Adaptive Wave Models for Sophisticated Option Pricing

Vladimir G. Ivancevic

.

DOI: 10.4236/jmf.2011.13006                

Abstract

Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black-Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equation. The nonlinear approach comes in two flavors: for the case of constant volatility, it is defined by a single adaptive nonlinear Schrödinger (NLS) equation, while for the case of stochastic volatility, it is defined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is defined in terms of de Broglie’s plane waves and free-particle Schrödinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black-Scholes data, as well as defining Greeks.

Keywords

Black-Scholes Option Pricing, Adaptive Nonlinear Schrödinger Equation, Adaptive Manakov System, Quantum-Mechanical Option Pricing, Market-Heat Potential

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	[1] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062
[2]	R. C. Merton, “Theory of Rational Option Pricing,” The Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143
[3]	M. F. M. Osborne, “Brownian Motion in the Stock Market,” Operations Research, Vol. 7, No. 2, 1959, pp. 145- 173. doi:10.1287/opre.7.2.145
[4]	K. It?, “On Stochastic Differential Equations,” Memoirs of the American Mathematical Society, Vol. 4, 1951, pp. 1-51.
[5]	J. Perello, J. M. Porra, M. Montero and J. Masoliver, Black-Scholes Option Pricing within It? and Stratonovich Conventions, Physica A: Statistical Mechanics and Its Applications, Vol. 278, No. 1-2, 2000, pp. 260-274. doi:10.1016/S0378-4371(99)00612-3
[6]	C. W. Gardiner, “Handbook of Stochastic Methods,” Sprin- ger, Berlin, 1983.
[7]	J. Voit, “The Statistical Mechanics of Financial Markets,” Springer, Berlin, 2005.
[8]	R. L. Stratonovich, “A New Representation for Stochastic Integrals and Equations,” SIAM Journal on Control and Optimization, Vol. 4, No. 2, 1966, pp. 362-371. doi:10.1137/0304028
[9]	M. Kelly, “Black-Scholes Option Model & European Op- tion Greeks,” The Wolfram Demonstrations Project, 2009. http://demonstrations.wolfram.com/EuropeanOptionGreeks
[10]	V. Ivancevic, “Adaptive-Wave Alternative for the Black- Scholes Option Pricing Model,” Cognitive Computation, Vol. 2, No. 1, 2010, pp. 17-30.
[11]	V. Ivancevic and T. Ivancevic, “Complex Dynamics: Advanced System Dynamics in Complex Variables,” Sprin- ger, Dordrecht, 2007.
[12]	V. Ivancevic and T. Ivancevic, “Quantum Leap: From Dirac and Feynman, Across the Universe, to Human Body and Mind,” World Scientific, Singapore, 2008. doi:10.1142/9789812819284
[13]	A. W. Lo and J. Portf, “The Adaptive Markets Hypothesis: Market Efficiency from an Evolutionary Perspective,” Journal of Portfolio Management, Vol. 30, 2004, pp. 15-29.
[14]	A. W. Lo and J. Inves, “Reconciling Efficient Markets with Behavioral Finance: The Adaptive Markets Hypothesis,” Journal of Investment Consulting, Vol. 7, 2005, pp. 21-44.
[15]	A. J. Frost, R. R. Prechter Jr., “Elliott Wave Principle: Key to Market Behavior,” 10th Edition, Wiley, New York, 1978.
[16]	P. Steven, “Applying Elliott Wave Theory Profitably,” Wiley, New York, 2003.
[17]	V. Ivancevic and E. Aidman, “Life-Space Foam: A Medium for Motivational and Cognitive Dynamics,” Physica A: Statistical Mechanics and its Applications, Vol. 382, No. 2, 2007, pp. 616-630. doi:10.1016/j.physa.2007.04.025
[18]	V. Ivancevic and T. Ivancevic, “Quantum Neural Computation,” Springer, Berlin, 2009.
[19]	H. Kleinert and H. Kleinert, “Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets,” 3rd Edition, World Scientific, Singapore, 2002.
[20]	S. Liu, Z. Fu, S. Liu and Q. Zhao, “Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations,” Physics Letters A, Vol. 289, No. 1-2, 2001, pp. 69-74. doi:10.1016/S0375-9601(01)00580-1
[21]	G.-T. Liu and T.-Y. Fan, “New Applications of Developed Jacobi Elliptic Function Expansion Methods,” Physics Letters A, Vol. 345, No. 1-3, 2005, pp. 161-166. doi:10.1016/j.physleta.2005.07.034
[22]	M. Abramowitz and I. A. Stegun, Eds., “Jacobian Elliptic Functions and Theta Functions. Chapter 16 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” 9th Edition, Dover, New York, 1972, pp. 567-581.
[23]	V. Ivancevic and T. Ivancevic,” Neuro-Fuzzy Associative Machinery for Comprehensive Brain and Cognition Modelling,” Springer, Berlin, 2007. doi:10.1007/978-3-540-48396-0
[24]	V. Ivancevic and T. Ivancevic, “Computational Mind: A Complex Dynamics Perspective,” Springer, Berlin, 2007. doi:10.1007/978-3-540-71561-0
[25]	V. Ivancevic and T. Ivancevic, “Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals,” Springer, Berlin, 2008.
[26]	F. Black, “Studies of Stock Price Volatility Changes,” Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economics Statistics Section, 1976, pp. 177-181.
[27]	H. E. Roman, M. Porto and C. Dose, “Skewness, Long-Time Memory, and Nonstationarity: Application to Leverage Effect in Financial Time Series,” Europhysics Letters, Vol. 84, No. 2, 2008, pp. 28001-28006.
[28]	B. Kosko, “Bidirectional Associative Memories,” IEEE Transactions on Systems, Man and Cybernetics, Vol. 18, No. 1, 1988, pp. 49-60. doi:10.1109/21.87054
[29]	B. Kosko, “Neural Networks, Fuzzy Systems, A Dynami- cal Systems Approach to Machine Intelligence,” Pren- tice Hall, Upper Saddle River, 1992.
[30]	S.-H. Hanm, I. G. Koh, “Stability of Neural Networks and Solitons of Field Theory,” Physical Review E, Vol. 60, No. 6, 1999, pp. 7608-7611. doi:10.1103/PhysRevE.60.7608
[31]	S. V. Manakov, “On the Theory of Two-Dimensional Stationary Self-Focusing of Electromagnetic Waves,” Soviet Physics JETP, Vol. 38, p. 248.
[32]	M. Haelterman and A. P. Sheppard, “Bifurcation Phenomena and Multiple Soliton-Bound States in Isotropic Kerr Media,” Physical Review E, Vol. 49, No. 4, 1994, pp. 3376-3381. doi:10.1103/PhysRevE.49.3376
[33]	J. Yang, “Classification of the Solitary Waves in Coupled Nonlinear Schr?dinger Equations,” Physica D: Nonlinear Phenomena, Vol. 108, No. 1-2, 1997, pp. 92-112. doi:10.1016/S0167-2789(97)82007-6
[34]	D. J. Benney and A. C. Newell, “The propagation of Nonlinear Wave Envelops,” Journal of Mathematical Physics, Vol. 46, 1967, pp. 133-139.
[35]	V. E. Zakharov, S. V. Manakov, S. P. Novikov and L. P. Pitaevskii, “Soliton Theory: Inverse Scattering Method. Nauka, Moscow, 1980.
[36]	A. Hasegawa and Y. Kodama, “Solitons in Optical Communications,” Clarendon, Oxford, 1995.
[37]	R. Radhakrishnan, M. Lakshmanan and J. Hietarinta, “Inelastic Collision and Switching of Coupled Bright Solitons in Optical Fibers,” Physical Review E, Vol. 56, No. 2, 1997, pp. 2213-2216. doi:10.1103/PhysRevE.56.2213
[38]	G. Agrawal, “Nonlinear Fiber Optics,” 3rd Edition, Academic Press, San Diego, 2001.
[39]	J. Yang, “Interactions of Vector Solitons,” Physical Review E, Vol. 64, No. 2, 2001, pp. 026607-026623. doi:10.1103/PhysRevE.64.026607
[40]	J. Elgin, V. Enolski and A. Its, “Effective Integration of the Nonlinear Vector Schr?dinger Equation,” Physica D: Nonlinear Phenomena, Vol. 225, No. 2, 2007, pp. 127- 152. doi:10.1016/j.physd.2006.10.005
[41]	D. J. Griffiths, “Introduction to Quantum Mechanics,’ 2nd Edition, Pearson Educ. Int., (2005)
[42]	B. Thaller, “Visual Quantum Mechanics,” Springer, New York, 2000.