IJPAM: Volume 115, No. 1 (2017)

Title

ANALYTICAL SOLUTIONS OF THE IVANCEVIC OPTION
PRICING MODEL WITH A NONZERO ADAPTIVE
MARKET POTENTIAL

Authors

S.O. Edeki$^1$, O.O. Ugbebor$^2$, O. González-Gaxiola$^3$
$^{1,2}$Department of Mathematics
Covenant University
Canaanland, Ota, NIGERIA
$^2$Department of Mathematics
University of Ibadan
Ibadan, NIGERIA
$^{3}$Departamento de Matemáticas Aplicadas y Sistemas
Universidad Autónoma Metropolitana-Cuajimalpa
Vasco de Quiroga 4871, Cuajimalpa, 05348, MEXICO

Abstract

The Ivancevic option pricing model (IOPM) is a nonlinear adaptive-wave alternative for the classical Black-Scholes option pricing model, representing a controlled Brownian motion in an adaptive setting relating to nonlinear Schrödinger equation. Despite the applicability and usefulness of the IOPM, analytical solutions of such model are barely found in literature. Therefore, this paper obtains analytical solutions of the IOPM by means of a proposed semi-analytical method referred to as projected differential transform method (PDTM). Cases of nonzero adaptive market potential are considered. The proposed method is proven to be direct, and effective as the obtained solutions tend rapidly to their exact forms.

History

Received: May 12, 2017
Revised: June 20, 2017
Published: June 29, 2017

AMS Classification, Key Words

AMS Subject Classification: 34G20, 93E35, 81Q05, 97M30
Key Words and Phrases: Ivancevic pricing model, nonlinear Black-Scholes model, option pricing, PDTM

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1
F. Black, M. Scholes, The pricing options and corporate liabilities, J. Political Econ., 81 (1973), 637-654.

2
V. G. Ivancevic, Adaptive Wave Models for Sophisticated Option Pricing, Journal of Mathematical Finance, 1 (2011), 41-49.

3
R. C. Merton, Theory of Rational Option Pricing, The Bell Journal of Economics and Management Science, 4 (1973), 141-183, doi: https://doi.org/10.2307/3003143.

4
O. González-Gaxiola, J. Ruíz de Chávez and J. A. Santiago, A Nonlinear Option Pricing Model Through the Adomian Decomposition Method, Int. J. Appl. Comput. Math., 2 (2016), 453-467, doi: https://doi.org/10.1007/s40819-015-0070-6.

5
S. O. Edeki, E. A. Owoloko, O. O. Ugbebor, The Modified Black-Scholes Model via Constant Elasticity of Variance for Stock Options Valuation, AIP Conference proceedings, 1705 (2016), 020041, doi: https://doi.org/10.1063/1.4940289.

6
G. Barles, H. M. Soner, Option pricing with transaction costs and a nonlinear Black-Scholes equation, Financ. Stoch., 2 (1998), 369-397.

7
M. R. Rodrigo, R. S. Mamon, An alternative approach to solving the Black-Scholes equation with time-varying parameters, Appl. Math. Lett., 19 (2006), 398-402.

8
S. O. Edeki, O. O. Ugbebor, E. A. Owoloko, Analytical Solutions of the Black-Scholes Pricing Model for European Option Valuation via a Projected Differential Transformation Method, Entropy, 17 (2015), 7510-7521.

9
R. Company, E. Navarro, J. R. Pintos, E. Ponsoda, Numerical solution of linear and nonlinear Black-Scholes option pricing equations, Comput. Math. Appl., 56 (2008), 813-821.

10
S. O. Edeki, O. O. Ugbebor, E. A. Owoloko, He's Polynomials for Analytical Solutions of the Black-Scholes Pricing Model for Stock Option Valuation, Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering 2016, WCE 2016, June 29 - July 1, London, U.K., (2016), 632-63.

11
O. Vukovic, On the Interconnectedness of Schrödinger and Black-Scholes Equation, Journal of Applied Mathematics and Physics, 3 (2015), 1108-1113.

12
B. E. Baaquie, Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press, Cambridge, (2004).

13
M. Contreras, R. Pellicer, M. Villena, A. Ruiz, A Quantum Model of Option Pricing: When Black-Scholes meets Schrödinger and Its Semi-Classical Limit, Physica A, 389 (2010), 5447-5459.

14
C. W. Gardiner, Handbook of Stochastic Methods, Springer Berlin, (1983).

15
J. Voit, The Statistical Mechanics of Financial Markets, Springer, Berlin, (2005).

16
O. González-Gaxiola, and J. Ruíz de Chávez, Solving the Ivancevic option pricing model using the Elsaki-Adomian decomposition method, Int. J. of Applied Math., 28 (2015), 515-525.

17
V. Ivancevic, Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model, Cognitive Computation, 2 (2010), 17-30.

18
R. Cont, Empirical properties of asset returns: stylized facts and statistical issues, Quant. Finance 1 (2001), 223-236.

19
J. Perello, R. Sircar, J. Masoliver, Option pricing under stochastic volatility: the exponential Ornstein-Uhlenbeck model, J. Stat. Mech. P06010, (2008).

20
S. O. Edeki, G. O. Akinlabi, S. A. Adeosun, On a modified transformation method for exact and approximate solutions of linear Schrödinger equations, AIP Conference Proceedings, 1705, 020048 (2016), doi: https://doi.org/10.1063/1.4940296.

21
A. S. V. Ravi Kanth, K. Aruna, “Comparison of two dimensional DTM and PTDM for solving time-dependent Emden-Fowler Type Equations, Int. J. of Nonlinear Science, 13 (2012), 228-239.

How to Cite?

DOI: 10.12732/ijpam.v115i1.14 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 115
Issue: 1
Pages: 187 - 198


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).