Detection of Mispricing in the Black–Scholes PDE Using the Derivative-Free Nonlinear Kalman Filter

Financial derivatives and option pricing models are usually described with the use of stochastic differential equations and diffusion-type partial differential equations (e.g., Black–Scholes models). Considering the latter case in this paper 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 model 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 transformed into a model of the linear canonical (Brunovsky) form through a change of coordinates (diffeomorphism) which is based on differential flatness theory. This transformation provides an extended description 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 provided 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.