Darboux Transformation in Quantum Black-Scholes Hamiltonian and Supersymmetry


In this paper, we consider the Black-Scholes (BS) equation for option pricing with constant volatility. Here, we construct the first-order Darboux transformation and the real valued condition of transformed potential for BS corresponding equation. In that case we also obtain the transformed of potential and wave function. Finally, we discuss the factorization method and investigate the supersymmetry aspect of such corresponding equation. Also we show that the first order equation is satisfied by commutative algebra.

Share and Cite:

J. Sadeghi, M. Rostami, A. Pourdarvish and B. Pourhassan, "Darboux Transformation in Quantum Black-Scholes Hamiltonian and Supersymmetry," Open Journal of Microphysics, Vol. 3 No. 2, 2013, pp. 43-46. doi: 10.4236/ojm.2013.32008.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] G. Darboux, “Sur une Proposition Relative aux équations Linéaires,” Comptes Rendus, Vol. 94, 1882, pp. 1456-1459.
[2] V. B. Matveev and M. A. Salle, “Darboux Transformations and Solitons,” Springer, Berlin, 1991. doi:10.1007/978-3-662-00922-2
[3] M. J. Ablowitz and H. Segur, “Solitons and the Inverse Scattering Transform,” SIAM, Philadelphia, 1981. doi:10.1137/1.9781611970883
[4] C. H. Gu, H. S. Hu and Z. X. Zhou, “Darboux Transformations in Integraable Systems,” Springer, Dordrecht, 2005.
[5] E. Witten, “Constraints on Supersymmetry Breaking,” Nuclear Physics B, Vol. 202, No. 2, 1982, pp. 253-316. doi:10.1016/0550-3213(82)90071-2
[6] Q. P. Liu and X. B. Hu, “Bilinerarization of N = 1 Supersymmetric Korteweg-de Vries Equation Revisited,” Journal of Physics A: Mathematical and General, Vol. 38, No. 28, 2005, pp. 6371-6378. doi:10.1088/0305-4470/38/28/009
[7] F. Cooper, A. Khare and U. Sukhatme, “Supersymmetry and Quntum Mechanics,” Physics Reports, Vol. 251, No. 5-6, 1995, pp. 267-285. doi:10.1016/0370-1573(94)00080-M
[8] A. R. Piastino, et al., “Supersymmetry Approach to Quantum Systems with Position-Dependent Effective Mass,” Physical Review A, Vol. 60, No. 6, 1999, pp. 4318-4325. doi:10.1103/PhysRevA.60.4318
[9] R. Kok and M. Koka, “A Systematic Study on the Exact Solution of the Position Dependent Mass Schrodinger Equation,” Journal of Physics A: Mathematical and General, Vol. 36, No. 29, 2003, p. 8105. doi:10.1088/0305-4470/36/29/315
[10] J. Sadeghi, “Superalgebras for Three Interacting Particles in an External Magnetic Field,” The European Physical Journal B, Vol. 50, No. 3, 2006, pp. 453-457. doi:10.1140/epjb/e2006-00150-9
[11] H. Fakhri and J. Sadeghi, “Supersymetry Approaches to The Bound States of the Generalized Wood-Saxon Potential,” Modern Physics Letters A, Vol. 19, No. 8, 2004, p. 615. doi:10.1142/S0217732304013313
[12] J. Sadeghi and M. Rostami, “The Supersymmetry Approaches to the Non-Central Kratzer Plus Ring-Shaped Potential,” International Journal of Theoretical Physics, Vol. 48, No. 10, 2009, pp. 2961-2970.
[13] J. Sadeghi, “Dirac Oscillator with Minimal Lengths and Free Particle on AdS2 and S2,” Journal of Mathematical Physics, Vol. 48, No. 11, 2007, Article ID: 113508. doi:10.1063/1.2804773
[14] B. G. Idis, M. M. Musakhanov and M. Sh. Usmanov, “Application of Supersymmetry and Factorization Methods to Solution of Dirac and Schrodinger Equations,” Theoretical and Mathematical Physics, Vol. 101, No. 1, 1984, pp. 1191-1199.
[15] R. M. Mantega and E. Stanley, “Introduction to Econophysics,” Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9780511755767
[16] V. Linetsky, “The Path Integral Approach to Financial Modeling and Options Pricing,” Computational Economics, Vol. 11, No. 1-2, 1998, pp. 129-163.
[17] M. Contreras, et al., “A Quantum Model of Option Pricing: When Black-Scholes Meets Schrodinger and Its Semi-Classical Limit,” Physica A: Statistical Mechanics and its Applications, Vol. 39, No. 23, 2010, pp. 5447-5459. doi:10.1016/j.physa.2010.08.018
[18] E. E. Haven, “A Discussion on Embedding the Black-Scholes Option Pricing Model in a Quantum Physics Setting,” Physica A: Statistical Mechanics and its Applications, Vol. 304, No. 3-4, 2002, pp. 507-524.
[19] E. E. Haven, “A Black-Scholes Schrodinger Option Price: ‘Bit’ versus ‘Qubit’,” Physica A: Statistical Mechanics and Its Applications, Vol. 324, No. 1-2, 2003, pp. 201-206.
[20] E. E. Haven, “The Wave-Equivalent of the Black-Scholes Option Price: An Interpretation,” Physica A: Statistical Mechanics and its Applications, Vol. 344, No. 1-2, 2004, pp. 142-145. doi:10.1016/j.physa.2004.06.105
[21] O. A. Choustova, “Quantum Bohmian Model for Financial Markets,” Physica A: Statistical Mechanics and its Applications, Vol. 374, No. 1, 2007, pp. 304-314. doi:10.1016/j.physa.2006.07.029
[22] J. Morales, J. J. Pena and J. L. Lopez-Bonilla, “Generalization of the Darboux Transform,” Journal of Mathematical Physics, Vol. 42, No. 2, 2001, p. 966. doi:10.1063/1.1334904
[23] J. J. Pena, G. Ovando, D. Morales-Guzman and J. Morales, “Solvable Quartic Potentials and Their Isospectral Partners,” International Journal of Quantum Chemistry, Vol. 85, No. 4-5, 2001, pp. 244-250. doi:10.1002/qua.10042
[24] J. Morales, J. J. Pena and A. Rubio-Ponce, “New Isospectral Generalized Potentials,” Theoretical Chemistry Accounts, Vol. 110, No. 6, 2003, pp. 403-409. doi:10.1007/s00214-003-0494-7
[25] A. Rubio-Ponce, J. J. Pena and J. Morales, “One-Parameter Isospectral Solutions for the Fokker-Planck Equation,” Physica A: Statistical Mechanics and its Applications, Vol. 339, No. 3-4, 2004, pp. 285-295. doi:10.1016/j.physa.2004.03.022
[26] A. Schulze-Halberg, “Effective Mass Hamiltonians with Linear Terms in the Momentum: Darboux Transformations and Form-Preserving Transformations,” International Journal of Modern Physics A, Vol. 22, No. 8-9, 2007, p. 1735. doi:10.1142/S0217751X07035021
[27] A. Schulze-Halberg, “Darboux Transformations for Effective Mass Schrodinger Equations with Energy-Dependent Potentials,” International Journal of Modern Physics A, Vol. 23, No. 3-4, 2008, p. 537. doi:10.1142/S0217751X0803807X

Copyright © 2021 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.