his chapter pulls together the various concepts you have learned, and shows you how to apply them to actual option markets.  We look at three applications: Stock Options (topic 13.2), Currency Forwards and Futures (topic 13.3), and Currency Options (topic 13.4).

One complication we will immediately encounter is that most of the options that are traded are American options, not European options.  Yet most of the theory you have learned deals with European options. 

Unfortunately, there are no pricing formulas for American options.  Instead, we use the same methodology, but use a numerical procedure to obtain the option value.

Several methods are commonly used to obtain numerical estimates of American option values.  Three examples are the Finite Difference Method (see topic 13.5),   the Barone-Adesi and Whaley (1987) Quadratic Approximation (see topic 13.6) and the Binomial Method (see topic 13.7).  The first solves the partial differential equation numerically.  The second method identifies a "stock cutoff" value for exercising the option, and the third uses the binomial option pricing model.  Online, the Option Calculator subject of Option Tutor, lets you choose which method you want to use to value options.

A second complication comes from dividends, especially when valuing stock options.  It is necessary to adjust the model to account for both discrete and continuous dividends.