ABSTRACT
Although far from being perfect, the Black-Scholes model is still useful. It demands a prerequisite of partial differential equations and the Laplace transform. The discrete model with a finite time interval in each step is not practicable because the large amount of data involved would be unmanageable. The solution is determined by first transforming the Black-Scholes equation into one-dimensional diffusion equation (heat equation), and then solving this equation by different methods. The chapter discusses the implied volatility, and where Black-Scholes goes wrong. Remember that prices are not set by the Black-Scholes options price. It is the markets that set prices, and according to some economists they set prices nearly perfectly.
