ABSTRACT
In Chapter 9, we discuss pricing derivatives via the Black-Scholes partial differential equation. As discussed in preceding chapters, prices of derivatives can be calculated by the expected value of the pay-off at maturity. It is well known that the calculation of such an expected value is equivalent to solving the corresponding Black-Scholes partial differential equation. Although the partial differential equation for a plain vanilla option has an analytic solution, the partial differential equations for exotic options in general do not have an analytic solution. For such cases, we can sometimes use an approximation of the partial differential equation. We first introduce the intuitive procedure of the approximation, called the explicit method. However, the explicit method is not necessarily stable. We also introduce, the more stable method called implicit method as an alternative.
