Stabilization of the multi-asset Black-Scholes PDE using differential flatness theory

A method for feedbsck control of the multi-asset Black-Scholes PDE is developed. By applying semi-discretization and a finite differences scheme the 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 to solve the associated control problem and to succeed stabilization of the options’ dynamics. For the local subsystems, into which the multi-asset Black-Scholes PDE is decomposed, it becomes possible to apply boundary-based feedback control. For each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the multi-asset Black-Scholes PDE system is found. The stability of the proposed control scheme is confirmed with the use of the Lyapunov method.