On The Black-Scholes Formula

Black-Scholes formula plays a very import role in asset pricing. Based on the formula, many new theories have been developed after modifying the original one. As is known to all, the assumptions of the formula are the defects that make the formula can not be applied into practice correctly, such as the volatility is constant. On the other hands, there are many new formulas based on the assumption that the volatility is stochastic. However, there is a detail that has been neglected in the formula, that is, the term (△t)2has been canceled in order to simplify the step. It is believed that the influence of (△t)2is too small to be taken into consideration. But the term may bring some error to the formula and the accumulative error is large. In this paper, it has been proved that the term (△t)2make the value is less than the true value. In order to make the conclusion more precise, finite difference method has been used to take the stochastic volatility into the formula.