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First steps towards an equilibrium theory for Lévy financial markets

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Abstract

For a continuous-time financial market with a single agent, we establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Lévy process. The agent is allowed to consume a lump at the terminal date; before that, only flow consumption is allowed. The agent’s utility function is assumed to be additive, defined via strictly increasing, strictly concave smooth felicity functions which are bounded below (thus, many CRRA and CARA utility functions are included). For technical reasons we require for our equilibrium existence result that only pathwise continuous trading strategies are permitted in the demand set. The resulting equilibrium asset price processes depend on the agent’s risk aversion (through the felicity functions). Even in our simple, straightforward economy, the equilibrium asset price processes will essentially only be (stochastic) exponential Lévy processes when they are already geometric Brownian motions. Our equilibrium asset pricing formulae can also be modified to obtain explicit equilibrium derivative pricing formulae.

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References

  • Albeverio S., Herzberg F.S.: A combinatorial infinitesimal representation of Lévy processes and an application to incomplete markets. Stochastics 78(5), 301–325 (2006a)

    Google Scholar 

  • Albeverio S., Herzberg F.S.: Lifting Lévy processes to hyperfinite random walks. Bulletin des Sciences Mathématiques 130(8), 697–706 (2006b)

    Article  Google Scholar 

  • Albeverio S., Høegh-Krohn R.J., Fenstad J.E., Lindstrøm T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Volume 122 of Pure and Applied Mathematics. Academic Press, Orlando (1986)

    Google Scholar 

  • Almendral A., Oosterlee C.: Accurate evaluation of European and American options under the CGMY process. SIAM J Sci Comput 29(1), 93–117 (2007a)

    Article  Google Scholar 

  • Almendral A., Oosterlee C.: On American options under the variance gamma process. Appl Math Finance 14(2), 131–152 (2007b)

    Article  Google Scholar 

  • Anderson R.M.: A non-standard representation for Brownian motion and Itô integration. Israel J Math 25(1–2), 15–46 (1976)

    Article  Google Scholar 

  • Anderson, R.M.: Non-standard analysis with applications to economics. In: Hildenbrand, W., Sonnenschein, H. (eds.) Handbook of Mathematical Economics. Handbooks in Economics, vol. IV, pp. 2145–2208. Amsterdam: North-Holland (1991)

  • Anderson R.M., Raimondo R.C.: Market clearing and derivative pricing. Econ Theory 25, 21–34 (2005)

    Article  Google Scholar 

  • Anderson R.M., Raimondo R.C.: Equilibrium in continuous-time financial markets: endogenously dynamically complete markets. Econometrica 76(4), 841–907 (2008)

    Article  Google Scholar 

  • Applebaum D.: Lévy processes—from finance to probability and quantum groups. Notices Am Math Soc 51(11), 1336–1347 (2004a)

    Google Scholar 

  • Applebaum D.: Lévy Processes and Stochastic Calculus of Cambridge Studies in Advanced Mathematics, vol. 93. Cambridge University Press, Cambridge (2004b)

    Book  Google Scholar 

  • Banaschewski B.: The power of the ultrafilter theorem. J Lond Math Soc Second Ser 27(2), 193–202 (1983)

    Article  Google Scholar 

  • Barndorff-Nielsen O.E.: Normal inverse Gaussian distributions and stochastic volatility modelling. Scand J Stat 24(1), 1–13 (1997)

    Article  Google Scholar 

  • Barndorff-Nielsen, O.E., Mikosch, T., Resnick, S.I. (eds.): Lévy Processes. Theory and Applications. Boston: Birkhäuser (2001)

  • Bick A.: On viable diffusion price processes of the market portfolio. J Finance 45(2), 673–689 (1990)

    Article  Google Scholar 

  • Black F., Scholes M.: The pricing of options and corporate liabilities. J Political Econ 81, 637–654 (1973)

    Article  Google Scholar 

  • Boyarchenko S.I., Levendorskiĭ S.Z.: Non-Gaussian Merton–Black–Scholes Theory. World Scientific, Singapore (2001)

    Google Scholar 

  • Brown D.J., Robinson A.: Nonstandard exchange economies. Econometrica 43, 41–55 (1975)

    Article  Google Scholar 

  • Carr P., Geman H., Madan D.B., Yor M.: The fine structure of asset returns: An empirical investigation. J Bus 75(2), 305–333 (2002)

    Article  Google Scholar 

  • Carr P., Geman H., Madan D.B., Yor M.: Stochastic volatility for Lévy processes. Math Finance 13(3), 345–382 (2003)

    Article  Google Scholar 

  • Chan T.: Pricing contingent claims on stocks driven by Lévy processes. Ann Appl Probab 9(2), 504–528 (1999)

    Article  Google Scholar 

  • Cox J.C., Ingersoll J.E., Ross S.A.: An intertemporal general equilibrium model of asset prices. Econometrica 53(2), 363–384 (1985)

    Article  Google Scholar 

  • Cutland N.J., Kopp P.E., Willinger W.: A nonstandard approach to option pricing. Math Finance 1(4), 1–38 (1991)

    Article  Google Scholar 

  • Cutland N.J., Kopp P.E., Willinger W.: A nonstandard treatment of options driven by Poisson processes. Stoch Stoch Rep 42(2), 115–133 (1993a)

    Article  Google Scholar 

  • Cutland N.J., Kopp P.E., Willinger W.: From discrete to continuous financial models: new convergence results for option pricing. Math Finance 3(2), 101–123 (1993b)

    Article  Google Scholar 

  • Cutland N.J., Kopp P.E., Willinger W.: From discrete to continuous stochastic calculus. Stoch Stoch Rep 52(3–4), 173–192 (1995)

    Google Scholar 

  • Cutland, N.J., Kopp, P.E., Willinger, W., Wyman, M.C.: Convergence of Snell envelopes and critical prices in the American put. In: Mathematics of derivative securities (Cambridge, 1995), Publications of the Newton Institute, vol. 15, pp. 126–140. Cambridge: Cambridge University Press (1997)

  • Cvitanić J., Polimenis V., Zapatero F.: Optimal portfolio allocation with higher moments. Ann Finance 4(1), 1–28 (2008)

    Article  Google Scholar 

  • Doléans C.: Existence du processus croissant naturel associé à un potentiel de la classe (D). Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 9(4), 309–314 (1968)

    Article  Google Scholar 

  • Doléans-Dade C.: Quelques applications de la formule de changement de variables pour les semi-martingales. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 16(3), 181–194 (1970)

    Article  Google Scholar 

  • Duffie C., Skiadas D.: Continuous-time security pricing. J Math Econ 23, 107–131 (1994)

    Article  Google Scholar 

  • Duffie D., Shafer W.: Equilibrium in incomplete markets. I. J Math Econ 14, 285–300 (1985)

    Article  Google Scholar 

  • Duffie D., Shafer W.: Equilibrium in incomplete markets. II. J Math Econ 15, 199–216 (1986)

    Article  Google Scholar 

  • Eberlein E., Keller U.: Hyperbolic distributions in finance. Bernoulli 1(3), 281–299 (1995)

    Article  Google Scholar 

  • Eberlein E., Kluge W., Papapantoleon A.: Symmetries in Lévy term structure models. Int J Theor Appl Finance 9(6), 967–986 (2006)

    Article  Google Scholar 

  • Fajardo, S., Keisler, H.J.: Model Theory of Stochastic Processes. Lecture Notes in Logic, vol. 14. Natick: A.K. Peters (2002)

  • Figueroa-López J.: Statistical estimation of Lévy-type stochastic volatility models. Ann Finance 8(2), 309–335 (2012)

    Article  Google Scholar 

  • Filipović D., Tappe S.: Existence of Lévy term structure models. Finance Stoch 12(1), 83–115 (2008)

    Article  Google Scholar 

  • Halpern, J.D., Levy, A.: The Boolean prime ideal theorem does not imply the axiom of choice. In: Axiomatic Set Theory, Part 1 of Proceedings of Symposia in Pure Mathematics, vol. XIII, pp. 83–134. Providence, RI: American Mathematical Society (1971)

  • He H., Leland H.: An intertemporal general equilibrium model of asset prices. Rev Financial Stud 6(3), 593–617 (1993)

    Article  Google Scholar 

  • Herzberg F.S.: A definable nonstandard enlargement. Math Logic Q 54(2), 167–175 (2008a)

    Article  Google Scholar 

  • Herzberg F.S.: Addendum to “A definable nonstandard enlargement”. Math Logic Q 54(6), 666–667 (2008b)

    Article  Google Scholar 

  • Herzberg F.S.: Perpetual Bermudan continuity corrections and a multi-dimensional Wiener-Hopf type result. Stoch Anal Appl 27(1), 176–195 (2009)

    Article  Google Scholar 

  • Herzberg F.S.: Hyperfinite stochastic integration for Lévy processes with finite-variation jump part. Bulletin des Sciences Mathématiques 134(4), 423–445 (2010)

    Article  Google Scholar 

  • Herzberg, F.S., Lindstrøm, T.: Corrigendum and addendum to “Hyperfinite Lévy processes” (Stochastics and Stochastics Reports 76(6), 517–548, 2004). Stochastics 81(6), 567–570 (2009)

  • Hoover D.N., Perkins E.: Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I. Trans Am Math Soc 275, 1–29 (1983)

    Google Scholar 

  • Kanovei, V., Reeken, M.: Nonstandard Analysis, Axiomatically. Springer Monographs in Mathematics. Berlin: Springer (2004)

  • Kanovei V., Shelah S.: A definable nonstandard model of the reals. J Symb Logic 69(1), 159–164 (2004)

    Article  Google Scholar 

  • Keisler H.J.: Getting to a competitive equilibrium. Econometrica 64(1), 29–49 (1996)

    Article  Google Scholar 

  • Khan M.A., Sun Y.: The capital asset-pricing model and arbitrage pricing theory: a unification. Proc Natl Acad Sci USA 94(8), 4229–4232 (1997)

    Article  Google Scholar 

  • Khan M.A., Sun Y.: Exact arbitrage and portfolio analysis in large asset markets. Econ Theory 22(3), 495–528 (2003)

    Article  Google Scholar 

  • Lindstrøm T.: Hyperfinite stochastic integration. II: comparison with the standard theory. Math Scand 46(2), 293–314 (1980)

    Google Scholar 

  • Lindstrøm T.: Hyperfinite Lévy processes. Stoch Stoch Rep 76(6), 517–548 (2004)

    Article  Google Scholar 

  • Lindstrøm T.: Nonlinear stochastic integrals for hyperfinite Lévy processes. Logic Anal 1(2), 91–129 (2008)

    Article  Google Scholar 

  • Loeb P.A.: Conversion from nonstandard to standard measure spaces and applications in probability theory. Trans Am Math Soc 211, 113–122 (1975)

    Article  Google Scholar 

  • Luxemburg, W.A.J.: What is nonstandard analysis? Am Math Mon 80(Supplement)(1), 38–67 (1973)

    Google Scholar 

  • Ma C.: Intertemporal recursive utility and an equilibrium asset pricing model in the presence of L évy jumps. J Math Econ 42(2), 131–160 (2006)

    Article  Google Scholar 

  • Madan D.B., Seneta E.: Chebyshev polynomial approximations and characteristic function estimation. J R Stat Soc Ser B Methodol 49(2), 163–169 (1987)

    Google Scholar 

  • Madan D.B., Seneta E.: Chebyshev polynomial approximations for characteristic function estimation: some theoretical supplements. J R Stat Soc Ser B Methodol 51(2), 281–285 (1989)

    Google Scholar 

  • Madan D.B., Seneta E.: The variance gamma (V.G.) model for share market returns. J Bus 63(4), 511–524 (1990)

    Article  Google Scholar 

  • Magill M., Quinzii M.: Theory of Incomplete Markets. Massachusetts Institute of Technology Press, Cambridge (1996)

    Google Scholar 

  • Merton R.C.: Option pricing when underlying stock returns are discontinuous. J Financial Econ 3(1–2), 125–144 (1976)

    Article  Google Scholar 

  • Métiviér M.: Reelle und vektorwertige Quasimartingale und die Theorie der Stochastischen Integration, Lecture Notes in Mathematics, vol. 607. Springer, Heidelberg (1977)

    Google Scholar 

  • Ng S.A.: Hypermodels in Mathematical Finance: Modelling Via Infinitesimal Analysis. World Scientific, Singapore (2003)

    Book  Google Scholar 

  • Ng S.-A.: A nonstandard Lévy-Khintchine formula and Lévy processes. Acta Mathematica Sinica 24(2), 241–252 (2008)

    Article  Google Scholar 

  • Radner R.: Existence of equilibrium of plans, prices and price expectiations in a sequence of markets. Econometrica 40(2), 289–303 (1972)

    Article  Google Scholar 

  • Raimondo R.C.: Market clearing, utility functions, and security prices. Econ Theory 25, 265–285 (2005)

    Article  Google Scholar 

  • Robinson A.: Non-standard Analysis. North-Holland, Amsterdam (1966)

    Google Scholar 

  • Sato K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  • Schoutens, W.: Lévy Processes in Finance: Pricing Financial Derivatices. Wiley Series in Probability and Statistics. Chichester: Wiley (2003)

  • Solovay R.M.: A model of set theory in which every set of reals is Lebesgue measurable. Ann Math 92(1), 1–56 (1970)

    Article  Google Scholar 

  • Stroyan K.D., Bayod J.M.: Foundations of Infinitesimal Stochastic Analysis, Studies in Logic and the Foundations of Mathematics, vol. 119. North-Holland, Amsterdam (1986)

    Google Scholar 

  • Veraart A., Veraart L.: Stochastic volatility and stochastic leverage. Ann Finance 8(2), 205–233 (2012)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematical Economics, Bielefeld University, Universitätsstraße 25, 33615, Bielefeld, Germany

    Frederik S. Herzberg

  2. Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich, Ludwigstraße 31, 80539, Munich, Germany

    Frederik S. Herzberg

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  1. Frederik S. Herzberg
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Correspondence to Frederik S. Herzberg.

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The greater part of this work originated while the author visited the University of California, Berkeley, on a research grant of the German Academic Exchange Service (Deutscher Akademischer Austauschdienst, DAAD) which is hereby acknowledged with thanks. I am deeply grateful to Professor Robert M. Anderson for numerous discussions and his hospitality. I would also like to thank Professor Frank Riedel, Professor Nicholas Yannelis and an anonymous referee for helpful comments.

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Herzberg, F.S. First steps towards an equilibrium theory for Lévy financial markets. Ann Finance 9, 543–572 (2013). https://doi.org/10.1007/s10436-012-0202-5

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