  It is well-known that the Black-Scholes formula has been derived under the
assumption of constant volatility in stocks. In spite of evidence that this
parameter is not constant, this formula is widely used by financial markets.
This paper addresses the question of whether an alternative model for stock
price exists for which the Black-Scholes or similar formulae hold. The results
obtained in this paper are very general as no assumptions are made on the
dynamics of the model, whether it be the underlying price process, the
volatility process or how they relate to each other. We show that if the
formula holds for a continuum of strikes and three terminal times, then the
volatility must be constant. However, when it only holds for finitely many
strikes, and three or more maturity times, we obtain a universal bound on the
variation of the volatility. This bound yields that the implied volatility is
constant when the sequence of strikes increases to cover the entire half-line.
This recovers the result for a continuum of strikes by a different approach.
