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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.

**Subsections**
**SYLLABUS** ** Previous:** 4.2.2 Barrier options
**Up:** 4 EUROPEAN OPTION PAYOFF
**Next:** 4.3.1 Transformation to log-normal