NO-ARBITRAGE SYMMETRIES

doi:10.1007/s10473-022-0407-2

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1373-1402.doi: 10.1007/s10473-022-0407-2

NO-ARBITRAGE SYMMETRIES

Iván DEGANO¹, Sebastián FERRANDO², Alfredo GONZÁLEZ³

1. Departamento de Matemática. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, CONICET, Funes 3350, Mar del Plata, 7600, Argentina;
2. Department of Mathematics, Ryerson University, 350 Victoria St., Toronto, Ontario, M5B 2K3, Canada;
3. Departamento de Matemática. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, Mar del Plata, 7600, Argentina

收稿日期:2020-09-18 修回日期:2021-04-26 出版日期:2022-08-25 发布日期:2022-08-23
通讯作者: Sebastián FERRANDO,E-mail:ferrando@ryerson.ca E-mail:ferrando@ryerson.ca
基金资助:
The research of S.E. Ferrando is supported in part by an NSERC grant. The research of I.L. Degano and A.L. González is supported in part by the National University of Mar del Plata, Argentina[EXA902/18].

NO-ARBITRAGE SYMMETRIES

Iván DEGANO¹, Sebastián FERRANDO², Alfredo GONZÁLEZ³

1. Departamento de Matemática. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, CONICET, Funes 3350, Mar del Plata, 7600, Argentina;
2. Department of Mathematics, Ryerson University, 350 Victoria St., Toronto, Ontario, M5B 2K3, Canada;
3. Departamento de Matemática. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, Mar del Plata, 7600, Argentina

Received:2020-09-18 Revised:2021-04-26 Online:2022-08-25 Published:2022-08-23
Contact: Sebastián FERRANDO,E-mail:ferrando@ryerson.ca E-mail:ferrando@ryerson.ca
Supported by:
The research of S.E. Ferrando is supported in part by an NSERC grant. The research of I.L. Degano and A.L. González is supported in part by the National University of Mar del Plata, Argentina[EXA902/18].

摘要/Abstract

摘要： The no-arbitrage property is widely accepted to be a centerpiece of modern financial mathematics and could be considered to be a financial law applicable to a large class of (idealized) markets. This paper addresses the following basic question: can one characterize the class of transformations that leave the law of no-arbitrage invariant? We provide a geometric formalization of this question in a non probabilistic setting of discrete time-the so-called trajectorial models. The paper then characterizes, in a local sense, the no-arbitrage symmetries and illustrates their meaning with a detailed example. Our context makes the result available to the stochastic setting as a special case.

关键词: No arbitrage symmetry, convexity preserving maps, non-probabilistic markets

Abstract: The no-arbitrage property is widely accepted to be a centerpiece of modern financial mathematics and could be considered to be a financial law applicable to a large class of (idealized) markets. This paper addresses the following basic question: can one characterize the class of transformations that leave the law of no-arbitrage invariant? We provide a geometric formalization of this question in a non probabilistic setting of discrete time-the so-called trajectorial models. The paper then characterizes, in a local sense, the no-arbitrage symmetries and illustrates their meaning with a detailed example. Our context makes the result available to the stochastic setting as a special case.

Key words: No arbitrage symmetry, convexity preserving maps, non-probabilistic markets

中图分类号:

91B24

Iván DEGANO, Sebastián FERRANDO, Alfredo GONZÁLEZ. NO-ARBITRAGE SYMMETRIES[J]. 数学物理学报(英文版), 2022, 42(4): 1373-1402.

Iván DEGANO, Sebastián FERRANDO, Alfredo GONZÁLEZ. NO-ARBITRAGE SYMMETRIES[J]. Acta mathematica scientia,Series B, 2022, 42(4): 1373-1402.

参考文献

[1] Bender C, Sottinen T, Valkeila E. Fractional processes as models in stochastic finance//Di Nunno G, Oksendal B. Advanced Mathematical Methods for Finance. Berlin, Heidelberg:Springer, 2010
[2] Bingham N H, Kiesel R. Risk-Neutral Valuation:Pricing and Hedging of Financial Derivatives. London:Springer-Verlag, 2004
[3] Britten-Jones M, Neuberger A. Arbitrage pricing with incomplete markets. Applied Mathematical Journal, 1996, 3(4):347-363
[4] Cutland N J, Roux A. Derivative Pricing in Discrete Time. London:Springer-Verlag, 2012
[5] Dalang R C, Morton A, Willinger W. Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch Stoch Rep, 1990, 29:185-201
[6] Degano I L, Ferrando S E, González A L. Trajectory Based Market Models. Evaluation of Minmax Price Bounds. Dynamics of Continuous, Discrete and Impulsive Systems Series B:Applications & Algorithms, 2018, 25(2):97-128
[7] Elliott R J, Kopp P E. Mathematics of Financi al Markets. New York:Springer-Verlag, 2005
[8] Ferrando S E, Fleck A, González A L, Rubtsov A. Trajectorial asset models with operational assumptions. Quantitative Finance and Economics, 2019, 3(4):661-708
[9] Ferrando S E, González A L, Degano I L, Rahsepar M. Trajectorial Market Models. Arbitrage and Pricing Intervals. Revista de la Unión Matemática Argentina, 2019, 60(1):149-185
[10] Ferrando S E, González A L. Trajectorial martingale transforms. Convergence and Integration. New York Journal of Mathematics, 2018, 24:702-738
[11] Föllmer H, Schied A. Stochastic Finance:An Introduction in Discrete Time. 3rd Edition. Berlin:De Gruyter, 2011
[12] Jarrow R A, Protter P, Sayit H. No arbitrage without semimartingales. The Annals of Applied Probability, 2009, 19(2):596-616
[13] Jacod J, Shirayev H. Local martingales and the fundamental asset pricing theorems in the discrete-time case. Finance & Stochastics, 1990, 2(3):259-273
[14] Kamara A, Miller T W. Daily and Intradaily Tests of European Put-Call Parity. The Journal of Financial and Quantitative Analysis, 1995, 30(4):519-539
[15] Musiela M, Rutkowski M. Martingale Methods in Financial Modelling. Second Edition. Berlin:SpringerVerlag, 2005
[16] Páles, Z. Characterization of segment and convexity preserving maps. 2012. https://arxiv.org/abs/1212.1268
[17] Pliska S R. Introduction to Mathematical Finance:Discrete Time Models. Wiley, 1997
[18] Rockafellar R T. Convex Analysis. Princeton University Press, 1970
[19] Vecer J. Stochastic Finance. A Numeraire Approach. CRC Pressm, 2011
[20] Whitney H. The Mathematics of Physical Quantities:Part I:Mathematical Models for Measurement. The American Mathematical Monthly, 1968, 75(2):115-138

NO-ARBITRAGE SYMMETRIES

NO-ARBITRAGE SYMMETRIES

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