Inverse parabolic problem with the Heaviside function arising in finance

The inverse problem arising in finance is both mathematically and practically interesting problem. First, we attempt to solve the inverse problem arising in binary option model. Using standard linearization and contraction arguments, we prove the stability and uniqueness of the solution of the inverse problem which originates from the diffusion equation with the initial condition given by the Heaviside function. Second, we numerically identify the local volatilities from given artificial prices of the binary option. In accordance with theory we propose the effective identification around the at-the-money level.