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 (PDE) with the terminal condition corresponding to the European option. We show that there exist innitely many solutions to the Black-Scholes PDE with the terminal condition. Such solutions include the Black-Scholes option valuation formula as a special case. This implies that the Black-Scholes formula for a European option price violates the well-known law of one price in economics. We consider the following questions including the non-uniqueness of the Black-Scholes PDE solution.

1. Is the current solution to the heat-transfer (aka diffusion) PDE appropriate?
2. Is the current solution to the Black-Scholes PDE appropriate?
3. Can we rely on the Newtonian paradigm for investigation of dynamic systems?
4. Is the Feyman-Kac formula approrpiate?
5. Can we rely on the Monte Carlo simulation approach for diffusion system problems?
6. Can we rely on the risk-neutral option pricing?
7. When we consider a diffusion PDE with an initial condition having a singularity point, should we assume the solution to be a continuous function of the state variable?
8. Should the fair value of a binary option be a continuous function of the underlying?
9. What are the basic assumptions for a stochstic calculus?