A Kalman filtering approach for detection of option mispricing in the Black-Scholes PDE model

The paper considers financial derivatives and option pricing models which are described with the use of diffusiontype partial differential equations (e.g. Black-Scholes models). Using this approach a new filtering method for distributed parameter systems is developed, for estimating option prices variations without knowledge of initial conditions. The proposed filtering method is the so-called Derivative-free nonlinear Kalman Filter and is based on a decomposition of the nonlinear partial-differential equation of the financial system into a set of ordinary differential equations with respect to time. Next, each one of the local models associated with the ordinary differential equations is written in the linear canonical form through a transformation which is based on differential flatness theory. This transformation provides a model of the nonlinear dynamics of the option pricing model for which state estimation is possible by applying the standard Kalman Filter recursion. Based on the obtained state estimate, validation of the Black-Scholes PDE model can be performed and the existence of inconsistent parameters in the Black-Scholes PDE model can be concluded.