstein, M.: Option pricing: a simplified approach. J. Financ. Econ. 7, 229–263 (1979) JE Anderson (159_CR2) 1987 JM Harrison (159_CR18) 1979; 20 Christian Bender (159_CR4) 2008; 12 JA Casteren van (159_CR25) 1994; 49 159_CR12 159_CR13 159_CR16 159_CR14 H Föllmer (159_CR17) 2002 MJ Machina (159_CR20) 1992; 60 JA Casteren van (159_CR24) 1979; 39 159_CR1 159_CR3 WH Summers (159_CR23) 1970; 151 159_CR6 159_CR8 A Bick (159_CR5) 1994; 50 159_CR7 F Delbaen (159_CR11) 1998; 312 H Föllmer (159_CR15) 1999; 308 RC Dalang (159_CR9) 1990; 29 D Kreps (159_CR19) 1981; 8 159_CR21 F Delbaen (159_CR10) 1994; 300 159_CR22 159_CR27 159_CR26
References_xml	– reference: DelbaenFSchachermayerWThe fundamental theorem of asset pricing for unbounded stochastic processesMath. Ann.199831221525010.1007/s002080050220 – reference: Föllmer, H.: Probabilistic Aspects of Financial Risk. Plenary Lecture at the Third European Congress of Mathematics. In: Proceedings of the European Congress of Mathematics, Barcelona 2000 (2001) – reference: BenderChristianSottinenTommiValkeilaEskoPricing by hedging and no-arbitrage beyond semimartingalesFinanc. Stochast.20081244146810.1007/s00780-008-0074-8 – reference: DalangRCMortonAWillingerWEquivalent martingale measures and no-arbitrage in stochastic securities market modelsStoch. Stoch. Rep.19902918520110.1080/17442509008833613 – reference: Bouchard, B., Nutz, M.: Arbitrage and Duality in Non-dominated Discrete-Time Models. Ann. Appl. Probab. (forthcoming) – reference: FöllmerHDer Zufall in den Wirtschaftswissenschaften: Zur Rolle der Wahrscheinlichkeitstheorie in der Theorie der FinanzmärkteNova Acta Leopold.1999308113125 – reference: FöllmerHSchiedAStochastic Finance. Studies in Mathematics2002BerlinWalter de Gruyter – reference: AndersonJENashPLinear Programming in Infinite-Dimensional Spaces1987New YorkWiley – reference: Epstein, L., Ji, S.: Ambiguous volatility and asset pricing in continuous time. Rev. Financ. Stud. 26, 1740–1786 (2013) – reference: MachinaMJSchmeidlerDA more robust definition of subjective probabilityEconometrcia19926074578010.2307/2951565 – reference: Vorbrink, Jörg: Financial Markets with Volatility Uncertainty. IMW Working Paper 441, http://www.imw.uni-bielefeld.de/papers/files/imw-wp-441 (2010) – reference: van CasterenJAStrictly positive functionals on vector latticesProc. Lond. Math. Soc.1979393517210.1112/plms/s3-39.1.51 – reference: KrepsDArbitrage and equilibrium in economies with infinitely many commoditiesJ. Math. Econ.19818153510.1016/0304-4068(81)90010-0 – reference: Beissner, P.: Viability and No-Arbitrage Under Uncertainty. IMW Working Paper (2012) – reference: Epstein, L., Ji, S.: Ambiguous volatility, possibility and utility in continuous time. J. Math. Econ. 50, 269–282 (2014) – reference: BickAWillingerWDynamic spanning without probabilitiesStoch. Process. Their Appl.19945034937410.1016/0304-4149(94)90128-7 – reference: Sondermann, D.: Introduction to Stochastic Calculus for Finance, volume 579 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (2006) – reference: Föllmer, H.: Calcul d’Itô sans probabilités, vol. 850. Séminaire de Probabilités XV 1979/80. Lecture Notes in Mathematics, pp. 143–150 (1981) – reference: Coviello R., Di Girolami, C., Russo, F.: On stochastic calculus related to financial assets without semimartingales. Bull. Sci. Math. 135, 733–774 (2011) – reference: Acciaio, B., Beiglbck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Financ. (forthcoming) – reference: van CasterenJAStrictly positive radon measuresJ. Lond. Math. Soc.199449110912310.1112/jlms/49.1.109 – reference: Yan, J.A.: Charactérisation d’une classe d’ensembles convexes de l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^1$$\end{document} ou h1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^1$$\end{document}. In: Séminaire de Probabilité XIV, volume 469 of Lecture Notes in Mathematics. Springer, Berlin, pp. 220–222 (1980) – reference: DelbaenFSchachermayerWA general version of the fundamental theorem of asset pricingMath. Ann.199430046352010.1007/BF01450498 – reference: Cox, John C., Ross, Stephen A., Rubinstein, M.: Option pricing: a simplified approach. J. Financ. Econ. 7, 229–263 (1979) – reference: SummersWHDual spaces of weighted spacesTrans. Am. Math. Soc.197015132333310.1090/S0002-9947-1970-0270129-5 – reference: Peng, S.: arXiv:0711.2834 (2007) – reference: HarrisonJMKrepsDMartingales and arbitrage in multi-period security marketsJ. Econ. Theory19792038140810.1016/0022-0531(79)90043-7 – ident: 159_CR14 doi: 10.1007/BFb0088364 – volume: 8 start-page: 15 year: 1981 ident: 159_CR19 publication-title: J. Math. Econ. doi: 10.1016/0304-4068(81)90010-0 – ident: 159_CR21 – volume: 39 start-page: 51 issue: 3 year: 1979 ident: 159_CR24 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/s3-39.1.51 – volume: 151 start-page: 323 year: 1970 ident: 159_CR23 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1970-0270129-5 – ident: 159_CR13 doi: 10.1016/j.jmateco.2013.09.005 – volume: 300 start-page: 463 year: 1994 ident: 159_CR10 publication-title: Math. 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SubjectTerms	Arbitrage Asset backed securities Black swan event Econometrics Economic theory Economic Theory/Quantitative Economics/Mathematical Methods Economics Economics and Finance Equilibrium Finance Financial economics Free markets Hedging Linear algebra Linear programming Management Market equilibrium Mathematical analysis Operations Research/Decision Theory Prices Probability Probability distribution Public Finance Random variables Sophistication Studies Uncertainty Volatility
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Title	Financial economics without probabilistic prior assumptions
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