Cost-efficient contingent claims with market frictions

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• 作者：Mario Ghossoub
• 关键词：Payoff distributional pricing ; Cost ; efficiency ; contingent claims ; Nonlinear pricing ; Bid ; ask spread ; Ambiguity ; Knightian uncertainty ; Non ; additive probability ; Choquet integral ; 91G10 ; 91G20 ; 91B42 ; 28A12 ; D89 ; G11
• 刊名：Mathematics and Financial Economics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：10
• 期：1
• 页码：87-111
• 全文大小：584 KB
• 参考文献：1.Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)MATH
  2.Amarante, M.: Foundations of neo-Bayesian statistics. J. Econ. Theory 144(5), 2146–2173 (2009)MATH MathSciNet CrossRef
  3.Araujo, A., Chateauneuf, A., Faro, J.H.: Pricing rules and Arrow–Debreu ambiguous valuation. Econ. Theory 49(1), 1–35 (2011)MathSciNet CrossRef
  4.Campi, L., Jouini, E., Porte, V.: Efficient portfolios in financial markets with proportional transaction costs. Math. Financ. Econ. 7(3), 281–304 (2013)MATH MathSciNet CrossRef
  5.Carlier, G., Dana, R.A.: Core of convex distortions of a probability. J. Econ. Theory 113(2), 199–222 (2003)MATH MathSciNet CrossRef
  6.Carlier, G., Dana, R.A.: Law invariant concave utility functions and optimization problems with monotonicity and comonotonicity constraints. Stat. Decis. 24(1), 127–152 (2006)MATH MathSciNet
  7.Carlier, G., Dana, R.A.: Two-persons efficient risk-sharing and equilibria for concave law-invariant utilities. Econ. Theory 36(2), 189–223 (2008)MATH MathSciNet CrossRef
  8.Carlier, G., Dana, R.A.: Optimal demand for contingent claims when agents have law invariant utilities. Math. Financ. 21(2), 169–201 (2011)MATH MathSciNet
  9.Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M.: Put-call parity and market frictions. J. Econ. Theory 157, 730–762 (2015)MathSciNet CrossRef
  10.Chateauneuf, A., Cornet, B.: Submodular Financial Markets with Frictions. Mimeo, New York (2015)
  11.Chateauneuf, A., Kast, R., Lapied, A.: Choquet pricing for financial markets with frictions. Math. Financ. 6(3), 323–330 (1996)MATH CrossRef
  12.Cohn, D.L.: Measure Theory. Birkhauser, Boston (1980)MATH CrossRef
  13.Denneberg, D.: Non-Additive Measure and Integral. Kluwer Academic Publishers, Dordrecht (1994)MATH CrossRef
  14.Doob, J.L.: Measure Theory. Springer, Berlin (1994)MATH CrossRef
  15.Dybvig, P.H.: Distributional analysis of portfolio choice. J. Bus. 61(3), 369–393 (1988)CrossRef
  16.Dybvig, P.H.: Inefficient dynamic portfolio strategies or how to throw away a million dollars in the stock market. Rev. Financ. Stud. 1(1), 67–88 (1988)CrossRef
  17.Ellsberg, D.: Risk, ambiguity, and the savage axioms. Q. J. Econ. 75(4), 643–669 (1961)MATH CrossRef
  18.Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. Walter de Gruyter, Berlin (2011)CrossRef
  19.Ghossoub, M.: Equimeasurable rearrangements with capacities. Math. Op. Res. 40(2), 429–445 (2015)MathSciNet CrossRef
  20.Ghossoub, M.: Vigilant measures of risk and the demand for contingent claims. Insurance 61, 27–35 (2015)MathSciNet
  21.Gilboa, I., Marinacci, M.: Ambiguity and the Bayesian paradigm. In: Acemoglu, D., Arellano, M., Dekel, E. (eds.) Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Cambridge University Press, Cambridge (2012)
  22.Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20(3), 381–408 (1979)MATH MathSciNet CrossRef
  23.He, X.D., Zhou, X.Y.: Portfolio choice via quantiles. Math. Financ. 21(2), 203–231 (2011)MATH MathSciNet
  24.Inada, K.: On a two-sector model of economic growth: comments and a generalization. Rev. Econ. Stud. 30(2), 119–127 (1963)CrossRef
  25.Jouini, E., Kallal, H.: Efficient trading strategies in the presence of market frictions. Rev. Financ. Stud. 14(2), 343–369 (2001)MathSciNet CrossRef
  26.Jouini, E., Porte, V.: Efficient Trading Strategies. Mimeo, New York (2005)
  27.Knight, F.H.: Risk, Uncertainty, and Profit. Houghton Mifflin, Boston (1921)
  28.Lehrer, E.: A new integral for capacities. Econ. Theory 39(1), 157–176 (2009)MATH MathSciNet CrossRef
  29.Marinacci, M., Montrucchio, L.: Introduction to the mathematics of ambiguity. In: Gilboa, I. (ed.) Uncertainty in Economic Theory: Essays in Honor of David Schmeidler 65th Birthday, pp. 46–107. Routledge, London (2004)CrossRef
  30.Pap, E.: Null-Additive Set Functions. Kluwer Academic Publishers, Dodretch (1995)MATH
  31.Rieger, M.O.: Co-monotonicity of optimal investments and the design of structured financial products. Financ. Stoch. 15(1), 27–55 (2010)MathSciNet CrossRef
  32.Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill Book Company, New York (1976)MATH
  33.Savage, L.J.: The Foundations of Statistics (2nd revised edition), vol. 1954, 1st edn. Dover Publications, New York (1972)
  34.Schied, A.: On the Neyman–Pearson problem for law-invariant risk measures and robust utility functionals. Ann. Appl. Probab. 14(3), 1398–1423 (2004)MATH MathSciNet CrossRef
  35.Schmeidler, D.: Subjective probability and expected utility without additivity. Econometrica 57(3), 571–587 (1989)MATH MathSciNet CrossRef
  36.von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1947)MATH
  
• 作者单位：Mario Ghossoub (1)
  
  1. Imperial College London, South Kensington Campus, London, SW7 2AZ, UK
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Quantitative Finance
  Finance and Banking
  Financial Economics
  Game Theory and Mathematical Methods
  Applications of Mathematics
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1862-9660

文摘

In complete frictionless securities markets under uncertainty, it is well-known that in the absence of arbitrage opportunities, there exists a unique linear positive pricing rule, which induces a state-price density (e.g., Harrison and Kreps in J Econ Theory 20(3):381–408, 1979). Dybvig (J Bus 61(3):369–393, 1988; Rev Financ Stud 1(1):67–88, 1988) showed that the cheapest way to acquire a certain distribution of a consumption bundle (or security) is when this bundle is anti-comonotonic with the state-price density, i.e., arranged in reverse order of the state-price density. In this paper, we look at extending Dybvig’s ideas to complete markets with imperfections represented by a nonlinear pricing rule (e.g., due to bid-ask spreads). We consider an investor in a securities market where the pricing rule is “law-invariant” with respect to a capacity (e.g., Choquet pricing as in Araujo et al. in Econ Theory 49(1):1–35, 2011; Chateauneuf et al. in Math Financ 6(3):323–330, 1996; Chateauneuf and Cornet in Submodular financial markets with frictions, 2015; Cerreia-Vioglio et al. in J Econ Theory 157:730–762, 2015). The investor holds a security with a random payoff X and his problem is that of buying the cheapest contingent claim Y on X, subject to some constraints on the performance of the contingent claim and on its level of risk exposure. The cheapest such claim is called cost-efficient. If the capacity satisfies standard continuity and a property called strong diffuseness introduced in Ghossoub (Math Op Res 40(2):429–445, 2015), we show the existence and monotonicity of cost-efficient claims, in the sense that a cost-efficient claim is anti-comonotonic with the underlying security’s payoff X. Strong diffuseness is satisfied by a large collection of capacities, including all distortions of diffuse probability measures. As an illustration, we consider the case of a Choquet pricing functional with respect to a capacity and the case of a Choquet pricing functional with respect to a distorted probability measure. Finally, we consider a simple example in which we derive an explicit analytical form for a cost-efficient claim. Keywords Payoff distributional pricing Cost-efficiency contingent claims Nonlinear pricing Bid-ask spread Ambiguity Knightian uncertainty Non-additive probability Choquet integral