When it was first introduced to the world, the Black-Scholes Option Pricing Model predicted option prices remarkably well. Later the accuracy of the model faltered. Indications of problems with the model, large volatility smiles, appeared with increased frequency and severity. It is shown that these are actually consistent with and related to the fundamental mathematics of the model. The analysis predicts when the model will have the least bias and be the most useful. Following the line of analysis leads to improved insight into the model and its problems. The analysis is consistent with the actual shapes of the volatility smile found in the various equity, currency and commodities markets.