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The true probability of a European call option to achieve positive return isinvestigated under the Black-Scholes model. It is found that the probability isdetermined by those market factors appearing in the BS formula, besides thegrowth rate of stock price. Our numerical investigations indicate that thebiases of BS formula is correlated with the growth rate of stock price. Analternative method to price European call option is proposed, which adopts anequilibrium argument to determine option price through the probability ofpositive return. It is found that the BS values are on average larger than thevalues of proposed method for out-of-the-money options, and smaller than thevalues of proposed method for in-the-money options. A typical smile shape ofimplied volatility is also observed in our numerical investigation. Thesetheoretical observations are similar to the empirical anomalies of BS values,which indicates that the proposed valuation method may have some merit.

Probabilities of Positive Returns and Values of Call Options