JMF  Vol.2 No.1 , February 2012
Dynamics and Controllability of Financial Derivatives: Towards Stabilization the Global Financial Systems Crisis
Author(s) Murad Shibli
ABSTRACT
This paper presents a new dynamic approach to control and stabilize the global financial derivatives. Since 2007 the Global Financial Economy has been experiencing what is said to be the worst financial crisis since the Great Depression in the 1930’s. The Bank of International Settlements (BIS) in Switzerland has recently reported that global outstanding derivatives have reached 1.14 quadrillion dollars: $548 Trillion in listed credit derivatives plus $596 trillion in notional OTC derivatives. Although the financial derivatives are governed by the celebrated parabolic partial differential Black- Scholes formula, but it is not clear how derivatives are controlled and stabilized. This paper investigates equilibrium, stability and control of financial derivatives. The analysis is based on the discretization of Balck-Scholes formula to a system of linear ordinary differential equations. It is found that such financial derivatives experience a drift which hardly can be brought to equilibrium state. Controllability and observability conditions of financial systems are proposed. Moreover, stability of such derivatives is tested by the virtue of Liapunov methodology. It is figured out that financial system should satisfy the quadratic form which can be interpreted as a conservation condition of financial instruments. Furthermore, a financial state-feedback control system is proposed. Such analysis shows that the financial derivatives system needs to be injected with cash to maintain its stability. These results may explain the shortfall of li-quidity needed to substitute for the 1.14 quadrillion dollars bubble. Finally, examples and simulation results are demonstrated to verify the effectiveness of the proposed approach.

Cite this paper
M. Shibli, "Dynamics and Controllability of Financial Derivatives: Towards Stabilization the Global Financial Systems Crisis," Journal of Mathematical Finance, Vol. 2 No. 1, 2012, pp. 54-65. doi: 10.4236/jmf.2012.21007.
References
[1]   Monetary and Economic Department, “OTC Derivatives Market Activity in the Second Half of 2008,” Bank for International Settlements, Basel, 2009.

[2]   M. Shibli, “The Fundamental Principle of Conservation of Physical Money: Its Violation and the Global Financial System Collapse,” Journal of iBusiness, Vol. 3, No.1, 2011, pp. 76-87. doi:10.4236/ib.2011.31013

[3]   K. Schwab, “The Future of the Global Financial System: A Near-Term Outlook and Long-Term Scenarios,” World Economic Forum’s World Scenarios Series, 2009.

[4]   P. Krugman, “How Did Economists Get It So Wrong?” New York Times, 2009.

[5]   E. P. Caldentey and M. Vernengo, “Modern Finance, Methodology and the Global Crisis,” Real-World Economics Review, No. 52, 2010, pp. 69-81.

[6]   Wharton School of the University of Pennsylvania, “Why Economists Failed to Predict the Financial Crisis,” Knowledge@Wharton, 2009. http://knowledge.wharton.upenn.edu/article.cfm?articleid=2234

[7]   F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economic, Vol. 81, No. 3, 1973, pp. 637-654. doi:10.1086/260062

[8]   J. Hull, “Options, Futures and Other Derivatives,” 5th Edition, Prentice Hall, Upper Saddle River, 2003.

[9]   ?. Ugur, “An Introduction to Computational Finance,” World Scientific Publishing, Singapor, 2008.

[10]   H. Lee and D. Sheen, “Laplace Trans-formation Method for the Blck-Scholes Equation,” International Journal of Numerical Analysis and Modeling, Vol. 6, No. 4, 2009, pp. 642-658.

[11]   C. C. W. Leentvaar and C. W. Oosterlee, “Multi-Asset Option Pricing Using a Parallel Fourier-Based Technique,” Journal of Computational and Applied Mathematics, Vol. 222, No. 1, 2009, pp.193-209. doi:10.1016/j.cam.2007.10.015

[12]   C. W. Oosterlee, “American Options with Discrete Dividends Solved by Highly Accurate Discretizations,” Ma- thematics in Industry, Vol. 8, Part 7, 2006, pp. 427-431.

[13]   C. W. Oosterlee, C. C. W. Leentvaar and A. A. Vazquez, “Pricing Options with Dividends by High Order Finite Difference and Grid Stretching,” European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), 2004.

[14]   C. C. W. Leentvaar, “Numerical Solution of the Black-Scholes Equation with a Small Number of Grid Points,” Master’s Thesis, Delft University of Technology, Delft, 2003.

[15]   Y. Y. Ye, “A Path to the Arrow-Debreu Competitive Market Equilibrium,” Journal of Math Programming, Vol. 111, No. 1-2, 2006, pp. 315-348. doi:10.1007/s10107-006-0065-5

[16]   Robert Almgren, “Financial Derivatives and Partial Differential Equations,” American Mathematical Monthly, Vol. 109, 2001, pp. 1-12. doi:10.2307/2695763

[17]   K. J. Arrow and J. Debreu, “Existence of an Equilibrium for a Competitive Economy,” Journal of Econometric Society, Vol. 22, No. 3, 1954, pp. 265-290.

[18]   K. J. Arrow and M. McManus, “A Not on Dynamic Sta-bility,” North-Holland Publishing, Amsterdam, 1958.

[19]   G. W. Recktenwald, “Finite-Difference Approximations to the Heat Equation,” Class Notes, 2004.

[20]   De Luca, G. Oriolo, “Modeling and Control of Nonholonomic Mechanical Systems,” In: J. A. Kecskemethy, Ed., Kinematics and Dynamics of Multi-Body Systems, CISM Courses and Lectures No. 360, pp. 277-342, Springer- Verlag, New York, 1995.

[21]   K. Ogata, “Modern Control Engineering,” Prentice Hall, Upper Saddle River, 1997.

 
 
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