Dynamics and Controllability of Financial Derivatives: Towards Stabilization the Global Financial Systems Crisis

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.