General solution of the Black-Scholes boundary-value problem

The Black-Scholes formula for a European option price, which resulted in the 1997 Nobel Prize in Economic Sciences, is known to be the unique solution of the boundary-value problem consisting of the Black-Scholes partial differential equation and the terminal condition defined by the European call option. This has been one of the most popular tools of finance in theory as well as in practice. Here we present infinitely many solutions of the boundary value problem, involving Hermite polynomials. This indicates that the Black-Scholes boundary-value problem violates the law of one price, which is one of the fundamental concepts in economics.