What is Options

The chapter analyzes differential flatness theory for the control of single asset and multi-asset option price dynamics, described by PDE models. Through these control methods, stabilization of distributed parameter (PDE modelled) financial systems is achieved and convergence to specific financial performance indices are made possible. The main financial model used in the chapter is the Black-Scholes PDE. By applying semi-discretization and a finite differences scheme the single-asset (equivalently multi-asset) Black-Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations, it is shown that differential flatness properties hold. This enables one to solve the associated control problem and to stabilize the options' dynamics. By showing the feasibility of control of the single-asset (equivalently multi-asset) Black-Scholes PDE, it is proven that through selected purchases and sales during the trading procedure, the price of options can be made to converge and stabilize at specific reference values.