Does the behaviour of the asset tell us anything about the option price formula? A cautionary tale

If Y = (Y 1,…,Y N) are the log-returns of an asset on succeeding days, then under the assumptions of the Black-Scholes option pricing formula, these are independent normal random variables with common mean and variance in the risk-neutral measure. If we can show empirically that Y does not have these properties in the realworld measure, does this mean that the Black-Scholes option pricing formula fails? It does not; as we show in this note, so long as the joint distribution of Y in the realworld measure has a strictly positive density, then the Black-Scholes option price formula may still be correct. We conclude that attempts to argue that the Black-Scholes formula must fail because observed log returns appear to be fat-tailed, or appear to have nonconstant volatility, or appear to have serial correlation are fallacious.