Abstract
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.
[1] K. Ilinski, Physics of Finance: Gauge Modeling in Non-equilibrium Pricing (Wiley, Chichester, 2001) Search in Google Scholar
[2] J.P. Bouchaud, M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000) Search in Google Scholar
[3] M. Constantin, S. Das Sarma, Phys. Rev. E 72, 051106 (2005) http://dx.doi.org/10.1103/PhysRevE.72.05110610.1103/PhysRevE.72.051106Search in Google Scholar
[4] B.G. Malkiel, A Random Walk Down Wall Street (W.W. Norton and Company, New York, 2003) Search in Google Scholar
[5] B.E. Baaquie, Quantum Finance (Cambridge University Press, Cambridge, 2004) http://dx.doi.org/10.1017/CBO978051161757710.1017/CBO9780511617577Search in Google Scholar
[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/9780198526650.001.000110.1093/acprof:oso/9780198526650.001.0001Search in Google Scholar
[7] R. Cont, J.-P. Bouchaud, Macroecon. Dyn. 4, 170 (2000) http://dx.doi.org/10.1017/S136510050001502910.1017/S1365100500015029Search in Google Scholar
[8] A.W. Lo, C. Mackinlay, Rev. Financ. Stud. 1, 41 (1998) http://dx.doi.org/10.1093/rfs/1.1.4110.1093/rfs/1.1.41Search in Google Scholar
[9] S. Drozdz, M. Forczek, J. Kwapien, P. Oswiecimka, R. Rak, Physica A 383, 59 (2007) http://dx.doi.org/10.1016/j.physa.2007.04.13010.1016/j.physa.2007.04.130Search in Google Scholar
[10] A. Dionisio, R. Menezes, D.A. Mendes, Physica A 382, 58 (2007) http://dx.doi.org/10.1016/j.physa.2007.02.00810.1016/j.physa.2007.02.008Search in Google Scholar
[11] A. Matacz, International Journal of Theoretical and Applied Finance 3, 143 (2000) http://dx.doi.org/10.1142/S021902490000007310.1142/S0219024900000073Search in Google Scholar
[12] X. Gabaix, P. Gopikrishnan, V. Plerou, H.E. Stanley, Nature 423, 267 (2003) http://dx.doi.org/10.1038/nature0162410.1038/nature01624Search in Google Scholar
[13] M. Fredrick, M.D. Johnson, Physica A 320, 525 (2003) http://dx.doi.org/10.1016/S0378-4371(02)01558-310.1016/S0378-4371(02)01558-3Search in Google Scholar
[14] J.P. Bouchaoud, Physica A 313, 238 (2002) http://dx.doi.org/10.1016/S0378-4371(02)01039-710.1016/S0378-4371(02)01039-7Search in Google Scholar
[15] R.N. Mantegna et al., Physica A 274, 216 (1999) http://dx.doi.org/10.1016/S0378-4371(99)00395-710.1016/S0378-4371(99)00395-7Search in Google Scholar
[16] J.D. Farmer, N. Zamani, Eur. Phys. J. B 55, 189 (2007) http://dx.doi.org/10.1140/epjb/e2006-00384-510.1140/epjb/e2006-00384-5Search in Google Scholar
[17] G. Bonanno, F. Lillo, R.N. Mantegna, Physica A 299, 16 (2001) http://dx.doi.org/10.1016/S0378-4371(01)00279-510.1016/S0378-4371(01)00279-5Search in Google Scholar
[18] J.V. Andersen, S. Gluzman, D. Sornette, Eur. Phys. J. B 14, 579 (2000) http://dx.doi.org/10.1007/s10051005106710.1007/s100510051067Search in Google Scholar
[19] Y. Bar-Yam, Dynamics of complex systems (Westview Press, Boulder, 1997) Search in Google Scholar
[20] H. Haken, Information and Self-Organization (Springer, Berlin, 1999) Search in Google Scholar
[21] N. Bocara, Modeling Complex systems (Springer, New York, 2004) Search in Google Scholar
[22] G. Mack, arXiv:hep-th/0011074 Search in Google Scholar
[23] A. Shleifer, W.R. Vishny, J. Financ. 52, 35 (1997) http://dx.doi.org/10.2307/232955510.2307/2329555Search in Google Scholar
[24] K. Ilinski, J. Phys. A: Math. Gen. 33, L5 (2000) http://dx.doi.org/10.1088/0305-4470/33/1/10210.1088/0305-4470/33/1/102Search in Google Scholar
[25] K. Ilinski, arXiv:cond-mat/9811197v1 Search in Google Scholar
[26] D. Sornette, Int. J. Mod. Phys. C 9, 505 (1998) http://dx.doi.org/10.1142/S012918319800040610.1142/S0129183198000406Search in Google Scholar
[27] J.C. Baez, J.W. Gilliam, Lett. Math. Phys. 31, 205 (1994) http://dx.doi.org/10.1007/BF0076171210.1007/BF00761712Search in Google Scholar
[28] J.B. Chen, H.Y. Guo, K. Wu, Appl. Math. Comput. 177, 226 (2006) http://dx.doi.org/10.1016/j.amc.2005.11.00210.1016/j.amc.2005.11.002Search in Google Scholar
[29] A.G. Lisi, arXiv:physics/0605068v1 Search in Google Scholar
[30] J.L. McCauley, Physica A 329, 199 (2003) http://dx.doi.org/10.1016/S0378-4371(03)00591-010.1016/S0378-4371(03)00591-0Search in Google Scholar
[31] D. Ruelle, Thermodynamic Formalism, The Mathematical Structures of Equilibrium Statistical Mechanics (Cambridge University Press, Cambridge, 2004) http://dx.doi.org/10.1017/CBO978051161754610.1017/CBO9780511617546Search in Google Scholar
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