Why does the Black-Scholes formula not work for historical volatility or expected drift?

Abstract

The Black-Scholes (B&S) formula often fails to match market valuations of European-style options when using historical (expected) volatility. This conclusion is based on a large body of empirical evidence. We consider that the presumption of Geometric Brownian Motion, which is used for the description of the price development of the underlying asset in the B&S model, is too inaccurate, and that is a grave reason why the B&S formula, if the historical or other reasonable volatility is used in the formula, does not systematically provide results comparable with a real market valuation of options. This conclusion is also supported by the fact that if we change the assumption for the way the market price of the underlying asset develops to a process that results in a price distribution that is very close to what we empirically observe, valuation will be almost equal to the market price of the option while using historical volatility. The paper proposes an algorithm for risk-neutral option valuation, implying that there is no long-term advantage to repeatedly opening an option speculative position. The research also includes a discussion about the impact of the absence of drift of the underlying assets in the B&S formula.