The Black-Scholes formula for a European option price, which resulted in the 1997 Nobel Prize in Economic Sciences, is known to be a unique solution to the Black-Scholes partial differential equation with the terminal condition corresponding to the European option. In this paper, we show that there exist infinitely many solutions to the Black-Scholes partial differential equation with the terminal condition. Such solutions include the Black-Scholes option valuation formula as a special one. Additionally, we show there are infinitely many solutions to the Black-Scholes partial differential equation subject to not only the terminal condition but also two additional conditions, which are commonly assumed when we solve the problem through a finite difference method. This implies that the Black-Scholes formula for a European option price violates the well-known law of one price in economics. We discuss this problem on mathematics points of view, particularly on the incompleteness of a boundary value problem as well as on the deficiency of Ito calculus as a practical tool.