4.3 Methods for European options: analytic formulation

A judicious combination of options and shares has been used in chapter 3.3.3 to eliminate the uncertainty in a portfolio, suggesting that the price of an option can be calculated by solving a partial differential equation. So far, this has been carried out using the VMARKET applet without paying much attention to the different methods that have to be employed. These methods are the subject of the rest of this chapter, showing with some mathematical details how to implement and use them within their validity limits. More advanced methods will be discussed later when dealing with bonds and American options, but could very well be used also for European options.

The first method uses analytical tools to produce an explicit solution in the form of the Black-Scholes formula. A considerable amount of algebra is required for the derivation and is only accessible to graduates from quantitative fields. The Black-Scholes formula, however, has a much broader appeal and is often used to calculate the implied volatility of prices that are traded on the markets. Unfortunately the analytical method is difficult to generalize beyond the simplest plain-vanilla or binary options.

• 4.3.1 Transformation to log-normal variables
• 4.3.2 Solution of the normalized diffusion equation
• 4.3.3 Black-Scholes formula