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Private information and the ‘Information function’: A survey of possible uses. (English) Zbl 1136.91438
Summary: Under certain conditions private information can be a source of trade. Arbitrage for instance can occur as a result of the existence of private information. In this paper we want to explicitly model information. To do so we define an ‘information function’. This information function is a mathematical object, also known as a so called ‘wave function’. We use the definition of wave function as it is used in quantum mechanics and we attempt to show the usefulness of this wave function in an economic context. We attempt to answer the following questions. How does the information function relate to private information? How can we use the information function to define the ‘quantity’ of information? How can we use the information function in arbitrage-based option pricing? How can the information function be used in the formulation of a so called Universal Brownian motion?

MSC:
91B28 Finance etc. (MSC2000)
91B24 Microeconomic theory (price theory and economic markets)
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[1] Arfi B. (2005), Resolving the trust predicament: a quantum game theoretic approach. Theory and Decision 59(2): 127–174 · Zbl 1119.91006 · doi:10.1007/s11238-005-8632-4
[2] Baaquie B. (2005) Quantum Finance. Cambridge University Press, Cambridge
[3] Bacciagaluppi G. (1999), Nelsonian mechanics revisited. Foundations of Physics Letters 12(1): 1–16 · doi:10.1023/A:1021622603864
[4] Back K., Baruch S. (2004), Information in securities markets: Kyle meets Glosten and Milgrom. Econometrica 72: 433–465 · Zbl 1130.91328 · doi:10.1111/j.1468-0262.2004.00497.x
[5] Black F., Scholes M. (1973), The pricing of options and corporate liabilities. Journal of Political Economy 81: 637–659 · Zbl 1092.91524 · doi:10.1086/260062
[6] Blackwell D. (1953), Equivalent comparisons of experiments. Annals of Mathematical Statistics 24: 265–272 · Zbl 0050.36004 · doi:10.1214/aoms/1177729032
[7] Bohm D. (1952), A suggested interpretation of the quantum theory in terms of ’hidden’ variables, Part I and II. Physical Review 85: 166–193 · Zbl 0046.21004 · doi:10.1103/PhysRev.85.166
[8] Bohm D. (1987), Hidden variables and the implicate order. In: Hiley B., Peat F. (eds), Quantum Implications: Essays in Honour of David Bohm. Routledge, New York · Zbl 0717.17003
[9] Bohm D., Hiley B. (1989), Non-locality and locality in the stochastic interpretation of quantum mechanics. Physics Reports 172(3): 93–122 · doi:10.1016/0370-1573(89)90160-9
[10] Bohm D., Hiley B. (1993), The Undivided Universe. Routledge, New York · Zbl 0990.81503
[11] Bowman G. (2005), On the classical limit in Bohm’s theory. Foundations of Physics 35(4): 605–625 · Zbl 1084.81009 · doi:10.1007/s10701-004-2013-7
[12] Broadie M., Detemple J. (2004), Option pricing: Valuation models and applications. Management Science 50(9): 1145–1177 · doi:10.1287/mnsc.1040.0275
[13] Brody, D.C., Hughston, L.P. and Macrina, A. (2006), Information based asset pricing, King’s College (London) (Department of Mathematics) and Imperial College (London) (Blackett Laboratory), Working paper (submitted), 1–32.
[14] Busemeyer J., Wang Z., Townsend J.T. (2006), Quantum dynamics of human decision making. Journal of Mathematical Psychology 50(3): 220–241 · Zbl 1186.91062 · doi:10.1016/j.jmp.2006.01.003
[15] Choustova, O. (2001), Pilot wave quantum model for the stock market, in Khrennikov A. (ed.), Quantum Theory: Reconsideration of Foundations. Växjö University Press (Sweden), Växjö, pp. 41–58.
[16] Choustova O. (2006), Quantum Bohmian model for financial markets. Physica A 374: 304–314 · doi:10.1016/j.physa.2006.07.029
[17] Choustova, O. (2007), Quantum modeling of nonlinear dynamics of prices of shares: Bohmian approach, Theoretical and Mathematical Physics, in press · Zbl 1186.91211
[18] Epstein, L. Schneider, M. (2006), Ambiguity, information quality and asset pricing, University of Rochester - Center for Economic Research (RCER), Working Paper: No. 519, 1–31.
[19] Falmagne, J.C., Regenwetter, M. and Grofman, B. (1997), A stochastic model for the evolution of preferences, in Marley, A.A.J. (eds.), Choice, Decision and Measurement: Essays in Honour of Duncan Luce. New Jersey, Earlbaum, pp. 113–131
[20] Fedotov S., Panayides S. (2005), Stochastic arbitrage returns and its implications for option pricing. Physica A 345: 207–217
[21] Georgescu-Roegen, N. (1999), The Entropy Law and the Economic Process. Harvard University Press.
[22] Grössing G. (2002), Quantum cybernetics: A new perspective for Nelson’s stochastic theory, non-locality, and the Klein-Gordon equation. Physics Letters A 296(1): 1–8 · Zbl 1039.81517 · doi:10.1016/S0375-9601(02)00071-3
[23] Haven E. (2005a), Pilot-wave theory and financial option pricing. International Journal of Theoretical Physics 44(11): 1957–1962 · Zbl 1094.81015 · doi:10.1007/s10773-005-8973-3
[24] Haven E. (2005b), Analytical solutions to the backward Kolmogorov PDE via an adiabatic approximation to the Schrödinger PDE. Journal of Mathematical Analysis and Applications 311: 439–444 · Zbl 1086.35512 · doi:10.1016/j.jmaa.2005.02.058
[25] Haven E. (2006), Bohmian mechanics in a macro-scopic quantum system. American Institute of Physics Conference Proceedings 810(1): 330–335 · Zbl 1110.81310
[26] Hiley B., Pylkkänen P. (1997) Active information and cognitive science - A reply to Kieseppä. In: Pylkkänen P., Pylkkö P., Hautamäki A. (eds). Brain, Mind and Physics. IOS Press, Amsterdam
[27] Holland, P. (1993), The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press.
[28] Ilinski, K. (2001), Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing. J. Wiley.
[29] Itô K. (1951), On stochastic differential equations, Memoirs. American Mathematical Society 4: 1–51
[30] Khrennikov A.Yu. (1999) Classical and quantum mechanics on information spaces with applications to cognitive, psychological, social and anomalous phenomena. Foundations of Physics 29: 1065–1098 · doi:10.1023/A:1018885632116
[31] Khrennikov, A. Yu. (2004), Information Dynamics in Cognitive, Psychological and Anomalous Phenomena, Series in the Fundamental Theories of Physics, v. 138, Kluwer, Dordrecht.
[32] Khrennikov A.Yu., Haven E. (2007) Does probability interference exist in social science?. American Institute of Physics Conference Proceedings 889: 299–310 · Zbl 1139.81304
[33] La Mura, P. (2003), Correlated equilibriae of classical strategic games with quantum signals, Working Paper No. 61, Leipzig Graduate School of Management.
[34] La Mura, P. (2006), Projective Expected Utility, Mimeo, Leipzig Graduate School of Management.
[35] Lambert Mogiliansky, A., Zamir, S. and Zwirn, H. (2003), Type indeterminacy: A model of the KT (Kahneman-Tversky) man, Discussion Paper 343, Centre for the Study of Rationality, The Hebrew University of Jerusalem - Israel. · Zbl 1178.91044
[36] Ma C. (2006) Intertemporal recursive utility and an equilibrium asset pricing model in the presence of Levy jumps. Journal of Mathematical Economics 42: 131–160 · Zbl 1137.91013 · doi:10.1016/j.jmateco.2005.08.003
[37] Milgrom P., Stokey N. (1982) Information, trade and common knowledge. Journal of Economic Theory 12: 112–128 · Zbl 0485.90018
[38] Morrison, M.A. (1990), Understanding Quantum Physics: A User’s Manual, Prentice- Hall.
[39] Nakata, H. (2006), Modelling choice of information acquisition, Working paper - Dept. of AFM, University of Essex.
[40] Nelson E. (1966) Derivation of the Schrödinger equation from Newtonian mechanics. Physical Review 150: 1079–1085 · doi:10.1103/PhysRev.150.1079
[41] Nelson, E. (1967), Dynamical Theories of Brownian Motion. Princeton University Press. · Zbl 0165.58502
[42] Otto M. (1999) Stochastic relaxational dynamics applied to finance: Towards non- equilibrium option pricing theory. European Physical Journal B 14: 383–394
[43] Panayides, S. (2005), Derivative pricing and hedging for incomplete markets: Stochastic arbitrage and adaptive procedure for stochastic volatility, Ph.D. Thesis, The School of Mathematics, University of Manchester.
[44] Paul, W. and Baschnagel, J. (2000), Stochastic Processes: From Physics to Finance. Springer Verlag. · Zbl 1295.60004
[45] Penrose, R., Shimony, A., Cartwright, N. and Hawking, S. (2000), The Large, the Small and the Human Mind. Cambridge University Press.
[46] Piotrowski E.W., Sladkowski J. (2002a) Quantum bargaining games. Physica A 308(1): 391–401 · Zbl 0995.91018 · doi:10.1016/S0378-4371(02)00592-7
[47] Piotrowski E.W., Sladkowski J. (2002b) Quantum market games. Physica A 312(1): 208–216 · Zbl 0997.91015 · doi:10.1016/S0378-4371(02)00842-7
[48] Piotrowski E.W., Sladkowski J. (2003a) Trading by quantum rules: Quantum anthropic principle. International Journal of Theoretical Physics 42(5): 1101–1106 · Zbl 1037.81021 · doi:10.1023/A:1025495128226
[49] Piotrowski E.W., Sladkowski J. (2003b) An invitation to quantum game theory. International Journal of Theoretical Physics 42(5): 1089–1099 · Zbl 1037.81020 · doi:10.1023/A:1025443111388
[50] Rényi, A. (1961), On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. I.J. Neyman (ed.), University of California Press. · Zbl 0106.33001
[51] Scheinkman, J. and Xiong, W. (2004), Heterogeneous beliefs, speculation and trading in financial markets, Paris-Princeton Lectures on Mathematical Finance, Lecture notes in Mathematics 1847, Springer-Verlag. · Zbl 1180.91138
[52] Segal W., Segal I.E. (1998) The Black-Scholes pricing formula in the quantum context. Proceedings of the National Academy of Sciences of the USA 95: 4072–4075 · Zbl 0903.90009 · doi:10.1073/pnas.95.7.4072
[53] Shubik M. (1987) What is an application and when is a theory a waste of time. Management Science 33(12): 1511–1522 · doi:10.1287/mnsc.33.12.1511
[54] Shubik M. (1999) Quantum economics, uncertainty and the optimal grid size. Economics Letters 64(3): 277–278 · Zbl 0973.91530 · doi:10.1016/S0165-1765(99)00095-6
[55] Sulganik E., Zilcha I. (1997) The value of information: The case of signal- dependent opportunity sets. Journal of Economic Dynamics and Control 21: 1615–1625 · Zbl 0897.90001 · doi:10.1016/S0165-1889(97)00039-0
[56] Tirole J. (1982) On the possibility of speculation under rational expectations. Econometrica 50: 1163–1181 · Zbl 0488.90026 · doi:10.2307/1911868
[57] Tversky A., Koehler D. (1994) Support theory: a nonexistential representation of subjective probability. Psychological Review 101: 547–567 · Zbl 02310229 · doi:10.1037/0033-295X.101.4.547
[58] Wilmott, P. (1998), Derivatives: The Theory and Practice of Financial Engineering. J. Wiley.
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