Model of discrete dynamics of asset price relations based on the minimal arbitrage principle

• Authors: Zvonko Kostanjčar , Kristian Hengster-Movrić , Branko Jeren
• Languages of publication: EN

Abstracts

EN

In this paper we present a deterministic and a probabilistic model of the dynamics of the price relations for a number of assets on the market. The formalism is based on the asset space introduced in a theory by Illinski. We derive, from an action functional for the system of price relations in that space, the corresponding difference equations, which constitute the deterministic description. Furthermore, we obtain the probability density function of the probabilistic model of market dynamics from the same action functional. The deterministic solution corresponds to a geometric sequence for the interest, whereas the derived probability density describes the probability of the next value of the price relations in dependence on their prior value. The formalism is completely developed for systems (markets) with two and three assets, but exactly the same approach is applicable to the systems consisting of an arbitrary number of assets.

Keywords

EN

complex systems; financial markets; equilibrium dynamics; variation principle

• Journal: Open Physics
• Year: 2011
• Volume: 9
• Issue: 3
• Pages: 865-873

Dates

published

1 - 6 - 2011

online

26 - 2 - 2011

Contributors

author

Zvonko Kostanjčar

• Department of Electronic Systems and Information Processing, University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, HR-10000, Zagreb, Croatia,

author

Kristian Hengster-Movrić

• Advanced Controls and Sensors Group, University of Texas at Arlington, 701 S. Nedderman Drive, Arlington, TX, 76019, USA,

author

Branko Jeren

• Department of Electronic Systems and Information Processing, University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, HR-10000, Zagreb, Croatia,

References

• [1] K. Ilinski, Physics of Finance: Gauge Modeling in Non-equilibrium Pricing (Wiley, Chichester, 2001)
• [2] J.P. Bouchaud, M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000)
• [3] M. Constantin, S. Das Sarma, Phys. Rev. E 72, 051106 (2005) http://dx.doi.org/10.1103/PhysRevE.72.051106[Crossref]
• [4] B.G. Malkiel, A Random Walk Down Wall Street (W.W. Norton and Company, New York, 2003)
• [5] B.E. Baaquie, Quantum Finance (Cambridge University Press, Cambridge, 2004) http://dx.doi.org/10.1017/CBO9780511617577[Crossref]
• [6] N.F. Johnson, P. Jeffreies, P.M. Hui, Financial Market Complexity (Oxford University Press, Oxford, 2003) http://dx.doi.org/10.1093/acprof:oso/9780198526650.001.0001[Crossref]
• [7] R. Cont, J.-P. Bouchaud, Macroecon. Dyn. 4, 170 (2000) http://dx.doi.org/10.1017/S1365100500015029[Crossref]
• [8] A.W. Lo, C. Mackinlay, Rev. Financ. Stud. 1, 41 (1998) http://dx.doi.org/10.1093/rfs/1.1.41[Crossref]
• [9] S. Drozdz, M. Forczek, J. Kwapien, P. Oswiecimka, R. Rak, Physica A 383, 59 (2007) http://dx.doi.org/10.1016/j.physa.2007.04.130[Crossref]
• [10] A. Dionisio, R. Menezes, D.A. Mendes, Physica A 382, 58 (2007) http://dx.doi.org/10.1016/j.physa.2007.02.008[Crossref]
• [11] A. Matacz, International Journal of Theoretical and Applied Finance 3, 143 (2000) http://dx.doi.org/10.1142/S0219024900000073[Crossref]
• [12] X. Gabaix, P. Gopikrishnan, V. Plerou, H.E. Stanley, Nature 423, 267 (2003) http://dx.doi.org/10.1038/nature01624[Crossref]
• [13] M. Fredrick, M.D. Johnson, Physica A 320, 525 (2003) http://dx.doi.org/10.1016/S0378-4371(02)01558-3[Crossref]
• [14] J.P. Bouchaoud, Physica A 313, 238 (2002) http://dx.doi.org/10.1016/S0378-4371(02)01039-7[Crossref]
• [15] R.N. Mantegna et al., Physica A 274, 216 (1999) http://dx.doi.org/10.1016/S0378-4371(99)00395-7[Crossref]
• [16] J.D. Farmer, N. Zamani, Eur. Phys. J. B 55, 189 (2007) http://dx.doi.org/10.1140/epjb/e2006-00384-5[Crossref]
• [17] G. Bonanno, F. Lillo, R.N. Mantegna, Physica A 299, 16 (2001) http://dx.doi.org/10.1016/S0378-4371(01)00279-5[Crossref]
• [18] J.V. Andersen, S. Gluzman, D. Sornette, Eur. Phys. J. B 14, 579 (2000) http://dx.doi.org/10.1007/s100510051067[Crossref]
• [19] Y. Bar-Yam, Dynamics of complex systems (Westview Press, Boulder, 1997)
• [20] H. Haken, Information and Self-Organization (Springer, Berlin, 1999)
• [21] N. Bocara, Modeling Complex systems (Springer, New York, 2004)
• [22] G. Mack, arXiv:hep-th/0011074
• [23] A. Shleifer, W.R. Vishny, J. Financ. 52, 35 (1997) http://dx.doi.org/10.2307/2329555[Crossref]
• [24] K. Ilinski, J. Phys. A: Math. Gen. 33, L5 (2000) http://dx.doi.org/10.1088/0305-4470/33/1/102[Crossref]
• [25] K. Ilinski, arXiv:cond-mat/9811197v1
• [26] D. Sornette, Int. J. Mod. Phys. C 9, 505 (1998) http://dx.doi.org/10.1142/S0129183198000406[Crossref]
• [27] J.C. Baez, J.W. Gilliam, Lett. Math. Phys. 31, 205 (1994) http://dx.doi.org/10.1007/BF00761712[Crossref]
• [28] J.B. Chen, H.Y. Guo, K. Wu, Appl. Math. Comput. 177, 226 (2006) http://dx.doi.org/10.1016/j.amc.2005.11.002[Crossref]
• [29] A.G. Lisi, arXiv:physics/0605068v1
• [30] J.L. McCauley, Physica A 329, 199 (2003) http://dx.doi.org/10.1016/S0378-4371(03)00591-0[Crossref]
• [31] D. Ruelle, Thermodynamic Formalism, The Mathematical Structures of Equilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 2004) http://dx.doi.org/10.1017/CBO9780511617546[Crossref]

Identifiers

DOI

10.2478/s11534-010-0093-x