CHAPTER 24Analysis of Black–Scholes
Aims
- To examine alternative ways of measuring and forecasting volatility.
- To test the validity of the Black–Scholes equation.
- To assess the limitations of the Black–Scholes equation for pricing European options.
24.1 VOLATILITY
In the Black–Scholes formula all variables are directly observable, except for the volatility of stock returns (over the life of the option). To price an option we therefore require a forecast of volatility. Typical values of the annual standard deviation might be in the range of 
for individual stocks and around 20% p.a. for stock indices (e.g. S&P 500). If stock returns are assumed to be independent (and identically distributed), the standard deviation over a horizon 
(measured in years or fractions of a year), is given by 
. For example, if 
the standard deviation over 3 months 
is 
. Hence if we use the -rule (‘root-tee’ rule) ...
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