# Essay on Introduction Of Black Scholes ' Pricing Theory

Many praises have been used to describe Black-Scholes’ pricing theory, which is the most successful and widely used in the application. Although the elegant Black-Scholes’ theory is, there is an absolute disadvantage(Heston 1993). The strong assumption of constant volatility cannot express high peak and flat tail character of derivatives, and the relative between the spot return and the variance. Therefore, there are several theories try to extend and modify the original Black-Scholes’ model. The Heston stochastic volatility model is the most popular extension to the Black-Scholes’ model(Tse & Wan 2013).

The first scheme widely accepted is Euler-Maruyama Scheme. This scheme is efficient, easy to implement, and almost can be used to any stochastic differential equations. However, this model cannot handle a situation, if the Feller condition is not satisfied. Actually, the variance of a step can become negative, when the Feller condition is violated. There are several authors done great work to modify the Euler scheme to extend the situation which the model can be applied. Unfortunately, there is still significant biased when the Feller condition is violated(Andersen 2007).

In another aspect, Broadie and Kaya present a method to exactly simulate from the Heston model. Their model based on acceptance-rejection sampling and introduce closed-form equations for the characteristics function. Despite elegant, the most serious problem of this model is the speed of…